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AMRS Financial Ratios - FinancialModelingPrep
\dfrac{Current Assets}{Current Liabilities}
\dfrac{Cash and Cash Equivalents + Short Term Investments + Account Receivables}{Current Liabilities}
\dfrac{Cash and Cash Equivalents}{Current Liabilities}
\dfrac{(Account Receivable (start) + Account Receivable (end))/2}{Revenue/365}
\dfrac{(Inventories (start) + Inventories (end))/2}{COGS/365}
\dfrac{DSO + DIO}{}
\dfrac{(Accounts Payable (start) + Accounts Payable (end))/2}{COGS/365}
\dfrac{DSO + DIO − DPO}{}
\dfrac{Gross Profit}{Revenue}
\dfrac{Operating Income}{Revenue}
\dfrac{Income Before Tax}{Revenue}
\dfrac{Net Income}{Revenue}
\dfrac{Provision For Income Taxes}{Income Before Tax}
\dfrac{Net Income}{Average Total Assets}
\dfrac{Net Income}{Average Total Equity}
\dfrac{EBIT}{Average Total Asset − Average Current Liabilities}
\dfrac{Net Income}{EBT}
\dfrac{EBT}{EBIT}
\dfrac{EBIT}{Revenue}
\dfrac{Total Liabilities}{Total Assets}
\dfrac{Total Debt}{Total Equity}
-6.66 This is a measurement of the percentage of the company’s balance sheet that is financed by suppliers, lenders, creditors and obligors versus what the shareholders have committed.
\dfrac{Long−Term Debt}{Long−Term Debt + Shareholders Equity}
\dfrac{Total Debt}{Total Debt + Shareholders Equity}
\dfrac{EBIT}{Interest Expense}
-25.69 The lower a company’s interest coverage ratio is, the more its debt expenses burden the company.
\dfrac{Operating Cash Flows}{Total Debt}
-0.55 The cash flow to debt ratio reveals the ability of a business to support its debt obligations from its operating cash flows.
\dfrac{Total Assets}{Total Equity}
-5.84 This is a measure of financial leverage.
\dfrac{Revenue}{NetPPE}
\dfrac{Revenue}{Total Average Assets}
\dfrac{Operating Cash Flow}{Revenue}
-1.99 Gives investors an idea of the company's ability to turn sales into cash.
\dfrac{Free Cash Flow}{Operating Cash Flow}
\dfrac{Operating Cash Flow}{Total Debt}
-0.55 The operating cash flow is simply the amount of cash generated by the company from its main operations, which are used to keep the business funded.
\dfrac{Operating Cash Flow}{Short-Term Debt}
-4.62 The short-term debt coverage ratio compares the sum of a company's short-term borrowings and the current portion of its long-term debt to operating cash flow.
\dfrac{Operating Cash Flow}{Capital Expenditure}
-5.75 The larger the operating cash flow coverage for these items, the greater the company's ability to meet its obligations, along with giving the company more cash flow to expand its business, withstand hard times, and not be burdened by debt servicing and the restrictions typically included in credit agreements.
\dfrac{Operating Cash Flow}{Dividend Paid + Capital Expenditure}
\dfrac{DPS (Dividend per Share)}{EPS (Net Income per Share Number}
\dfrac{Stock Price per Share}{Equity per Share}
-4.70 The price-to-book value ratio, expressed as a multiple (i.e. how many times a company's stock is trading per share compared to the company's book value per share), is an indication of how much shareholders are paying for the net assets of a company.
\dfrac{Stock Price per Share}{Operating Cash Flow per Share}
\dfrac{Stock Price per Share}{EPS}
\dfrac{Price Earnings Ratio}{Expected Revenue Growth}
\dfrac{Stock Price per Share}{Revenue per Share}
\dfrac{Dividend per Share}{Stock Price per Share}
\dfrac{Entreprise Value}{EBITDA}
\dfrac{Stock Price per Share}{Intrinsic Value}
-4.70 Helps investors determine whether a stock is trading at, below, or above its fair value estimate,A price/fair value ratio below 1 suggests the stock is trading at a discount to its fair value, while a ratio above 1 suggests it is trading at a premium to its fair value.
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Squeeze Tickets 2022 | Tour Dates | Concerts Schedule
Home > Artists > Squeeze Tickets
Squeeze Tour 2022
The tickets for Squeeze concerts are already available.
It would be difficult to find a better way to enjoy music. This is exactly why fans keep travelling together with the band to many destinations around the world. This has to be experienced at least once and then you will understand why events like these happen everywhere. The Squeeze tour is something truly special, and if you don’t want to miss it then check our deals as soon as possible.
We make sure to offer the most competitive prices for tickets for different concerts. Moreover, you can choose your tickets based on seat preference. Just follow the Squeeze schedule carefully and make sure to book your tickets in advance. The most amazing experience is guaranteed!
When your favorite band arrives in your hometown, we will make sure to provide you with the best offers. Here you will find the Squeeze 2022 tour dates and all the details related to their live shows.
Squeeze Tickets 2022
Squeeze VIP Packages 2022
In mechanics, compression is the application of balanced inward ("pushing") forces to different points on a material or structure, that is, forces with no net sum or torque directed so as to reduce its size in one or more directions. It is contrasted with tension or traction, the application of balanced outward ("pulling") forces; and with shearing forces, directed so as to displace layers of the material parallel to each other. The compressive strength of materials and structures is an important engineering consideration.
In uniaxial compression, the forces are directed along one direction only, so that they act towards decreasing the object's length along that direction. The compressive forces may also be applied in multiple directions; for example inwards along the edges of a plate or all over the side surface of a cylinder, so as to reduce its area (biaxial compression), or inwards over the entire surface of a body, so as to reduce its volume.
Technically, a material is under a state of compression, at some specific point and along a specific direction
{displaystyle x}
, if the normal component of the stress vector across a surface with normal direction
{displaystyle x}
is directed opposite to
{displaystyle x}
. If the stress vector itself is opposite to
{displaystyle x}
, the material is said to be under normal compression or pure compressive stress along
{displaystyle x}
. In a solid, the amount of compression generally depends on the direction
{displaystyle x}
, and the material may be under compression along some directions but under traction along others. If the stress vector is purely compressive and has the same magnitude for all directions, the material is said to be under isotropic or hydrostatic compression at that point. This is the only type of static compression that liquids and gases can bear.
In a mechanical longitudinal wave, or compression wave, the medium is displaced in the wave's direction, resulting in areas of compression and rarefaction.
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Rng_(algebra) Knowpia
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term "rng" (IPA: /rʊŋ/) is meant to suggest that it is a "ring" without "i", that is, without the requirement for an "identity element".
(R, +) is an abelian group,
(R, ·) is a semigroup,
f(x · y) = f(x) · f(y)
for all x and y in R.
Example: even integersEdit
Example: finite quinary sequencesEdit
{\displaystyle {\mathcal {T}}=\bigoplus _{i=1}^{\infty }\mathbf {Z} /5\mathbf {Z} }
equipped with coordinate-wise addition and multiplication is a rng with the following properties:
Its idempotent elements form a lattice with no upper bound.
Every element x has a reflexive inverse, namely an element y such that xyx = x and yxy = y.
For every finite subset of
{\displaystyle {\mathcal {T}}}
, there exists an idempotent in
{\displaystyle {\mathcal {T}}}
that acts as an identity for the entire subset: the sequence with a one at every position where a sequence in the subset has a non-zero element at that position, and zero in every other position.
Ideals, quotient rings, and modules can be defined for rngs in the same manner as for rings.
Working with rngs instead of rings complicates some related definitions, however. For example, in a ring R, the left ideal (f) generated by an element f, defined as the smallest left ideal containing f, is simply Rf, but if R is only a rng, then Rf might not contain f, so instead
{\displaystyle (f)=Rf+\mathbf {Z} f=\{af+nf:a\in R\;\mathrm {and} \;n\in \mathbf {Z} \}}
where nf must be interpreted using repeated addition/subtraction since n need not represent an element of R. Similarly, the left ideal generated by elements f1, ..., fm of a rng R is
{\displaystyle (f_{1},\ldots ,f_{m})=\{a_{1}f_{1}+\cdots +a_{m}f_{m}+n_{1}f_{1}+\cdots n_{m}f_{m}:a_{i}\in R\;\mathrm {and} \;n_{i}\in \mathbf {Z} \},}
a formula that goes back to Emmy Noether.[1] Similar complications arise in the definition of submodule generated by a set of elements of a module.
Some theorems for rings are false for rngs. For example, in a ring, every proper ideal is contained in a maximal ideal, so a nonzero ring always has at least one maximal ideal. Both these statements fail for rngs.
A rng homomorphism f : R → S maps any idempotent element to an idempotent element.
If f : R → S is a rng homomorphism from a ring to a rng, and the image of f contains a non-zero-divisor of S, then S is a ring, and f is a ring homomorphism.
Adjoining an identity element (Dorroh extension)Edit
n · 1 + r
(n1 + r1) · (n2 + r2) = n1n2 + n1r2 + n2r1 + r1r2.
(n1, r1) + (n2, r2) = (n1 + n2, r1 + r2),
(n1, r1) · (n2, r2) = (n1n2, n1r2 + n2r1 + r1r2).
Given any ring S and any rng homomorphism f : R → S, there exists a unique ring homomorphism g : R^ → S such that f = gj.
The map g can be defined by g(n, r) = n · 1S + f(r).
Properties weaker than having an identityEdit
Rings with enough idempotents: A rng R is said to be a ring with enough idempotents when there exists a subset E of R given by orthogonal (i.e. ef = 0 for all e ≠ f in E) idempotents (i.e. e2 = e for all e in E) such that R = ⊕e∈E eR = ⊕e∈E Re.
Rings with local units: A rng R is said to be a ring with local units in case for every finite set r1, r2, ..., rt in R we can find e in R such that e2 = e and eri = ri = rie for every i.
s-unital rings: A rng R is said to be s-unital in case for every finite set r1, r2, ..., rt in R we can find s in R such that sri = ri = ris for every i.
Firm rings: A rng R is said to be firm if the canonical homomorphism R ⊗R R → R given by r ⊗ s ↦ rs is an isomorphism.
Idempotent rings: A rng R is said to be idempotent (or an irng) in case R2 = R, that is, for every element r of R we can find elements ri and si in R such that
{\displaystyle r=\Sigma _{i}r_{i}s_{i}}
Rings are rings with enough idempotents, using E = {1}. A ring with enough idempotents that has no identity is for example the ring of infinite matrices over a field with just a finite number of nonzero entries. The matrices that have just 1 over one element in the main diagonal and 0 otherwise are the orthogonal idempotents.
Rings with enough idempotents are rings with local units just taking finite sums of the orthogonal idempotents to satisfy the definition.
Rings with local units are in particular s-unital; s-unital rings are firm and firm rings are idempotent.
Rng of square zeroEdit
A rng of square zero is a rng R such that xy = 0 for all x and y in R.[2] Any abelian group can be made a rng of square zero by defining the multiplication so that xy = 0 for all x and y;[3] thus every abelian group is the additive group of some rng. The only rng of square zero with a multiplicative identity is the zero ring {0}.[4]
Unital homomorphismEdit
Given two unital algebras A and B, an algebra homomorphism
(x, r) ∗ (y, s) = (xy + sx + ry, rs)
^ Noether (1921), p. 30, §1.2.
^ See Bourbaki, p. 102, where it is called a pseudo-ring of square zero. Some other authors use the term "zero ring" to refer to any rng of square zero; see e.g. Szele (1949) and Kreinovich (1995).
^ Bourbaki, p. 102.
^ Zariski and Samuel, p. 133.
Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7.
Dorroh, J. L. (1932). "Concerning Adjunctions to Algebras". Bull. Amer. Math. Soc. 38 (2): 85–88. doi:10.1090/S0002-9904-1932-05333-2.
Kreinovich, V. (1995). "If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring". Algebra Universalis. 33 (2): 237–242. doi:10.1007/BF01190935. MR 1318988. S2CID 122388143.
Herstein, I. N. (1996). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-36879-3.
McCrimmon, Kevin (2004). A taste of Jordan algebras. Springer. ISBN 978-0-387-95447-9.
Noether, Emmy (1921). "Idealtheorie in Ringbereichen" [Ideal theory in rings]. Mathematische Annalen (in German). 83 (1–2): 24–66. doi:10.1007/BF01464225. S2CID 121594471.
Szele, Tibor (1949). "Zur Theorie der Zeroringe". Mathematische Annalen. 121: 242–246. doi:10.1007/bf01329628. MR 0033822. S2CID 122196446.
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Almost-universal quadratic forms: An effective solution of a problem of Ramanujan
15 March 2009 Almost-universal quadratic forms: An effective solution of a problem of Ramanujan
J. Bochnak, B.-K. Oh
J. Bochnak,1 B.-K. Oh2
1Department of Mathematics, Vrije Universiteit
2Department of Mathematical Sciences, Seoul National University
In this article we investigate almost-universal positive-definite integral quaternary quadratic forms, that is, those representing sufficiently large positive integers. In particular, we provide an effective characterization of all such forms. In this way we obtain the final solution to a problem first addressed by Ramanujan in [12]. Special attention is given to
2
-anisotropic almost-universal quaternaries
J. Bochnak. B.-K. Oh. "Almost-universal quadratic forms: An effective solution of a problem of Ramanujan." Duke Math. J. 147 (1) 131 - 156, 15 March 2009. https://doi.org/10.1215/00127094-2009-008
J. Bochnak, B.-K. Oh "Almost-universal quadratic forms: An effective solution of a problem of Ramanujan," Duke Mathematical Journal, Duke Math. J. 147(1), 131-156, (15 March 2009)
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Revision as of 14:05, 28 April 2015 by MathAdmin (talk | contribs) (Created page with "'''Question:''' Find f(5) for f(x) given in problem 1. Note: The function f(x) from problem 1 is: <math>f(x) = \log_3(x+3)-1</math> {| class="mw-collapsible mw-collapsed" s...")
Question: Find f(5) for f(x) given in problem 1.
Note: The function f(x) from problem 1 is:
{\displaystyle f(x)=\log _{3}(x+3)-1}
{\displaystyle {\begin{array}{rcl}f(5)&=&2(5)+1\\&=&11\end{array}}}
{\displaystyle {\begin{array}{rcl}f(5)&=&\log _{3}(5+3)-1\\&=&\log _{3}(8)-1\end{array}}}
{\displaystyle \log _{3}(8)-1}
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Health threat caused by fungi of medical interest: where are we in 2021?
Guillaume Desoubeaux, Adélaïde Chesnay
(This article belongs to the Special Issue Fungal Diseases)
Gas chromatography-mass spectrometry metabolic profiling, molecular simulation and dynamics of diverse phytochemicals of Punica granatum L. leaves against estrogen receptor
Talambedu Usha, Sushil Kumar Middha, Dhivya Shanmugarajan, Dinesh Babu, ... Kora Rudraiah Sidhalinghamurthy
Molecular subtype classification of breast cancer using established radiomic signature models based on
{}^{18}
F-FDG PET/CT images
Jianjing Liu, Haiman Bian, Yufan Zhang, Yongchang Gao, ... Wengui Xu
(This article belongs to the Special Issue Cancer Metabolism and the Tumor Microenvironment)
Paeonol alleviates migration and invasion of endometrial stromal cells by reducing HIF-1
\alpha
-regulated autophagy in endometriosis
Conghui Pang, Zhijuan Wu, Xiaoyan Xu, Wenxiu Yang, ... Yinghua Qi
Clinical features, treatments, and outcomes of patients with anti-N-methyl-d-aspartate encephalitis—a single-center, retrospective analysis in China
Qianhui Xu, Yong Zhu, Qian Wang, Jing Han, ... Ying Huang
Influences of blood flow parameters on temperature distribution during liver tumor microwave ablation
Jinying Wang, Shuicai Wu, Zeyi Wu, Hongjian Gao, Shengyang Huang
PLA-based core-shell structure stereocomplexed nanoparticles with enhanced loading and release profile of paclitaxel
Yuemin Wang, Siyuan Cui, Bing Wu, Quanxing Zhang, Wei Jiang
(This article belongs to the Special Issue Biomacromolecules, Biomaterials, Biosensors, and Biomedical Devices)
Effects of selenium application on biochemical characteristics and biofortification level of kohlrabi (Brassica oleracea L. var. gongylodes) produce
Marina Antoshkina, Nadezhda Golubkina, Agnieszka Sekara, Alessio Tallarita, Gianluca Сaruso
Foliar pathogen-induced assemblage of beneficial rhizosphere consortia increases plant defense against Setosphaeria turcica
Lin Zhu, Songhua Wang, Haiming Duan, Xiaomin Lu
(This article belongs to the Special Issue Diverse mechanisms of plant adaptation to abiotic or biotic stress)
Assessing psychosocial interventions for informal caregivers of older people with early dementia: a systematic review of randomized controlled evidence
Shanshan Wang, Johanna de Almeida Mello, Anja Declercq
Current strategies and technologies for finding drug targets of active components from traditional Chinese medicine
Feng Lu, Dan Wang, Ruo-Lan Li, Li-Ying He, ... Chun-Jie Wu
The role of astrocytes in brain metastasis at the interface of circulating tumour cells and the blood brain barrier
Layla Burn, Nicholas Gutowski, Jacqueline Whatmore, Georgios Giamas, Md Zahidul Islam Pranjol
(This article belongs to the Special Issue The dynamic roles of glial and stromal cells in neurological diseases)
Methylation alterations and advance of treatment in lymphoma
Meng-Ke Liu, Xiao-Jian Sun, Xiao-Dong Gao, Ying Qian, ... Wei-Li Zhao
(This article belongs to the Special Issue Epigenetics in development and cancer)
The renin-angiotensin system in central nervous system tumors and degenerative diseases
Simon Haron, Ethan J Kilmister, Paul F Davis, Stanley S Stylli, ... Agadha C Wickremesekera
An overview of chia seed (Salvia hispanica L.) bioactive peptides’ derivation and utilization as an emerging nutraceutical food
Roshina Rabail, Moazzam Rafiq Khan, Hafiza Mahreen Mehwish, Muhammad Shahid Riaz Rajoka, ... Rana Muhammad Aadil
(This article belongs to the Special Issue Role of Trace Elements and Nutrients in Preventing Various Disease States)
Advances in the regulation of plant development and stress response by miR167
Xia Liu, Sheng Huang, Hongtao Xie
Pregnancy, a unique case of heterochronic parabiosis and peripartum cardiomyopathy
Pascal J. Goldschmidt-Clermont, Corinne Hubinont, Alexander J.P. Goldschmidt, Darcy L. DiFede, Ian A. White
(This article belongs to the Special Issue Stem cell and stem cell-derived therapy in the 21st century)
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UpSample - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Sampling Rate : UpSample
up-sample a signal
UpSample( A, factor, phase )
posint, up-sample factor
The UpSample( A, factor, phase ) command up-samples the signal in the Array A.
The effect of this command is to place factor - 1 zeroes between each pair of samples in the Array A.
If the container=C option is provided, then the results are put into C and C is returned. With this option, no additional memory is allocated to store the result. The container must be an Array of size N * factor, where
N
is the number of elements of A, and it must have datatype float[8] if A is real and complex[8] if A is complex.
The SignalProcessing[UpSample] command is thread-safe as of Maple 17.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{interface}\left('\mathrm{rtablesize}'=20\right):
A≔\mathrm{Array}\left([1,2,3,4,5,6],'\mathrm{datatype}'='\mathrm{float}'[8]\right)
\left[\begin{array}{cccccc}1.0& 2.0& 3.0& 4.0& 5.0& 6.0\end{array}\right]
\mathrm{UpSample}\left(A,1\right)
\left[\begin{array}{cccccc}1.0& 2.0& 3.0& 4.0& 5.0& 6.0\end{array}\right]
\mathrm{UpSample}\left(A,2\right)
\left[\begin{array}{cccccccccccc}1.0& 0.0& 2.0& 0.0& 3.0& 0.0& 4.0& 0.0& 5.0& 0.0& 6.0& 0.0\end{array}\right]
\mathrm{UpSample}\left(A,3\right)
\left[\begin{array}{cccccccccccccccccc}1.0& 0.0& 0.0& 2.0& 0.0& 0.0& 3.0& 0.0& 0.0& 4.0& 0.0& 0.0& 5.0& 0.0& 0.0& 6.0& 0.0& 0.0\end{array}\right]
\mathrm{UpSample}\left(A,3,0\right)
\left[\begin{array}{cccccccccccccccccc}1.0& 0.0& 0.0& 2.0& 0.0& 0.0& 3.0& 0.0& 0.0& 4.0& 0.0& 0.0& 5.0& 0.0& 0.0& 6.0& 0.0& 0.0\end{array}\right]
\mathrm{UpSample}\left(A,3,1\right)
\left[\begin{array}{cccccccccccccccccc}0.0& 1.0& 0.0& 0.0& 2.0& 0.0& 0.0& 3.0& 0.0& 0.0& 4.0& 0.0& 0.0& 5.0& 0.0& 0.0& 6.0& 0.0\end{array}\right]
\mathrm{UpSample}\left(A,3,2\right)
\left[\begin{array}{cccccccccccccccccc}0.0& 0.0& 1.0& 0.0& 0.0& 2.0& 0.0& 0.0& 3.0& 0.0& 0.0& 4.0& 0.0& 0.0& 5.0& 0.0& 0.0& 6.0\end{array}\right]
C≔\mathrm{Array}\left(1..18,'\mathrm{datatype}'='\mathrm{float}'[8]\right):
\mathrm{UpSample}\left(A,3,'\mathrm{container}'=C\right)
\left[\begin{array}{cccccccccccccccccc}1.0& 0.0& 0.0& 2.0& 0.0& 0.0& 3.0& 0.0& 0.0& 4.0& 0.0& 0.0& 5.0& 0.0& 0.0& 6.0& 0.0& 0.0\end{array}\right]
C
\left[\begin{array}{cccccccccccccccccc}1.0& 0.0& 0.0& 2.0& 0.0& 0.0& 3.0& 0.0& 0.0& 4.0& 0.0& 0.0& 5.0& 0.0& 0.0& 6.0& 0.0& 0.0\end{array}\right]
A≔\mathrm{Array}\left([I,-I,1+I,I-1],\mathrm{datatype}=\mathrm{complex}[8]\right)
\left[\begin{array}{cccc}0.0+1.0\textcolor[rgb]{0,0,1}{}I& 0.0-1.0\textcolor[rgb]{0,0,1}{}I& 1.0+1.0\textcolor[rgb]{0,0,1}{}I& -1.0+1.0\textcolor[rgb]{0,0,1}{}I\end{array}\right]
C≔\mathrm{Array}\left(1..8,'\mathrm{datatype}'='\mathrm{complex}'[8]\right):
\mathrm{UpSample}\left(A,2,1,'\mathrm{container}'=C\right)
\left[\begin{array}{cccccccc}0.0+0.0\textcolor[rgb]{0,0,1}{}I& 0.0+1.0\textcolor[rgb]{0,0,1}{}I& 0.0+0.0\textcolor[rgb]{0,0,1}{}I& 0.0-1.0\textcolor[rgb]{0,0,1}{}I& 0.0+0.0\textcolor[rgb]{0,0,1}{}I& 1.0+1.0\textcolor[rgb]{0,0,1}{}I& 0.0+0.0\textcolor[rgb]{0,0,1}{}I& -1.0+1.0\textcolor[rgb]{0,0,1}{}I\end{array}\right]
C
\left[\begin{array}{cccccccc}0.0+0.0\textcolor[rgb]{0,0,1}{}I& 0.0+1.0\textcolor[rgb]{0,0,1}{}I& 0.0+0.0\textcolor[rgb]{0,0,1}{}I& 0.0-1.0\textcolor[rgb]{0,0,1}{}I& 0.0+0.0\textcolor[rgb]{0,0,1}{}I& 1.0+1.0\textcolor[rgb]{0,0,1}{}I& 0.0+0.0\textcolor[rgb]{0,0,1}{}I& -1.0+1.0\textcolor[rgb]{0,0,1}{}I\end{array}\right]
The SignalProcessing[UpSample] command was introduced in Maple 17.
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Symbol representing a property or relation in logic
In logic, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula
{\displaystyle P(a)}
{\displaystyle P}
is a predicate which applies to the individual constant
{\displaystyle a}
. Similarly, in the formula
{\displaystyle R(a,b)}
{\displaystyle R}
is a predicate which applies to the individual constants
{\displaystyle a}nd
{\displaystyle b}
In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula
{\displaystyle R(a,b)}
would be true on an interpretation if the entities denoted by
{\displaystyle a}nd
{\displaystyle b}
stand in the relation denoted by
{\displaystyle R}
. Since predicates are non-logical symbols, they can denote different relations depending on the interpretation used to interpret them. While first-order logic only includes predicates which apply to individual constants, other logics may allow predicates which apply to other predicates.
Predicates in different systems
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Get Positive Results With Negative Basis Trades
It always seems like there is a trade du jour that certain market conditions, new products, or security liquidity issues can make particularly profitable. The negative basis trade has represented such a trade for single corporate issuers. In this article, we explain why these opportunities exist and outline a basic way to execute a negative basis trade.
Basis has traditionally meant the difference between the spot (cash) price of a commodity and its future's price (derivative). This concept can be applied to the credit derivatives market where basis represents the difference in spread between credit default swaps (CDS) and bonds for the same debt issuer and with similar, if not exactly equal, maturities. In the credit derivatives market, basis can be positive or negative. A negative basis means that the CDS spread is smaller than the bond spread.
When a fixed-income trader or portfolio manager refers to spread, this represents the difference between the bid and ask price over the treasury yield curve (treasuries are generally considered a riskless asset). For the bond portion of the CDS basis equation, this refers to a bond's nominal spread over similar-term treasuries, or possibly the Z-spread. Because interest rates and bond prices are inversely related, a larger spread means the security is cheaper.
Fixed-income participants refer to the CDS portion of a negative basis trade as synthetic (because a CDS is a derivative) and the bond portion as cash. So you might hear a fixed-income trader mention the difference in spread between synthetic and cash bonds when they are talking about negative basis opportunities.
Executing a Negative Basis Trade
To capitalize on the difference in spreads between the cash market and the derivative market, the investor should buy the "cheap" asset and sell the "expensive" asset, consistent with the adage "buy low, sell high." If a negative basis exists, it means that the cash bond is the cheap asset and the credit default swap is the expensive asset (remember from above that the cheap asset has a greater spread). You can think of this as an equation:
\text{CDS basis} = \text{CDS spread} - \text{bond spread}
CDS basis=CDS spread−bond spread
It is assumed that at or near bond maturity, the negative basis will eventually narrow (heading toward the natural value of zero). As the basis narrows, the negative basis trade will become more profitable. The investor can buy back the expensive asset at a lower price and sell the cheap asset at a higher price, locking in a profit.
The trade is usually done with bonds that are trading at par or at a discount, and a single-name CDS (as opposed to an index CDS) of a tenor equal to the maturity of the bond (the tenor of a CDS is akin to maturity). The cash bond is purchased, while simultaneously the synthetic (single-name CDS) is shorted.
When you short a credit default swap, this means you have bought protection much like an insurance premium. While this might seem counterintuitive, remember that buying protection means you have the right to sell the bond at par value to the seller of the protection in the event of default or another negative credit event. So, buying protection is equal to a short.
While the basic structure of the negative basis trade is fairly simple, complications arise when trying to identify the most viable trade opportunity and when monitoring that trade for the best opportunity to take profits.
Market Conditions Create Opportunities
There are technical (market-driven) and fundamental conditions that create negative basis opportunities. Negative basis trades are usually done based on technical reasons as it is assumed that the relationship is temporary and will eventually revert to a basis of zero.
Many people use the synthetic products as part of their hedging strategies, which can cause valuation disparities versus the underlying cash market, especially during times of market stress. At these times, traders prefer the synthetic market because it is more liquid than the cash market. Holders of cash bonds may be unwilling or unable to sell the bonds they hold as part of their longer-term investment strategies. Therefore, they might look to the CDS market to buy protection on a specific company or issuer rather than simply sell their bonds. Magnify this effect during a crunch in the credit markets, and you can see why these opportunities exist during market dislocations.
Since market dislocations or "credit crunches" create the conditions for a negative basis trade to be possible, it is very important for the holders of this trade to monitor the marketplace constantly. The negative basis trade will not last forever. Once market conditions revert back to historical norms, spreads also go back to normal, and liquidity returns to the cash market, the negative basis trade will no longer be attractive. But as history has taught us, another trading opportunity is always around the corner. Markets quickly correct inefficiencies, or create new ones.
|
Solid-State Microrefrigeration in Conjunction With Liquid Cooling | J. Electron. Packag. | ASME Digital Collection
Younes Ezzahri,
e-mail: younes@soe.ucsc.edu
e-mail: ali@soe.ucsc.edu
Ezzahri, Y., and Shakouri, A. (September 8, 2010). "Solid-State Microrefrigeration in Conjunction With Liquid Cooling." ASME. J. Electron. Packag. September 2010; 132(3): 031002. https://doi.org/10.1115/1.4001853
Thermal design requirements are mostly driven by the peak temperatures. Reducing or eliminating hot spots could alleviate the design requirement for the whole package. Combination of solid-state and liquid cooling will allow removal of both hot spots and background heating. In this paper, we analyze the performance of thin film
Bi2Te3
microcooler and the 3D SiGe-based microrefrigerator, and optimize the maximum cooling and cooling power density in the presence of a liquid flow. Liquid flow and heat transfer coefficient will change the background temperature of the chip but they also affect the performance of the solid-state coolers used to remove hot spots. Both Peltier cooling at interfaces and Joule heating inside the device could be affected by the fluid flow. We analyze conventional Peltier coolers as well as 3D coolers. We study the impact of various parameters such as thermoelectric leg thickness, thermal interface resistances, and geometry factor on the overall system performance. We find that the cooling of a conventional Peltier cooler is significantly reduced in the presence of fluid flow. On the other hand, 3D SiGe cooler can be effective to remove high power density hot spots up to
500 W/cm2
. 3D microrefrigerators can have a significant impact if the thermoelectric figure-of-merit,
ZT
, could reach 0.5 for a material grown on silicon substrate. It is interesting to note that there is an optimum microrefrigerator active region thickness that gives the maximum localized cooling. For liquid heat transfer coefficient between 5000 and
20,000 W m−2 K−1
, the optimum is found to be between
10 μm
20 μm
bismuth compounds, contact resistance, cooling, Ge-Si alloys, heat transfer, microfluidics, micromechanical devices, Peltier effect, refrigeration, refrigerators, semiconductor device models, semiconductor devices, semiconductor materials, semiconductor thin films, thermoelectric devices, thin film devices
Cooling, Temperature, Thermoelectric coolers, Thin films
Handbook of Thermoelectrics
A New Approach to Optimum Design in Thermoelectric Cooling Systems
Optimum Design of a Thermoelectric Device
Investigation of Thermal Contact Effect on Thermoelectric Coolers
Direct Liquid Cooling of High Flux Micro and Nano Electronic Components
Hybrid Solid-State/Fluidic Cooling for Hot Spot Removal
Proceedings of the ITherm 2008
Thermal Quadrupoles: Solving the Heat Equation Through Integral Transforms
Dilhaire
Patino-Lopez
Dynamical Behavior and Cut-Off Frequency of Si/SiGe Microcoolers
A Comparison of Thin Film Microrefrigerators Based on Si/SiGe Superlattice and Bulk SiGe
Harmonic Regime Analysis and Inverse Methods Applied to the Simultaneous Determination of Thermoelectric Properties
, Vienna, Austria, Aug. 6–10.
Patiño-Lopez
, 2004, Ph.D. thesis, University Bordeaux 1, France.
Etude du Transport des Phonons dans les Microréfrigérateurs à base de Super-réseaux Si/SiGe
,” Ph.D. thesis, University Bordeaux 1, France.
Multilayer Thermionic Refrigerator and Generator
“ Nanoparticle-in-Alloy” Approach to Efficient Thermoelectric: Silicides in SiGe
Computational Modeling of Transverse Peltier Coolers
|
Solid element with properties derived from external file - MATLAB - MathWorks 日本
File Solid
Exporting Geometry Properties
Solid element with properties derived from external file
The File Solid block models a solid element with geometry, inertia, color, and reference frame derived from an external file. The file must be of a part model, which is to say that it contains at least solid geometry data. Some formats may provide color and inertia data, though such properties can be specified manually if need be.
Among the supported formats are those native to CATIA (V4, V5, and V6), Creo, Inventor, Unigraphics NX, Solid Edge, SolidWorks, and Parasolid (all CAD applications common in industry and academia). These include CATPART, PRT, IPT, SLDPRT, and X_T (and its binary version, X_B). Other valid formats, not associated with a specific application but common in 3-D modeling, include SAT (often referred to as ACIS), JT, STL, and STEP.
(CAD drawing and assembly files, which do not contain the necessary data for a solid element, cannot be imported to the block.)
For part model files with density data, the block gives the option to (automatically) set the mass, center of mass, and inertia tensor of the solid from calculation. This behavior is enabled by default (through the Type and Based On parameters under the Inertia node, which, in their original states, will read Calculate from Geometry and Density from File).
If the imported file does not contain density data, you must specify it (or, equivalently, mass) for the calculations to be made. Set the Based On parameter to Custom Density or Custom Mass to enter the missing data.
Alternatively, if you have the complete mass properties of the imported part—often provided, for CAD models, by the CAD application itself—you can enter them directly as block parameters. Set the inertia Type parameter to Custom in order to do this.
Note that the frame in which the moments and products of inertia are defined will vary among CAD applications. In this block, the origin of that frame is assumed to be at the center of mass (and its axes parallel to those of the reference frame). This frame is referred to here as the inertia resolution frame. (The center of mass, on the other hand, is defined in the reference frame.) For more information, see Specifying Custom Inertias.
If the mass properties are computed from geometry, you can view their values in the block dialog box. To do so, expand the Derived Values node under Inertia and click Update. (This feature, as it is specified to computed properties, requires that the inertia Type setting be Calculated from Geometry.) If a geometry or inertia block parameter changes, click the Update button once again to display the new mass properties. All values are in SI units of length (m) and mass (kg).
Like most components, the solid connects through frames, of which it has at least one. The default frame, which serves as its reference and is associated with port R, gets its origin and axes from the data in the imported file. (The origin is generally the zero coordinate of the CAD model or, if such technology is used, the 3-D scan, contained in the file.)
For those cases in which the reference frame is ill-placed for connection, or in which multiple connection frames are needed, the block comes with a frame creation tool. Treat this tool as an interactive alternative to the Rigid Transform block (the latter a numerical means to add and translate as well as rotate frames, though one that keeps the frames separate from the solid).
You can create (and edit) frames using geometry features as constraints—placing the frame origin on, and orienting the frame axes along, selected vertices, edges, and faces. You can also use the reference frame origin and its axes, as well as the center of mass and the principal inertia axes, to define the new frames. Each frame adds to the block a new frame port (its label derived from the name given in the frame creation pane).
To create or edit a frame, first expand the Frames node in the block dialog box. Click the button to create a frame or the button to edit a frame (if one, other than the reference frame, already exists). The frame definitions depend on a mix of geometry and inertia data, so you must have previously imported a part geometry file. If a block parameter changes, you must refresh the visualization pane (by clicking the button) in order to create or edit a frame.
A custom frame is fully defined when its origin and axes are too. Of these, the axes require the most care. You must specify two axes, one primary and one secondary. The primary axis defines the plane (that normal to it) on which the other axes must lie. The secondary axis is merely the projection of a selected direction—axis or geometry feature—on that plane.
The remaining (and unspecified) axis is set by requiring that all three be perpendicular and ordered according to the right-hand rule. Naturally, the secondary axis must have a vector component perpendicular to the primary axis. If the two are parallel, the frame is invalid. If the frame is then saved, its orientation is set to that of the reference frame.
To use a geometry feature for the frame origin or axis definitions:
In the frame creation pane, select the Based on Geometric Feature radio button.
In the solid visualization pane, click a vertex, edge, or face. Zoom in, if necessary, to more precisely select a feature.
Again in the frame creation pane, click the Use Selected Feature button.
It is common in a model to parameterize blocks in terms of MATLAB variables. Instead of a scalar, vector, or string, for example, a block parameter will have in its field the name of a variable. The variable is defined elsewhere, often in a subsystem mask or in the model workspace, sometimes by reference to an external M file.
This approach suits complex models in which multiple blocks must share the same parameter value—a common density, say, or color, if defined as an RGB vector. When the MATLAB variable definition then changes, so do all block parameters that depend on it. Consider using MATLAB variables here if a parameter is likely to be shared by several blocks in a large model.
(For a simple example with solid blocks parameterized in terms of workspace variables, open the sm_compound_body model)
The File Solid block can generate a convex hull geometry representation of an imported CAD file in the Simscape Multibody environment. This geometric data can be used to model spatial contact forces.
As shown in the figure, the convex hull geometry is an approximation of the true geometry. Note that the block calculates the physical properties, such as mass and inertia, based on its true geometry.
Frame by which to connect the solid in a model. The frame node to which this port connects—generally another frame port or a frame junction—determines the position and orientation of the solid relative to other components. Add a Rigid Transform block between the port and the node if the frames they represent must be offset from one another.
File Name — Name of the part model file to import
custom character vector
Name and extension of the part model file to import. If the file is not on the MATLAB path, the file location must be specified. The file location can be specified as an absolute path, starting from the root directory of the file system—e.g., 'C:/Users/JDoe/Documents/myShape.STEP'. It can also be specified as a relative path, starting from a folder on the MATLAB path—e.g., 'Documents/myShape.STEP'.
Unit Type — Source for solid geometry units
From File (default) | Custom
Source of the solid geometry units. Select From File to use the units specified in the imported file. Select Custom to specify your own units.
Unit — Length units in which geometry coordinates are specified in the imported file
m (default) | cm | mm | km | in | ft
Length units in which to interpret the geometry defined in a geometry file. Changing the units changes the scale of the imported geometry.
Convex Hull — Generate a convex hull representation of the true geometry
Select Convex Hull to generate a convex hull representation of the true geometry. This convex hull can be used for contacts by connecting the Spatial Contact Force block.
To enable this option, select Convex Hull under the Export.
Inertia parameterization to use. Select Point Mass to model a concentrated mass with negligible rotational inertia. Select Custom to model a distributed mass with the specified moments and products of inertia. The default setting, Calculate from Geometry, enables the block to automatically calculate the rotational inertia properties from the solid geometry and either density or mass.
Density from File (default) | Custom Density | Custom Mass
Parameter to use in inertia calculation. The block calculates the inertia tensor from the solid geometry and the parameter selected.
Use the default setting of Density from File to base the calculations on the density obtained from the imported file. (Note that only some formats can carry density data. Of those that do, only some will actually carry it. Often this data is specified in a CAD application before saving or exporting the part model file.)
Use Custom Density to specify a density other than that obtained from the imported file. Use Custom Mass to instead specify the total mass of the solid.
\left(\begin{array}{ccc}{I}_{xx}& & \\ & {I}_{yy}& \\ & & {I}_{zz}\end{array}\right),
{I}_{xx}=\underset{m}{â«}\left({y}^{2}+{z}^{2}\right)\text{â}dm
{I}_{yy}=\underset{m}{â«}\left({x}^{2}+{z}^{2}\right)\text{â}dm
{I}_{zz}=\underset{m}{â«}\left({x}^{2}+{y}^{2}\right)\text{â}dm
\left(\begin{array}{ccc}& {I}_{xy}& {I}_{zx}\\ {I}_{xy}& & {I}_{yz}\\ {I}_{zx}& {I}_{yz}& \end{array}\right),
{I}_{yz}=â\underset{m}{â«}yz\text{â}dm
{I}_{zx}=â\underset{m}{â«}zx\text{â}dm
{I}_{xy}=â\underset{m}{â«}xy\text{â}dm
Derived Values — Display of calculated values of mass properties
Simple (default) | Advanced | From File
Parameterization for specifying visual properties. Select Simple to specify color and opacity. Select Advanced to add specular highlights, ambient shadows, and self-illumination effects. Select From File if the imported file has color data and you want to use it in the model.
(Only some file formats allow color data. In those that do, that data is often optional. If your file does not specify color, the solid will take on a gray hue (the default solid color). Select another parameterization to customize color in such cases.)
Brick Solid | Cylindrical Solid | Ellipsoidal Solid | Extruded Solid | Revolved Solid | Spherical Solid | Variable Brick Solid | Variable Cylindrical Solid | Variable Spherical Solid | Rigid Transform
|
8
-dozen chocolate-chip muffins for the Food Fair at school. The recipe she is using makes
3
-dozen muffins. If the original recipe calls for
16
ounces of chocolate chips, how many ounces of chocolate chips does she need for her new amount? (Allie buys her chocolate chips in bulk and can measure them to the nearest ounce.)
\frac{8}{3}=\frac{\textit{x}}{16}
|
Inquisitive_semantics Knowpia
Inquisitive semantics is a framework in logic and natural language semantics. In inquisitive semantics, the semantic content of a sentence captures both the information that the sentence conveys and the issue that it raises. The framework provides a foundation for the linguistic analysis of statements and questions.[1][2] It was originally developed by Ivano Ciardelli, Jeroen Groenendijk, Salvador Mascarenhas, and Floris Roelofsen.[3][4][5][6][7]
Basic notionsEdit
The essential notion in inquisitive semantics is that of an inquisitive proposition.
An information state (alternately a classical proposition) is a set of possible worlds.
An inquisitive proposition is a nonempty downward-closed set of information states.
Inquisitive propositions encode informational content via the region of logical space that their information states cover. For instance, the inquisitive proposition
{\displaystyle \{\{w\},\emptyset \}}
encodes the information that {w} is the actual world. The inquisitive proposition
{\displaystyle \{\{w\},\{v\},\emptyset \}}
encodes that the actual world is either
{\displaystyle w}
{\displaystyle v}
An inquisitive proposition encodes inquisitive content via its maximal elements, known as alternatives. For instance, the inquisitive proposition
{\displaystyle \{\{w\},\{v\},\emptyset \}}
has two alternatives, namely
{\displaystyle \{w\}}
{\displaystyle \{v\}}
. Thus, it raises the issue of whether the actual world is
{\displaystyle w}
{\displaystyle v}
while conveying the information that it must be one or the other. The inquisitive proposition
{\displaystyle \{\{w,v\},\{w\},\{v\},\emptyset \}}
encodes the same information but does not raise an issue since it contains only one alternative.
The informational content of an inquisitive proposition can be isolated by pooling its constituent information states as shown below.
The informational content of an inquisitive proposition P is
{\displaystyle \operatorname {info} (P)=\{w\mid w\in t{\text{ for some }}t\in P\}}
Inquisitive propositions can be used to provide a semantics for the connectives of propositional logic since they form a Heyting algebra when ordered by the subset relation. For instance, for every proposition P there exists a relative pseudocomplement
{\displaystyle P^{*}}
, which amounts to
{\displaystyle \{s\subseteq W\mid s\cap t=\emptyset {\text{ for all }}t\in P\}}
. Similarly, any two propositions P and Q have a meet and a join, which amount to
{\displaystyle P\cap Q}
{\displaystyle P\cup Q}
respectively. Thus inquisitive propositions can be assigned to formulas of
{\displaystyle {\mathcal {L}}}
Given a model
{\displaystyle {\mathfrak {M}}=\langle W,V\rangle }
where W is a set of possible worlds and V is a valuation function:
{\displaystyle [\![p]\!]=\{s\subseteq W\mid \forall w\in s,V(w,p)=1\}}
{\displaystyle [\![\neg \varphi ]\!]=\{s\subseteq W\mid s\cap t=\emptyset {\text{ for all }}t\in [\![\varphi ]\!]\}}
{\displaystyle [\![\varphi \land \psi ]\!]=[\![\varphi ]\!]\cap [\![\psi ]\!]}
{\displaystyle [\![\varphi \lor \psi ]\!]=[\![\varphi ]\!]\cup [\![\psi ]\!]}
The operators ! and ? are used as abbreviations in the manner shown below.
{\displaystyle !\varphi \equiv \neg \neg \varphi }
{\displaystyle ?\varphi \equiv \varphi \lor \neg \varphi }
Conceptually, the !-operator can be thought of as cancelling the issues raised by whatever it applies to while leaving its informational content untouched. For any formula
{\displaystyle \varphi }
, the inquisitive proposition
{\displaystyle [\![!\varphi ]\!]}
expresses the same information as
{\displaystyle [\![\varphi ]\!]}
, but it may differ in that it raises no nontrivial issues. For example, if
{\displaystyle [\![\varphi ]\!]}
is the inquisitive proposition P from a few paragraphs ago, then
{\displaystyle [\![!\varphi ]\!]}
is the inquisitive proposition Q.
The ?-operator trivializes the information expressed by whatever it applies to, while converting information states that would establish that its issues are unresolvable into states that resolve it. This is very abstract, so consider another example. Imagine that logical space consists of four possible worlds, w1, w2, w3, and w4, and consider a formula
{\displaystyle \varphi }
{\displaystyle [\![\varphi ]\!]}
contains {w1}, {w2}, and of course
{\displaystyle \emptyset }
. This proposition conveys that the actual world is either w1 or w2 and raises the issue of which of those worlds it actually is. Therefore, the issue it raises would not be resolved if we learned that the actual world is in the information state {w3, w4}. Rather, learning this would show that the issue raised by our toy proposition is unresolvable. As a result, the proposition
{\displaystyle [\![?\varphi ]\!]}
contains all the states of
{\displaystyle [\![\varphi ]\!]}
, along with {w3, w4} and all of its subsets.
Rising declarative
^ "What is inquisitive semantics?". Institute for Logic, Language and Computation, University of Amsterdam.
^ Ciardelli, Ivano; Groenendijk, Jeroen; Roelofsen, Floris (2019). Inquisitive Semantics (PDF). Oxford University Press.
^ Ciardelli, Ivano; Roelofsen, Floris (2009). "Generalized inquisitive logic: completeness via intuitionistic Kripke models" (PDF). Proceedings of the 12th Conference on Theoretical Aspacts of Rationality and Knowledge. ACM: 71–80.
^ Jeroen Groenendijk (2009). "Inquisitive semantics: Two possibilities for disjunction" (PDF). Proceedings of the 7th International Tbilisi Symposium on Language, Logic, and Computation. Springer: 80–94.
^ Groenendijk, Jeroen; Roelofsen, Floris (2009). "Inquisitive semantics and pragmatics" (PDF). Proceedings of the ILCLI International Workshop on Semantics, Pragmatics and Rhetoric: 41–72.
^ Mascarenhas, Salvador (2009). "Inquisitive semantics and logic" (PDF). Master Thesis, ILLC University of Amsterdam.
Ciardelli, Ivano; Groenendijk, Jeroen; and Roelofsen, Floris (2019) Inquisitive Semantics. Oxford University Press. ISBN 9780198814788
https://projects.illc.uva.nl/inquisitivesemantics/
|
Home1-D kinematics
In physics, the position, the position vector or the location vector of a body with respect to a coordinate system is defined as the vector that links the location of the body with the origin of the coordinate system. In Cartesian coordinates, it is expressed as:
\stackrel{\to }{r}=x\stackrel{\to }{i}+y\stackrel{\to }{j}+z\stackrel{\to }{k}
\stackrel{\to }{r}
: Is the position vector
x, y, z : Are the coordinates of the position vector
\stackrel{\to }{i}\text{,}\stackrel{\to }{j}\text{,}\stackrel{\to }{k}
: Are the unit vectors in the directions of axes OX, OY and OZ respectively.
The unit of measurement for position in the International System is meter [m]. Like all vectors, the position vector in physics has direction and magnitude (also known as size, modulus or length of the vector). The magnitude of the position vector is the distance of the body to the origin of the reference system. To calculate it you can use the following formula:
\left|\stackrel{\to }{r}\right|=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}
For those problems where you are working in fewer dimensions, you can simplify the previous formula by eliminating unnecessary terms. This way, the position equation:
In two dimensions becomes
\stackrel{\to }{r}=x\stackrel{\to }{i}+y\stackrel{\to }{j}+\overline{)z\stackrel{\to }{k}}=x\stackrel{\to }{i}+y\stackrel{\to }{j}
,and its magnitude
\left|\stackrel{\to }{r}\right|=\sqrt{{x}^{2}+{y}^{2}+\overline{){z}^{2}}}=\sqrt{{x}^{2}+{y}^{2}}
, since z=0
In one dimension becomes
\stackrel{\to }{r}=x\stackrel{\to }{i}+\overline{)y\stackrel{\to }{j}}+\overline{)z\stackrel{\to }{k}}=x\stackrel{\to }{i}
, and its magnitude
\left|\stackrel{\to }{r}\right|=\sqrt{{x}^{2}+\overline{){y}^{2}}+\overline{){z}^{2}}}=\sqrt{{x}^{2}}=x
, since y=0 and z=0
We have represented a position vector in three dimensions (left) and another in two dimensions (right) in the following figure:
Position vector and its magnitude
Slide the body within the coordinate system and observe how the magnitude is as as the coordinates and direction of its position vector change.
Notice that as the body approaches the origin of coordinates (0,0) the modulus of the vector, which is expressed as |
\stackrel{\to }{r}
|, decreases. What happens if you move it away? , is reduced. What if you move it away?
NOTE: The dotted line is not usually drawn, we show it here so you can see more clearly the direction of the vector.
\stackrel{\to }{r}
= ·
\stackrel{\to }{i}
\stackrel{\to }{j}
\left|\stackrel{\to }{r}\right|
Find the position vector and its magnitude for the following points:
Now... ¡Test yourself!
Contents of Position Vector are closely related to:
Author: José L. Fernández
|
IsEulerian - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : IsEulerian
test if graph is Eulerian
IsEulerian(G)
IsEulerian(G, T)
FindEulerianPath(G)
The IsEulerian command returns true if the input graph is an Eulerian graph, i.e there exists a closed walk in the graph that uses each edge exactly once. It returns false otherwise.
An optional second argument T is assigned an Eulerian Trail of the graph if such a trail exists, and FAIL otherwise.
The FindEulerianPath command returns a list corresponding to an Eulerian trail if one exists, and NULL otherwise.
The algorithm used to construct the Eulerian trail is depth-first-search. The complexity is
\mathrm{O}\left(n+m\right)
where n=|V| and m=|E|.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{IsEulerian}\left(\mathrm{CompleteGraph}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsEulerian}\left(\mathrm{CompleteGraph}\left(5\right),'T'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
T
\textcolor[rgb]{0,0,1}{\mathrm{Trail}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{FindEulerianPath}\left(\mathrm{SpecialGraphs}:-\mathrm{BookGraph}\left(3\right)\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]
The GraphTheory[FindEulerianPath] command was introduced in Maple 2020.
|
plz solve this question (-2)*(-4)/6 - Maths - Rational Numbers - 10627599 | Meritnation.com
\mathrm{We} \mathrm{have},\phantom{\rule{0ex}{0ex}}\frac{\left(-2\right)\left(-4\right)}{6}\phantom{\rule{0ex}{0ex}}=\frac{8}{6}\phantom{\rule{0ex}{0ex}}=\frac{4}{3}
Sree Charan answered this
Rujuta Shastri answered this
Shivansh Shekhar answered this
Devesh answered this
Ritika Choudhary answered this
srry to crrct u shivansh but 4/3 is 1.3333
Nisarg answered this
Pratik Singh Chouhan answered this
Prapti Shetty answered this
|
Rudabányaite, a new mineral with a [Ag2Hg2]4+ cluster cation from the Rudabánya ore deposit (Hungary) | European Journal of Mineralogy | GeoScienceWorld
Rudabányaite, a new mineral with a [Ag2Hg2]4+ cluster cation from the Rudabánya ore deposit (Hungary)
Herta Effenberger;
Herta Effenberger
, Althanstrasse 14, 1090Vienna,
Corresponding author, e-mail: herta.silvia.effenberger@univie.ac.at
Sándor Szakáll;
Sándor Szakáll
Institute of Mineralogy and Geology, University of Miskolc
, 3515Miskolc-Egyetemváros,
Béla Fehér;
Department of Mineralogy, Herman Ottó Museum
, Kossuth u. 13, 3525Miskolc,
, Konkoly-Thege Miklós út 29–33, 1121Budapest,
Norbert Zajzon
Herta Effenberger, Sándor Szakáll, Béla Fehér, Tamás Váczi, Norbert Zajzon; Rudabányaite, a new mineral with a [Ag2Hg2]4+ cluster cation from the Rudabánya ore deposit (Hungary). European Journal of Mineralogy 2019;; 31 (3): 537–547. doi: https://doi.org/10.1127/ejm/2019/0031-2830
Rudabányaite was found in cavities of siliceous sphaerosiderite and limonite rocks at the Adolf mine area of the Rudabánya ore deposit (northern-East Hungary). The new mineral forms small crystals up to 0.6 mm and aggregates of a few mm across. Usually they have a xenomorphic shape, only occasionally cubic symmetry is morphologically discernible; the crystal forms {110} and {100} were recognized. The crystals are transparent, have yellowish-orange to brownish-yellow colour and a lemon-yellow streak, the lustre is adamantine. The Mohs’ hardness is 3–4. No cleavage was observed. Rudabányaite is optically isotropic. The density could not be measured due to lack of material; ρ(calc.) = 8.04 g/cm3. Electron-microprobe analyses gave the average composition (in wt%) Ag2O 29.39, Hg2O 52.62, As2O5 13.69, Cl 4.62, SO3 0.19, O=Cl −1.04, sum 99.47. The empirical formula based on four oxygen atoms is (Ag2.06Hg2.05)Σ=4.11(As0.97S0.02)Σ=0.99O4Cl1.06; the idealized formula as derived from chemical analyses and crystal-structure investigation is [Ag2Hg2][AsO4]Cl. The crystal-structure investigation was performed on single-crystal X-ray data; the refinements on F2 converged at wR2(F2) = 0.068 and R1(F) = 0.031 for all 972 unique data and 53 variable parameters. Rudabányaite crystallizes in space group
F4¯3c
, a = 17.360(3) Å, V = 5231.8 Å3, Z = 32. The crystal structure is characterised by two crystallographically different [M4]4+ cluster cations forming tetrahedra; M = (Ag,Hg) with a ratio Ag:Hg ~ 1:1. There is not any evidence for an order between the Ag and Hg atoms. Small amounts of the M atoms are displaced by ~0.5 Å. Topologically, the barycentres of the [M4]4+ clusters and the As atom positions of the crystal structure of rudabányaite form a cubic primitive lattice with a′ = ½a = 8.68 Å; half of the voids are occupied by Cl atoms.
Rudabanya Deposit
rudabanyaite
Adolf Mine
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Teresa Bermúdez — 1997
Chain-finite operators and locally chain-finite operators.
Teresa Bermúdez; Antonio Martinón — 1999
The problem we are concerned with in this research announcement is the algebraic characterization of chain-finite operators (global case) and of locally chain-finite operators (local case).
Acotabilidad de la función resolvente local.
Teresa Bermúdez; Manuel González — 1998
On Neumann operators.
Almost regular operators are regular.
De conejos y números. La sorprendente sucesión de Fibonacci.
Ángel Alonso; Teresa Bermúdez — 2002
Operators with an ergodic power
Teresa Bermúdez; Manuel González; Mostafa Mbekhta — 2000
We prove that if some power of an operator is ergodic, then the operator itself is ergodic. The converse is not true.
On operators T such that f(T) is hypercyclic.
Teresa Bermúdez; Vivien G. Miller — 2000
Local ergodic theorems.
Stability of the local spectrum.
Teresa Bermúdez; Manuel González; Antonio Martinón — 1995
On the poles of the local resolvent.
Properties and applications of the local functional calculus.
Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces
Teresa Bermúdez; Antonio Bonilla; José A. Conejero; Alfredo Peris — 2005
Our aim in this paper is to prove that every separable infinite-dimensional complex Banach space admits a topologically mixing holomorphic uniformly continuous semigroup and to characterize the mixing property for semigroups of operators. A concrete characterization of being topologically mixing for the translation semigroup on weighted spaces of functions is also given. Moreover, we prove that there exists a commutative algebra of operators containing both a chaotic operator and an operator which...
Powers of m-isometries
Teresa Bermúdez; Carlos Díaz Mendoza; Antonio Martinón — 2012
A bounded linear operator T on a Banach space X is called an (m,p)-isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X,
{\sum }_{k=0}^{m}{\left(-1\right)}^{k}\left(\genfrac{}{}{0pt}{}{m}{k}\right)||{T}^{k}x{||}^{p}=0
. We prove that any power of an (m,p)-isometry is also an (m,p)-isometry. In general the converse is not true. However, we prove that if
{T}^{r}
{T}^{r+1}
are (m,p)-isometries for a positive integer r, then T is an (m,p)-isometry. More precisely, if
{T}^{r}
is an (m,p)-isometry and
{T}^{s}
is an (l,p)-isometry, then
{T}^{t}
is an (h,p)-isometry, where t = gcd(r,s) and h = min(m,l)....
10 Extracta Mathematicae
14 Bermúdez, T
7 González, M
6 Martinón, A
2 Mbekhta, M
1 Alonso, Á
1 Bonilla, A
1 Conejero, JA
1 Mendoza, CD
1 Miller, VG
1 Peris, A
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Interview Query | Running Dog - Statistics Problem
A man and a dog stand at opposite ends of a football field that is 100 feet long. Both start running towards each other.
Let’s say that the man runs at
x
ft/s and the dog runs at twice the speed of the man. Each time the dog reaches the man, the dog runs back to the end of the field where it started and then back to the man and repeat.
What is the total distance the dog covers once the man finally reaches the end of the field?
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The dimension of a cuboid area in the ratio 4:5:6 and the total surface area is 5,328cm2 Find the - Maths - Visualising Solid Shapes - 7009208 | Meritnation.com
The dimension of a cuboid area in the ratio 4:5:6 and the total surface area is 5,328cm2. Find the volume and the cost of painting inner surface area without the upper face, if the rate of painting is Rs. 25 per square meter.
The dimension of cuboid are in the ratio 4 : 5 : 6.
So, suppose the length, width and height of cylinder are 4x , 5x and 6x respectively.
Then its total surface area =
2\left(lw+wh+lh\right)\quad =\quad 2\left(4x\times 5x+5x\times 6x+4x\times 6x\right)\quad =\quad 2\times 74{x}^{2}\quad =\quad 148{x}^{2}
And total surface area given is 5328 sq. cm. So we have;
148{x}^{2}\quad =\quad 5328\phantom{\rule{0ex}{0ex}}\Rightarrow {x}^{2}\quad =\quad \frac{5328}{148}\quad =\quad 36\phantom{\rule{0ex}{0ex}}\Rightarrow x\quad =\quad \sqrt{36}\quad =\quad 6
So, length of the cuboid =
4\times 6\quad =\quad 24\quad \mathrm{cm}
Width of cuboid =
5\times 6\quad =\quad 30\quad \mathrm{cm}
Height of cuboid =
6\times 6\quad =\quad 36\quad \mathrm{cm}
So volume of cuboid =
lwh\quad =\quad 24\times 30\times 36\quad =\quad 25920\quad {\mathrm{cm}}^{3}
Inner surface area without the upper face = lateral surface area + area of base = 2h(l + w) + lw
=\quad 2\times 36\left(24+30\right)+24\times 30\phantom{\rule{0ex}{0ex}}=\quad 2\times 36\times 54+720\phantom{\rule{0ex}{0ex}}=\quad 4608\quad {\mathrm{cm}}^{2}
So total area to be painted =
4608\quad {\mathrm{cm}}^{2}\quad =\quad \frac{4608}{10000}\quad {\mathrm{m}}^{2}\quad =\quad 0.4608\quad {\mathrm{m}}^{2}
Cost of painting per square metre = Rs.25
Therefore total cost of painting the inner surface without upper face =
25\times 0.4608\quad
= Rs.11.52
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Analysis of vibration based windmill coupled micromachined energy harvester | JVE Journals
Pavan R1 , Shyamsundar P I2 , Venkatesh K P Rao3
1, 2, 3Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani, Rajasthan, 333031, India
Copyright © 2019 Pavan R, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The present work exploits the centripetal, Coriolis and Euler forces generated in a rotating windmill. The MEMS device is placed on the blade of a windmill to harvest the energy. Modal analysis is carried out to optimize the dimensions of the structure to match the desired conditions. The real time response of the structure and the voltage generated in the piezoelectric layer are evaluated using transient analysis. It was noticed that Euler and Coriolis forces have a significant contribution in the initial time when the wind turbine accelerates from rest. The later portion is dominated by the Coriolis and Euler forces, and in some instances they cancel out each other. However, there is always a steady contribution from the centripetal force which is proportional to the magnitude of angular velocity of the wind turbine.
Keywords: MEMS, piezoelectric materials, energy harvesting, natural frequency, transient analysis.
With the concerns over the depletion of fossil fuels as an energy source, renewable energy is gaining popularity. Wind energy has the potential to become a significant source of energy in the future. In [1], authors have given a detailed account on harnessing energy using piezoelectric materials. An idea of harnessing electricity using piezoelectric material through various devices are reported in [2-6]. This manuscript describes a new approach to increase energy production from a windmill using a MEMS energy harvester. The MEMS device consists of a proof mass coupled to four Z-shaped cantilevers. This setup is placed on the blade of a windmill.
The idea is to exploit the centripetal, Coriolis, and Euler forces generated in a rotating windmill, which otherwise goes unutilized since they increase the strain energy in the blades. This also gets subjected to random vibration, due to the wind as well as the vibrations of the whole structure. The device consists of a proof mass connected to thin beams and can move in all directions with different stiffness. When the windmill rotates, the centripetal acceleration initiates the motion in a direction. As this is a rotating frame, Coriolis force gets generated, and it acts in the direction perpendicular to rotation vector and velocity vector. These forces are supplemented by Euler forces created because of angular acceleration or deceleration of the windmill.
All these forces, along with the random vibrations, produce stresses in the beams which can be converted into electricity with the help of piezoelectric material, which otherwise goes unutilized. This can supplement the conventional wind energy produced to a reasonable extent.
Our device aims to harvest the strain energy created in the blades of the windmill with the help of a MEMS device. The device consists of a proof mass supported by four arms, which is driven by Euler, Centripetal and Coriolis forces created due to the windmill rotation and proof mass movement. This motion of the proof mass is utilized to create stress in the supporting arms, and the points of highest stress are identified. The so-identified points are made of piezoelectric materials, which are used to create potential differences and hence generate power.
Fig. 1 shows the final geometry of the energy harvester, has a uniform thickness of 5 µm. This is the optimum geometry with dimensions such that the beams are not too stiff, and the Coriolis and centripetal accelerations of the windmill rotor can cause vibrations in the proof mass, and stresses are generated in the beams so that energy can be harvested using a piezoelectric material. The material used in this study is Ga-As. Table 1 shows the geometric and material properties of the structure. The material GaAs has a Zinc blende structure with piezoelectric constant,
{d}_{14}
, 2.63 PC/N.
Table 1. Material and geometric properties of the structure
Length of the beam,
{l}_{b}
500 × 500 µm2
Fig. 1. The structure of the energy harvester
The study involves geometric parameterization, modal analysis and transient structural analysis of windmill coupled energy harvester. As the structure is of uniform thickness, a 2-D model was generated in ANSYS, and modal analysis was carried out. The proof mass length, beam thickness, and structural thickness were taken as parameters for optimization of the natural frequencies
The optimal geometry was used to analyze the stresses generated due to the action of centripetal, Coriolis and tangential forces. These stresses generate potential on the surface of the piezoelectric material. Transient structural analysis was carried out to account for the stresses induced due to the combined loading (centripetal, Coriolis and Euler forces) that can arise in the device when mounted on a rotating body such as the windmill. COMSOL Multiphysics was used to simulate this environment – The device is hinged at a distance of 88 meters from a rotating frame. This resembles the device being attached to a windmill blade, at 8 meters from the center of the rotor and is constrained from performing free motion in any direction except for rotation with respect to the center (not the proof mass). Two different profiles of angular velocity of the wind turbine were given as inputs to study the resultant stresses and piezo surface potentials generated.
A hypothetical angular velocity profile, which shows a steep and sudden increase and decrease in the angular velocities, was studied. This study was done in order to qualitatively account for the contribution of Euler force and Coriolis force in stress generation. Whereas the realistic angular velocity profile [7] depicting the working of a wind turbine on any normal day was taken to account for the real-life stresses produced. It represents the angular velocity profile of a wind turbine starting from rest and running at an almost constant angular velocity with mild periodic variations.
3.1.1. Mode shapes
Using the primitive geometry (Fig. 1), modal analysis was carried out to determine the mode shapes and the corresponding natural frequencies. The modes of interest are given in Fig. 2. The four ends of the beams are fixed. Since the axial stiffness in the
Y
-direction is highest, the natural frequency of the
Y
-mode is the highest, which is evident from Fig. 2. Given these results, the modes in which various forces act are as follows:
Z
-direction – Euler,
Y
direction – Coriolis,
X
direction – Centripetal.
Fig. 2. Mode shapes of interest: a) out of plane mode at 10.85 kHz, b)
X
-direction mode at 24.408 kHz, c)
Y
-direction mode at 30.799 kHz
Further, parametric studies were carried out to reduce the natural frequencies to the necessary limits and hence to reduce the force required to produce motion and stresses in every mode described.
3.1.2. Geometric parameterization
The geometrical dimensions of the structure such as the proof mass size, structure thickness, and beam thickness were chosen as parameters to optimize the natural frequencies of the structure. For this, the variation of natural frequencies with each of the aforementioned parameters is shown in Fig. 3. It can be noticed that the natural frequencies for all modes decrease with increase in the size of the proof mass, and the natural frequency for all modes increases with increase in beam thickness. It is also well established that the natural frequency increases with the increase in structure thickness.
3.2. Transient structural analysis
3.2.1. Hypothetical profile
The maximum von mises stresses induced in the device were extracted from the transient analysis and results are as shown in Fig. 4.
The corresponding maximum surface potentials produced is compared to the input angular velocity profile and is shown in Fig. 5. It can be noticed that the Euler forces and Coriolis forces have significant contributions in the time range 0-3 seconds. This can be seen by the higher peak of the surface potential graph in this time range. The rest of the surface potential line follows the angular velocity line proportionally which implies that the centripetal force makes majority of contribution at other time ranges.
Fig. 3. The variation of natural frequency with various geometric parameters
Fig. 4. Maximum von mises stress developed a function of time (hypothetical profile)
Fig. 5. Maximum surface potentials developed with time (hypothetical profile)
3.2.2. Realistic profile
The maximum von mises stresses produced in the volume of the device concerning time were computed, and results are as shown in Fig. 6.
The corresponding maximum surface potentials produced is compared to the input angular velocity profile and is shown in Fig. 7. Euler forces and Coriolis forces have a significant contribution in the time range 0-5 seconds when the wind turbine accelerates from rest. After 5 seconds, when there is a mild periodic variation in the angular velocity, in some instances the Coriolis force and Euler force add up (such as the time range 10-12 seconds), and in some instances they cancel out each other (such as the time range 7-10 seconds). However, there is always a steady contribution from the centripetal force which is proportional to the magnitude of angular velocity of the wind turbine.
Fig. 6. Maximum von mises stress developed with time (realistic profile)
Fig. 7. Maximum surface potentials and angular velocity vs time (realistic profile)
In the present study, a new concept of harnessing wind energy is introduced. Transient analysis simulation was carried out and it was observed that potential difference of the order 1 mV develops. The mass of each device is around 10 nanograms which is negligible compared to the mass of the windmill blade and hence not affecting the basic windmill operations. When several such devices are coupled in series, significant power can be harnessed.
Future scope of the study can include experimental verification, fatigue analysis, efficiency and economic viability.
Cook-Chennault, Kimberly Ann, et al. Piezoelectric energy harvesting: a green and clean alternative for sustained power production. Bulletin of Science, Technology and Society, Vol. 28, Issue 6, 2008, p. 496-509. [Publisher]
Priya Shashank Piezoelectric Windmill Apparatus. U.S. Patent No. 8,294,336, 2012. [Search CrossRef]
Zhu Qiang, Peng Zhangli Mode coupling and flow energy harvesting by a flapping foil. Physics of Fluids, Vol. 21, Issue 3, 2009, p. 033601. [Publisher]
Lockhart Robert, et al. A wearable system of micromachined piezoelectric cantilevers coupled to a rotational oscillating mass for on-body energy harvesting. 27th International Conference on Micro Electro Mechanical Systems, 2014. [Search CrossRef]
Alper Erturk, Inman Daniel J. Piezoelectric Energy Harvesting. John Wiley and Sons, 2011. [Publisher]
Kim Heung Soo, Kim Joo-Hyong, Kim Jaehwan A review of piezoelectric energy harvesting based on vibration. International Journal of Precision Engineering and Manufacturing, Vol. 12, Issue 6, 2011, p. 1129-1141. [Publisher]
Neto Maria Augusta, Wenbin Yu, Ambrósio Jorge A. C., Leal Rogério Pereira Design blades of a wind turbine using flexible multibody modelling. International Conference on Renewable Energies and Power Quality, Valencia, Spain, 2009. [Publisher]
Harne Ryan L., Wang K. W. A review of the recent research on vibration energy harvesting via bistable systems. Smart Materials and Structures, Vol. 22, Issue 2, 2013, p. 023001. [Publisher]
Bressers Scott, et al. Small-scale modular windmill. American Ceramics Society Bulletin, Vol. 89, Issue 8, 2010, p. 34-40. [Search CrossRef]
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BlackmanHarrisWindow - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Windowing Functions : BlackmanHarrisWindow
multiply an array of samples by a Blackman-Harris windowing function
BlackmanHarrisWindow(A)
The BlackmanHarrisWindow(A) command multiplies the Array A by the Blackman-Harris windowing function and returns the result in an Array having the same length.
The Blackman-Harris windowing function
w\left(k\right)
N
0.35875-0.48829\mathrm{cos}\left(\frac{2k\mathrm{\pi }}{N}\right)+0.14128\mathrm{cos}\left(\frac{4k\mathrm{\pi }}{N}\right)-0.1168\mathrm{cos}\left(\frac{6k\mathrm{\pi }}{N}\right)
The SignalProcessing[BlackmanHarrisWindow] command is thread-safe as of Maple 18.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
N≔1024:
a≔\mathrm{GenerateUniform}\left(N,-1,1\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628415008457060}}
\mathrm{BlackmanHarrisWindow}\left(a\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628414856839164}}
c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right):
\mathrm{BlackmanHarrisWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right),'\mathrm{container}'=c\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628414856815308}}
u≔\mathrm{`~`}[\mathrm{log}]\left(\mathrm{FFT}\left(c\right)\right):
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{listplot}\left(\mathrm{ℜ}\left(u\right)\right),\mathrm{listplot}\left(\mathrm{ℑ}\left(u\right)\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
The SignalProcessing[BlackmanHarrisWindow] command was introduced in Maple 18.
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Correct state and state estimation error covariance using extended or unscented Kalman filter, or particle filter and measurements - MATLAB correct - MathWorks Australia
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k\right]
\mathrm{mu}
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k-1|k-1\right]
x\left[k\right]=\sqrt{x\left[k-1\right]+u\left[k-1\right]}+w\left[k-1\right]
y\left[k\right]=x\left[k\right]+2*u\left[k\right]+v\left[k{\right]}^{2}
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k-1|k-1\right]
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k\right]
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Use the triangles below right to answer the questions that follow.
What is the scale factor from A to B?
What scale factor value times
6
ft would result in
5
ft? Consider the variable equation with
x
as the scale factor:
6x=5
After isolating
x
, what do you get as the scale factor?
\frac{5}{6}
What is the scale factor from B to A?
If the change from B to A is just the reverse of the change from A to B, what would the scale factor be? Create an expression like above if necessary.
What is the relationship of the scale factors?
The scale factors are reciprocals.
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Three naive methods for solving systems of ordinary differential equations
by Elias Hernandis • 02 April 2020
We present three naive iterative methods for solving systems of ordinary differential equations which are widely discussed in undergraduate level scientific computing courses. Implementations are given in MATLAB code.
There are several strategies for solving the systems of ODEs. Some systems, such as the linear ones, can relatively easily be solved analytically. Others, mandate that we use numerical methods. We shall first instroduce a method based on an analytic result that will allow us to validate the other methods, at least on the linear case. Then we move on to other methods which support arbitrary systems of ODEs.
This is a work in progress post. This means I have probably not read it even twice. It also means that maybe it will never be completed.
The exponential matrix method
This method is based on the following result from ODE theory. Consider a linear system of ordinary differential equations of the form
\begin{cases} \dot x (t) = A x(t),\qquad t \in [0, T] \\ x(0) = x_0 \end{cases},
x : \mathbb{R} \to \mathbb{R}^n
x_0 \in \mathbb{R}^n
A \in \mathbb{R}^{n \times n}
is a real-valued matrix. Then the solution to the previous initial value problem is given by the matrix exponential:
x(t) = e^{t A} x_0,\qquad t \in [0, T].
It is very convenient that MATLAB provides the function expm which finds the value of the matrix exponential numerically. Hence, implementing this method only requires defining the initial condition vector
x_0
, the system matrix
A
and calculating the matrix exponential.
\dot u (t) = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} u(t), \qquad u(0) = \begin{pmatrix} \sin 1 \\ \sin 2 \\ \sin 3 \end{pmatrix}.
The corresponding MATLAB translation and numerical solution can be computed by
u0 = sin(1:3)
u = @(t) expm(t .* A) * u0'
A handy way of working with the solution is to define it as a MATLAB function handle. A function handle is the way to define a mathematical function in MATLAB. It allows us to evaluate an expression that depends on one or more variables without having to retype the expression each time. In the previous example, we can find the value of
u
at say time
t = 0.1
by evaluating the expression u(0.1).
In the context of the method of lines, the matrix
A
is some finite difference matrix obtained by approximating the space derivative in the first step of the method. Further examples are given in the application section.
In terms of stability, this method does not present any problems although some systems may yield solutions with very big numbers which overflow MATLAB’s floating point representation of numerical values. Since we will only consider small time intervals, this will not be a problem. On the other hand, a downside of this method is that it can only be used when the ODE system at hand is linear.
The Euler-Forward method
Another way of solving systems of ODEs is to apply the so-called Euler forward method. The main idea behind this method is to approximate the derivative by a finite difference
\dot u (t) \approx \frac{u^{n + 1} - u^n}{\Delta t}.
In the context of (linear) systems of ODEs we have that
\dot{\mathbf u} \approx \frac{\mathbf u^{n + 1} - \mathbf u^n}{\Delta t} = A \mathbf u,
\mathbf u = (u_0, \dots, u_i, \dots, u_N)^T
is a column vector made from the approximations
u_i(t)
obtained in the first part of the method of lines. Since there are no restrictions on what we can have on the right hand side of the equality so this very simple method works for non-linear systems as well. To put it more formally, given an initial value problem of the form
\begin{cases} \dot{\mathbf u} (t) = \mathbf f(t, \mathbf u(t)),\qquad t \in [0, T] \\ \mathbf u(0) = \mathbf u_0 \end{cases},
the iteration of the Euler forward method is given by
\mathbf u^{n + 1} = \mathbf u^n + \Delta t\, \mathbf f(t, \mathbf u^n),\qquad \mathbf u^0 = \mathbf u_0.
Notice how we have arranged the iteration: given
\mathbf u^n
we have an explicit formula for obtaining the next step
\mathbf u^{n + 1}
. This makes implementing this method extremely simple and also gives it an alternative name, the explicit method.
However, this simplicity comes at a price. The EF method is only stable under some circumstances. We will not dive into the general stability considerations of the EF method in this report.
The Euler-Backward method
The stability problems of the EF method lead us to this slightly more advanced version: the Euler backward method. Again, it is based on approximating the derivative by a finite difference, only this time we choose the index
n + 1
for the right-hand side. In the context of a system of the form described in the previous section, the iteration is given by
\mathbf u^{n + 1} - \Delta t\, \mathbf f(t, \mathbf u^{n + 1}) = \mathbf u^n.
Contrarily to the EF case, the next step in the iteration
\mathbf u^{n + 1}
appears implicitly in the iteration. This gives this method its alternative name, the implicit method. This means that computing an iteration will be much more complicated: depending on
\mathbf f
we may even have to solve a non-linear system of equations. If we restrict ourselves to the case of linear systems, then
\mathbf f(t, \mathbf u^n) = A \mathbf u^n
and thus the iteration becomes
(I - \Delta t\, A)\mathbf u^{n + 1} = \mathbf u^n,
i.e. a system of linear equations. Matlab makes this very easy to implement using the backslash ("\") operator. Solving a linear system is not as trivial as evaluating an expression as we did in EF, but this method has the benefit that it is always stable for the applications we consider.
The IMEX method
When a model is described by a non-linear system of ODEs, it is almost impossible to apply the EB method since it would require a system of non-linear equations for every step of the iteration. In this case, EF is still a possibility, but, on the other hand, we still would like to have some stability warranties. We introduce the IMEX method. The IMEX method owes its name to the fact that it is a combination of the EB (or IMplicit) and the EF (or EXplicit) methods.
To keep things simple we shall focus on ODE systems of the particular form
\dot{\mathbf u (t)} = A \mathbf u(t) + \mathbf f(\mathbf u(t)),
where we can think of the right-hand side as having a linear part
A \mathbf u
and a potentially non-linear part
\mathbf f(\mathbf u)
. The IMEX method uses EB for the “easy” linear part and EF for the “hard” non-linear part. Hence the iteration is given by
(I - \Delta t\, A)\mathbf u^{n + 1} = \mathbf u^n + \Delta t\, \mathbf f(\mathbf u^n).
This iteration still has an implicit form, but at least we know the right-hand side from the previous step and hence the system is always linear and can be solved with the MATLAB backslash operator. Stability consideration in for this method are much, much more elaborate and are not discussed here.
© Elias Hernandis 2022
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ChiSquareIndependenceTest - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Tests : ChiSquareIndependenceTest
ChiSquareIndependenceTest(X, options)
Matrix of categorized data
(optional) equation(s) of the form option=value where option is one of level, output, or summarize; specify options for the ChiSquareIndependenceTest function
The ChiSquareIndependenceTest function computes the chi-square test for independence in a matrix. This test attempts to determine if two factors can be considered to be independent of one another for purposes of analysis.
The first parameter X is a matrix of categorized data samples.
This option is used to specify the level of the analysis (minimum criteria for a data set to be considered independent). By default this value is 0.05.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔\mathrm{Matrix}\left([[32,12],[14,22],[6,9]]\right):
Y≔\mathrm{Matrix}\left([[2,4],[4,9],[7,12]]\right):
Perform the independence test on the first sample.
\mathrm{ChiSquareIndependenceTest}\left(X,\mathrm{level}=0.05,\mathrm{summarize}=\mathrm{embed}\right):
\textcolor[rgb]{0,0,1}{3.}
\textcolor[rgb]{0,0,1}{95.}
\textcolor[rgb]{0,0,1}{\mathrm{ChiSquare}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)
\textcolor[rgb]{0,0,1}{10.7122}
\textcolor[rgb]{0,0,1}{0.00471928}
\textcolor[rgb]{0,0,1}{5.99146}
Perform the independence test on the second sample.
\mathrm{ChiSquareIndependenceTest}\left(Y,\mathrm{level}=0.05,\mathrm{summarize}=\mathrm{true}\right)
Computed Statistic: .1289151874
\textcolor[rgb]{0,0,1}{\mathrm{hypothesis}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{criticalvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5.99146454710798}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{distribution}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{ChiSquare}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{pvalue}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.937575872647938}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{statistic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.1289151874}
The Statistics[ChiSquareIndependenceTest] command was updated in Maple 2016.
|
Numerical analysis of single grit grinding on aluminum workpiece | JVE Journals
Srinivasa Reddy Bode1 , Ashish Kumar2 , Baij Nath Singh3
3Indian Institute of Technology Dhanbad, Dhanbad, 826004, Jharkhand, India
Copyright © 2019 Srinivasa Reddy Bode, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Grinding is a finishing process used in almost every industry. The purpose of this research is to study the effects of grinding on aluminum. Finite Element Method (FEM) is used to numerically simulate grinding by using microscale modeling. It is found that though under grinding, stress mises developed do not exceed yield stress. The force variation and displacement parameters have been correspondingly discussed. The effects of grinding on the workpiece with and without lubrication.
Keywords: grit grinding, microscale modelling, friction coefficient, stress analysis.
Surface finishing process like grinding is complex because of multiple cutting edges. The quality of workpiece often depends on workers skill and experience. Slightest error would make workpiece incompatible to use. Especially in fields like aerospace and bio medical engineering where precision is of utmost importance. Many researches [1] were dedicated to find optimal conditions of performing grinding operation for better surface finish.
Experimental analysing of this process under confined conditions is good for validation of grinding process. Performing experimental analysis under different conditions would involve a lot of time and money. Finite element analysis would be a fine approach to this problem. Results obtained through numerical simulation would be same as what we obtained in experimental procedure. This could save both time and money.
The quality of a simulation depends upon the model we have chosen. Models are of two types physical model and empirical model. Physical or mathematical models are based on the applications of physical laws like grinding force models, chip thickness models, grinding models etc. On the other hand, empirical models are based on the measured values. Physical models have broader application compared to empirical models but they are not always feasible in real life scenarios. Both physical and empirical models are used in this simulation. Generally grinding is modelled as a mechanical process in which we analyse the interaction between grinding wheel and workpiece, this is called macro scale modelling. There is one more approach in which single grit machining is modelled, this is known as micro scale modelling [2-5].
In this paper we intend to demonstrate the grinding process using both physical and empirical model with the help of finite element modelling. By using finite element modelling we simulate interaction of grinding grit and workpiece in Abaqus/Standard. Materials we use are grinding grit made of steel and aluminium workpiece [6].
Abaqus consists of different components to completely define, analyse and obtain results of a problem.
2.1. Discretized geometry
Both workpiece and grit are designed on Abaqus itself. Workpiece is designed by extruding a rectangle while grit is designed by rotating a quadrant of circle. Dimension of workpiece is 1×1×1 mm and radius of grit is 0.05 mm. Hexahedral elements are the recommended solid(continuum) elements. Three-dimensional hexahedral element. Strains are more accurate at integration points and integration point of the C3D8R element is located in the middle of an element. So, small size elements are needed to capture stress concentration at the boundaries.
Fig. 1. Work piece geometry
Fig. 2. Grit geometry
For this simulation, the Hex type mesh is used. The hex form meshed is chosen because it gives the best result and also at lower cost and processing time. In the contact area, there is a fine mesh along the cutting path to ensure the best result. Structured meshing is applied to specific model topologies using pre-established mesh patterns. Nevertheless, in order to use this method, complex structures typically need to be separated into simpler regions.
Grit material (steel): density (
\rho
) = 7700 kg/m3, Poissons ratio (
v
) = 0.28, Youngs modulus (
E
) = 215 GPa.
Work piece material (aluminum): density (
\rho
) = 2,710 kg/m3, Poissons ratio (
v
E
) = 69 GPa.
Johnson cook is a empirical relation developed by Jonson and cook, It is used to model damage evolution and predict failure in a material.
Table 1. Johnson cook damage of steel
A
B
n
m
General contact allows us to define contact between many or all regions of a model with a single interaction. Interactions typically include all bodies in the model. Very few restrictions on the types of surfaces involved for contact constrained we use penalty method. The friction coefficient take into account is 0.15. General contact allows you to define contact between many or all regions of a model with single interaction. The surfaces that can interact with one another comprise the contact domain and can span many disconnected regions of a model. Contact pairs describe contact between two surfaces. Requires more careful definition of contact. Every possible contact pair interaction must be defined.
Abaqus can carry out two types of simulations static and dynamic analyses. In this problem we use dynamic analysis.
The stress produced during this simulation is studied below. The Contour below shows the distribution of Stress. The depth of cut is 0.05 mm and is kept constant for complete simulation. We can see that the stress of 1748 N/mm2 or 1748 MPa is developed by grinding.
The deformation in the workpiece is also studied and shown in Figures below.
Fig. 3. Stress mises contour in MPa
Fig. 4. Deformation contours U1, U2 and U3 in mm
Fig. 5. Figure showing reaction forced developed due to deflection and Shear stress caused due to frictional contact
Fig. 6. Figure showing load deflection curve
Fig. 7. Figure showing total energy curve
This research manifests the importance of lubricant in grinding. Grinding of aluminum under dry conditions resulted in higher grinding forces and high specific energy requirement. Consequently, grinding damages are more if the aluminum is ground under dry condition.
We also get the stress developed in workpiece. The defects that can affect the fatigue life of the components in service as these may act as areas of stress concentration and favorable sites for earlier initiation of fatigue cracks.
Tönshoff H. K., Peters J., Inasaki I., Paul T. Modeling and simulation of grinding processes. Annals of the CIRP, Vol. 41, Issue 2, 1992, p. 677-688. [Publisher]
Park H. W., Liang S. Y. Micro grinding force predictive modelling based on microscale single grain interaction analysis. Manufacturing Technology and Management, Vol. 12, 2007, p. 25-28. [Publisher]
Klocke F. Modelling and simulation of grinding processes. 1st European Conference on Grinding, 2003. [Search CrossRef]
Doman D. A., Bauer R., Warkentin A. Experimentally validated finite element model of the rubbing and ploughing phases in scratch tests. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, Vol. 223, 2009, p. 1519-1527. [Publisher]
Takenaka N. A study on the grinding action by single grit. Annals of the CIRP, Vol. 13, 1966, p. 183-190. [Search CrossRef]
Tahsin Opoz, Chen Xun Numerical simulation of single grit grinding. Proceeding of the 16th International Conference on Automation and Computing, Birmingham, UK, 2010. [Search CrossRef]
|
Algebra Problem on Linear Diophantine Equations - Problem Solving: Indeterminate System of Linear Equations - Steven Zheng | Brilliant
Indeterminate System of Linear Equations
\begin{aligned} y & = 2{x}_{1}+{x}_{2} \\ y & = 3{x}_{2}+{x}_{3} \\ y & = 4{x}_{3}+{x}_{4} \\ y & = 5{x}_{4}+{x}_{5} \\ y & = 6{x}_{5}+{x}_{1}. \end{aligned}
If all of the variables are integers, what is the minimum positive integer value of
\left(\sum_{i=1}^{5}{x}_{i}\right) - y ?
|
Proper length - Wikipedia
Length of an object in the object's rest frame
Proper length[1] or rest length[2] is the length of an object in the object's rest frame.
1 Proper length or rest length
2 Proper distance between two events in flat space
3 Proper distance along a path
Proper length or rest length[edit]
{\displaystyle L_{0}=\Delta x.}
{\displaystyle L={\frac {L_{0}}{\gamma }}.}
{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}-c^{2}\Delta t^{2}}}.}
Proper distance between two events in flat space[edit]
{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}},}
{\displaystyle \Delta \sigma ={\sqrt {\Delta x^{2}+\Delta y^{2}+\Delta z^{2}-c^{2}\Delta t^{2}}},}
Proper distance along a path[edit]
{\displaystyle L=c\int _{P}{\sqrt {-g_{\mu \nu }dx^{\mu }dx^{\nu }}},}
In the equation above, the metric tensor is assumed to use the +−−− metric signature, and is assumed to be normalized to return a time instead of a distance. The − sign in the equation should be dropped with a metric tensor that instead uses the −+++ metric signature. Also, the
{\displaystyle c}
should be dropped with a metric tensor that is normalized to use a distance, or that uses geometrized units.
^ a b c Moses Fayngold (2009). Special Relativity and How it Works. John Wiley & Sons. ISBN 978-3527406074.
^ a b Franklin, Jerrold (2010). "Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity". European Journal of Physics. 31 (2): 291–298. arXiv:0906.1919. Bibcode:2010EJPh...31..291F. doi:10.1088/0143-0807/31/2/006. S2CID 18059490.
^ Poisson, Eric; Will, Clifford M. (2014). Gravity: Newtonian, Post-Newtonian, Relativistic (illustrated ed.). Cambridge University Press. p. 191. ISBN 978-1-107-03286-6. Extract of page 191
^ Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011). Relativistic Celestial Mechanics of the Solar System. John Wiley & Sons. p. 136. ISBN 978-3-527-63457-6. Extract of page 136
Retrieved from "https://en.wikipedia.org/w/index.php?title=Proper_length&oldid=1042245362"
|
Transverse mode - Wikipedia
(Redirected from Transverse electric and magnetic mode)
Find sources: "Transverse mode" – news · newspapers · books · scholar · JSTOR (November 2009) (Learn how and when to remove this template message)
A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular (i.e., transverse) to the radiation's propagation direction. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.[1]
1 Types of modes
1.1.1 Optical fibers
Types of modes[edit]
Modes in waveguides can be classified as follows:
Transverse electromagnetic (TEM) modes
Neither electric nor magnetic field in the direction of propagation.
Transverse electric (TE) modes
No electric field in the direction of propagation. These are sometimes called H modes because there is only a magnetic field along the direction of propagation (H is the conventional symbol for magnetic field).
Transverse magnetic (TM) modes
No magnetic field in the direction of propagation. These are sometimes called E modes because there is only an electric field along the direction of propagation.
Non-zero electric and magnetic fields in the direction of propagation. See also Planar transmission line § Modes.
Optical fibers[edit]
See also: Equilibrium mode distribution, Mode volume, and Cladding mode
The number of modes in an optical fiber distinguishes multi-mode optical fiber from single-mode optical fiber. To determine the number of modes in a step-index fiber, the V number needs to be determined:
{\textstyle V=k_{0}a{\sqrt {n_{1}^{2}-n_{2}^{2}}}}
{\displaystyle k_{0}}
is the wavenumber,
{\displaystyle a}
is the fiber's core radius, and
{\displaystyle n_{1}}
{\displaystyle n_{2}}
are the refractive indices of the core and cladding, respectively. Fiber with a V-parameter of less than 2.405 only supports the fundamental mode (a hybrid mode), and is therefore a single-mode fiber whereas fiber with a higher V-parameter has multiple modes.[4]
{\displaystyle I_{pl}(\rho ,\varphi )=I_{0}\rho ^{l}\left[L_{p}^{l}(\rho )\right]^{2}\cos ^{2}(l\varphi )e^{-\rho }}
{\displaystyle E_{mn}(x,y,z)=E_{0}{\frac {w_{0}}{w}}H_{m}\left({\frac {{\sqrt {2}}x}{w}}\right)H_{n}\left({\frac {{\sqrt {2}}y}{w}}\right)\exp \left[-(x^{2}+y^{2})\left({\frac {1}{w^{2}}}+{\frac {jk}{2R}}\right)-jkz-j(m+n+1)\zeta \right]}
{\displaystyle w_{0}}
{\displaystyle w(z)}
{\displaystyle R(z)}
{\displaystyle \zeta (z)}
are the waist, spot size, radius of curvature, and Gouy phase shift as given for a Gaussian beam;
{\displaystyle E_{0}}
is a normalization constant; and
{\displaystyle H_{k}}
is the k-th physicist's Hermite polynomial. The corresponding intensity pattern is
{\displaystyle I_{mn}(x,y,z)=I_{0}\left({\frac {w_{0}}{w}}\right)^{2}\left[H_{m}\left({\frac {{\sqrt {2}}x}{w}}\right)\exp \left({\frac {-x^{2}}{w^{2}}}\right)\right]^{2}\left[H_{n}\left({\frac {{\sqrt {2}}y}{w}}\right)\exp \left({\frac {-y^{2}}{w^{2}}}\right)\right]^{2}}
^ "Transverse electromagnetic mode"
^ F. R. Connor, Wave Transmission, pp.52-53, London: Edward Arnold 1971 ISBN 0-7131-3278-7.
^ U.S. Navy-Marine Corps Military Auxiliary Radio System (MARS), NAVMARCORMARS Operator Course, Chapter 1, Waveguide Theory and Application, Figure 1-38.—Various modes of operation for rectangular and circular waveguides.
^ Kahn, Joseph M. (Sep 21, 2006). "Lecture 3: Wave Optics Description of Optical Fibers" (PDF). EE 247: Introduction to Optical Fiber Communications, Lecture Notes. Stanford University. p. 8. Archived from the original (PDF) on June 14, 2007. Retrieved 27 Jan 2015.
^ Paschotta, Rüdiger. "Modes". Encyclopedia of Laser Physics and Technology. RP Photonics. Retrieved Jan 26, 2015.
^ K. Okamoto, Fundamentals of Optical Waveguides, pp. 71–79, Elsevier Academic Press, 2006, ISBN 0-12-525096-7.
^ Svelto, O. (2010). Principles of Lasers (5th ed.). p. 158.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Transverse_mode&oldid=1082244321"
|
Anand Parameters for SAC305 Alloys After Prolonged Storage up to 1-Year | InterPACK | ASME Digital Collection
US Army AMRDEC, Huntsville, AL
Lall, P, Zhang, D, Suhling, J, & Locker, D. "Anand Parameters for SAC305 Alloys After Prolonged Storage up to 1-Year." Proceedings of the ASME 2017 International Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Microsystems collocated with the ASME 2017 Conference on Information Storage and Processing Systems. ASME 2017 International Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Microsystems. San Francisco, California, USA. August 29–September 1, 2017. V001T05A009. ASME. https://doi.org/10.1115/IPACK2017-74300
Electronics in automotive underhood environments may be subjected to high temperatures in the neighborhood of 175°C while subjected to high strain rate mechanical loads of vibration. Portable products such as smartphones and tablets stay in the powered on condition for a majority of their operational life during which time the device internals are maintained at higher than ambient temperature. Thus, it would be expected for interconnects in portable products to be at a temperature high than room temperature when subjected to accidental drop or shock. Furthermore, electronics in missile-applications may be subjected to high strain rates after prolonged period of storage often at high temperature. Electronics systems including interconnects may experience high strain rates in the neighborhood of 1–100 per sec during operation at high temperature. However, the material properties of SAC305 leadfree solders at high strain rates and high operating temperatures are scarce after long-term storage. Furthermore, the solder interconnects in simulation of product drop are often modeled using elastic-plastic properties or linear elastic properties, neither of which accommodate the effect of operating temperature on the solder interconnect deformation at high operating temperature. SAC305 solders have been shown to demonstrate the significant degradation of mechanical properties including the tensile strength and the elastic modulus after exposure to high temperature storage for moderate periods of time. Previously, Anand’s viscoplastic constitutive model has been widely used to describe the inelastic deformation behavior of solders in electronic components under thermo-mechanical deformation. Uniaxial stress-strain curves have been plotted over a wide range of strain rates (
ε.
= 10, 35, 50, 75 /sec) and temperatures (T = 25, 50, 75, 100, 125, 150, 175, 200°C). Anand viscoplasticity constants have been calculated by non-linear fitting procedures. In addition, the accuracy of the extracted Anand constants has been evaluated by comparing the model prediction and experimental data.
Transportation: Autonomous and Electric Vehicles
Alloys, Storage, High temperature, Solders, Temperature, Deformation, Electronics, Operating temperature, Constitutive equations, Elastic moduli, Elasticity, Electronic components, Fittings, Lead-free solders, Materials properties, Mechanical properties, Missiles, Shock (Mechanics), Simulation, Stress, Stress-strain curves, Tensile strength, Thermomechanics, Vibration, Viscoplasticity
High Strain Rate Mechanical Properties of SAC-Q With Sustained Elevated Temperature Storage at 100°C
Strain Rate Effects and Rate-Dependent Constitutive Models of Lead-Based and Lead-Free Solders
|
Vibration based diagnosis of wheel defects of metro train sets using one period analysis on the wayside | JVE Journals
Onur Kılınç1 , Jakub Vágner2
1, 2Department of Transport Means and Diagnostics, Faculty of Transport Engineering, University of Pardubice, Pardubice, Czech Republic
This research examines two different methods; Wavelet Packet Energy (WPE) and Time-domain Features (TDF) which are effective in faulty signal feature extraction of metro wheels in wayside level using vibration sensors. Signals of each wheelset passing of a trainset with both healthy and faulty wheels are recorded by the vibration sensors which are mounted on both left and right rails and a novel one-period sampling is performed at 51.2 kHz sample rate. Retrieved signal samples are used in the construction of a database which is consistent of healthy and faulty cases. Since the database has insufficient number of faulty samples, the database is balanced by a method so called Adaptive Synthetic Sampling (ADASYN) so that each class has the same number of observations. Two state-of art classifiers; Support Vector Machines (SVM) and Fisher Linear Discriminant Analysis (FLDA) are employed by utilizing 16-fold cross validation to solve the two-class problem. Referring to the results, SVM-I-TDF outperforms by classifying all samples with a success rate of 100 % and other methods have also promising results. Proposed methods may be used in the condition monitoring of metro wheelsets effectively by means of not only performance but also cost-efficiency.
Keywords: wavelet packet energy, time domain features, wheelset fault diagnosis, one period analysis, Fisher linear discriminant analysis, support vector machine.
Condition monitoring of railway vehicles is advantageous due to providing cost efficient maintenance especially when a wayside diagnosis approach is used. The main challenge is to make the diagnosis of the related component without significant number of false positives and false negatives; otherwise irreversible and high costs may occur.
Diagnosis of faults of metro wheelsets is vital to prevent the damage on other structures of the vehicle and rails which may threaten the safety of the run. Detection of these faults provides cost effective maintenance outperforming the scheduled maintenance by means of cost efficiency. The most successful detection of such cases can be performed by mounting sensors of each wheelsets which is too costly. For that reason, wayside approaches have been focused by the researches in recent years.
In condition monitoring of railway vehicles, three main categories are existed; model based dynamical techniques including various Kalman [1, 2] and particle filter [3], signal based techniques including band-pass filter, spectral analysis, wavelet analysis and Fast Fourier Transform [4], and wayside sensor based approaches which are generally practical in wheel defects according to a recent review [5]; using accelerometer and piezoelectric sensors on rail and using wavelet based methods and thresholding to identify the degree of the wheel flat [6], using a high speed camera to identify the wheel profile when the railway vehicle is passing by a low speed (10 mph) and image analysis, using optical sensors, accelerometers, load cells and strain gages to measure vertical deflection of the rail which makes it able to identify wheel defects like; out-of round, flat, shelling, measuring lateral force to determine bogie performance using hunting track detectors [7], acoustic bearing defect detectors which is based on statistical processing of the data, ultrasonic cracked wheel detection and many others [8].
The main drawback of wayside diagnosis approaches is noise since acting of multiple components produce irrelevant vibration and environmental noise which is relatively heavy in most cases. Moreover, when it comes to vibration analysis of metro wheelsets, the difficulty of distinguishing faulty data can be more challenging since the signals are badly affected by both noise and insufficiency of the sampling time due to the fact that the wheelset which is wanted to be diagnosed is overlapping to other sampling windows of nearby wheels of the same bogie.
This study presents several methods which are efficient in wheel defect detection of metro train sets. Throughout the research, measurements from two vibration sensors which are accompanied by two optical gates to detect vehicle speed and wheelset positions are recorded on a passage between two metro stations which are present in Prague metro line. Two feature extraction techniques are proposed which are considered as efficient in non-stationary diagnosis besides two cutting edge classifiers to maintain a stable condition monitoring framework.
The organization of this paper is as follows: Test environment and signal acquisition as well as sampling by one period analysis are told in Section 2. Feature extraction procedures for each proposed method is presented in Section 3. Experimental results and discussion is given in Section 4. Lastly, conclusion part is presented in Section 5.
2. Database and segmentation
In this research, a vibration based measurement system is installed in a metro tunnel on a wayside passage between Dejvická and Bořislavka which is located in Prague metro line-A. Two vibration sensors
{Z}_{1}
{Z}_{2}
are accompanied by two optical gates
{G}_{A}
{G}_{B}
operating at 500 Hz for determining wheelset positions and vehicle speed.
The wayside passage, which all measurements are done, provides a stable speed of the run which is almost constant. Thanks to the instrumentation device NI-cDAQ-9234 which records the signals in all channels at 51.2 kHz sampling rate during all day when metros are in operation. Measurements are done in the project; Competence Centre of Railway Vehicles, No. TAČR TE01020038.
Metro train sets of type 81-71M are identical by means of bogies and passenger cars (4-axle, five passenger cars). During one day, eight passing of a train set (ID-108) which both have wheel defects and normal wheelsets. The wayside measurement system is shown in Fig. 1. The sampling window is chosen so that each wheel is rotated only one period according to the wheel diameter information from maintenance of Prague metro and exact operational speed.
Fig. 1. Wayside diagnostic measurement system nearby Dejvická metro station
Two main datasets are prepared in order to evaluate efficiency of proposed methods; A1 has only measurement signals of normal and faulty cases of wheels while SA1 is consistent of both measured faulty samples and over sampled vectors of faulty wheels. Table 1 describes the datasets.
Table 1. Description of classified datasets of measured and synthesized datasets
Train ID.
Measured samples
Synth. samples
Wheelset interval
In this section, the information about proposed techniques for extracting faulty characteristics of the measured signals, which are retrieved in wayside passage by vibration sensors, are told. Furthermore, an adaptive oversampling algorithm for more reliable model evaluation is also presented.
3.1. Wavelet packet energy
This method is based on Short Time Fourier Transform (STFT) which is used in frequency domain signal processing applications.
In Wavelet Packet Transform (WPT) Fourier spectrum of the signal is divided into desired number of frequency bands which provides custom frequency resolution. Respect to the number of level
L
, number of components in WPT is
n={2}^{L}
. After determining maximum number of level, filtering for each frequency band is carried out on the signal to obtain low and high frequency components which is followed by down sampling to achieve next level. Referring to Fig. 2, since WPT is lack of being time-invariant [11], energy property of each wavelet packet is calculated to increase its performance in detecting transients in non-stationary signal processing which leads to Wavelet Packet Energy (WPE) feature extraction (Fig. 3).
In this research, Haar wavelets [12] is preferred in the decomposition of using WPE. Five level WPE is calculated which results size of 32×1 feature vectors for each signal sample.
Fig. 2. a) Transient signals that are shifted in time domain, b) resultant three level WPT
3.2. Time domain features
Statistically, faulty signals have similar behaviour not only in frequency domain but also in time domain. Statistical time-domain feature extraction methods have already been proposed for recognition of numerous faulty conditions of different components in rotating machinery [13, 14].
Proposed Time Domain Features (TDF) are consistent of several individual properties (Eq. (1)) of the discrete signals which is preferred as in the research [10]:
TDF=\left[energy,mean,\mathrm{ }std.\mathrm{ }dev,max,\mathrm{ }min,\mathrm{ }kurtosis,skewness,crest\mathrm{ }factor\mathrm{ }{\right]}_{8×1}.
Fig. 3. Resultant three level WPE for the signals in Fig. 2
3.3. Adaptive synthetic sampling
In practical applications of wayside fault diagnosis subjected to railway vehicles, number of healthy cases superimpose the faulty ones in the measurement. However, most classifiers in pattern recognition requires balanced database which means the same number of observations in each class. When the faulty observations are insufficient, which means the number of features in each vector is shorter than the number of faulty samples, model evaluation cannot be performed in a stable way. The most straightforward way to deal with this problem is to interpolate samples and generate new ones.
According to recent literature, a synthetic minority oversampling technique (SMOTE) is proposed [15] which interpolates the existing feature vectors of the minority class linearly and create new feature vectors. Nevertheless, SMOTE approach has limited to focusing only on minority class. Being lack of consideration of the boundaries between classes, over sampling with SMOTE has unrealistic results.
Fig. 4. Feature oversampling from minority class to balance datasets using ADASYN (only two dimensions are shown)
In the proposed method for generating additional feature vectors, Adaptive Synthetic Sampling (ADASYN) [16], which is the further improvement of SMOTE is used. ADASYN methodology creates more samples between boundaries of each class adaptively for each feature, thus provides more stable oversampling.
The oversampling operation of time domain features which are extracted from samples of wheel defects from 16 observations (minority class) to the number of normal observations (majority class that has 128 observations) is shown in Fig. 4. It is clear that, the outlier effect is diminished and samples are created in a more plausible way.
After having extracted the features with proposed WPE and TDF algorithms, two state of art classifiers; Support Vector Machine with linear kernel (SVM-I) and Fisher Linear Discriminant Analysis (FLDA) are employed in the analyses of measured and synthetically oversampled datasets of normal and faulty cases using different number of cross validations that leads to best results. All classification results are shown in Table 2.
Referring to the results, in case of insufficient data, WPE approach is nominated to be not very promising whereas after oversampling the classification accuracy increased significantly. Best results are obtained by TDF with SVM-I classifier in both insufficient and over sampled datasets.
Table 2. Classification performance of measured (A1) and synthesized classes (SA1)
Average classification accuracies (%)
Num. of cross validation
SVM-I
In this study, two techniques that are efficient in non-stationary signal processing are used in order to diagnose wheel defects on metro train sets.
Measurement of real faulty data is so rare in real operation of metro trainsets unless a real-time measurement is performed. After having utilized an adaptive over sampling method (ADSAYN), more faulty samples are acquired which make it able to understand if more normal samples can be classified correctly.
With respect to results, an efficient model is developed after over sampling data and all normal cases are managed to be classified by both methods efficiently up to 100 % (SVM-I-TDF). This research may aid the maintenance specialists by providing an efficient framework for wayside condition monitoring of railway vehicles.
Tipireddy R., Nasrellah H. A., Manohar C. S. A Kalman filter based strategy for linear structural system identification based on multiple static and dynamic test data. Probabilistic Engineering Mechanics, Vol. 24, Issue 1, 2009, p. 60-74. [Search CrossRef]
Hide C., Moore T., Smith M. Adaptive Kalman filtering algorithms for integrating GPS and low cost INS. Position Location and Navigation Symposium, 2004, p. 227-233. [Publisher]
Charles G., Goodall R., Dixon R. Model-based condition monitoring at the wheel-rail interface. Vehicle System Dynamics, Vol. 46, Issue 1, 2008, p. 415-430. [Search CrossRef]
Bruni S., Goodall R., Mei T. X., Tsunashima H. Control and monitoring for railway vehicle dynamics. Vehicle System Dynamics, Vol. 45, Issues 7-8, 2007, p. 743-779. [Search CrossRef]
Ngigi R. W., Pislaru C., Ball A., Gu F. Modern techniques for condition monitoring of railway vehicle dynamics. Journal of Physics: Conference Series, Vol. 364, 2012, p. 012016. [Search CrossRef]
Belotti V., Crenna F., Michelini R. C., Rossi G. B. Wheel-flat diagnostic tool via wavelet transform. Mechanical Systems and Signal Processing, Vol. 20, Issue 8, 1953, p. 1966-2006. [Search CrossRef]
Barke D., Chiu W. K. Structural health monitoring in the railway industry: a review. Structural Health Monitoring, Vol. 4, Issue 1, 2005, p. 81-93. [Search CrossRef]
Brickle B., Morgan R., Smith E., Brosseau J., Pinney C. Identification of Existing and New Technologies for Wheelset Condition Monitoring. RSSB Report T607 UK P-07-005, TTCI Ltd., 2008. [Search CrossRef]
Lou X., Loparo K. A. Bearing fault diagnosis based on wavelet transform and fuzzy inference. Mechanical Systems and Signal Processing, Vol. 18, Issue 5, 2004, p. 1077-1095. [Search CrossRef]
Yu X., Ding E., Chen C., Liu X., Li L. A novel characteristic frequency bands extraction method for automatic bearing fault diagnosis based on Hilbert-Huang transform. Sensors, Vol. 15, Issue 11, 2015, p. 27869-27893. [Search CrossRef]
Yen G. G., Lin K.-C. Wavelet packet feature extraction for vibration monitoring. IEEE Transactions on Industrial Electronics, Vol. 47, Issue 3, 2000, p. 650-667. [Search CrossRef]
Li L., Qu L., Liao X. Haar wavelet for machine fault diagnosis. Mechanical Systems and Signal Processing, Vol. 21, Issue 4, 2007, p. 1773-1786. [Search CrossRef]
Fu S., Liu K., Xu Y., Liu Y. Rolling bearing diagnosing method based on time domain analysis and adaptive fuzzy C-means clustering. Shock and Vibration, Vol. 1, 2016, p. 2016-8. [Search CrossRef]
Wang M., Hu N.-Q., Hu L., Gao M. Feature optimization for bearing fault diagnosis. International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, 2013, p. 1738-1741. [Publisher]
Chawla N. V., Bowyer K. W., Hall L. O., Kegelmeyer W. P. SMOTE: synthetic minority over-sampling technique. Journal of Artificial Intelligence Research, Vol. 16, 2002, p. 321-357. [Search CrossRef]
He H., Bai Y., Garcia E. A., Li S. ADASYN: Adaptive synthetic sampling approach for imbalanced learning. IEEE International Joint Conference on Neural Networks, IEEE World Congress on Computational Intelligence, 2008, p. 1322-1328. [Search CrossRef]
Automatic Recognition and Positioning of Wheel Defects in Ultrasonic B-Scan Image Using Artificial Neural Network and Image Processing
Meng Yuan, Jinlong Li, Yingwei Liu, Xiaorong Gao
|
15 February 2014 Kesten’s theorem for invariant random subgroups
Miklós Abért, Yair Glasner, Bálint Virág
An invariant random subgroup of the countable group
\Gamma
is a random subgroup of
\Gamma
whose distribution is invariant under conjugation by all elements of
\Gamma
. We prove that for a nonamenable invariant random subgroup
H
, the spectral radius of every finitely supported random walk on
\Gamma
is strictly less than the spectral radius of the corresponding random walk on
\Gamma /H
. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that, for a Cayley graph
G
of a linear group with no amenable normal subgroups, any sequence of finite quotients of
G
that spectrally approximates
G
G
in Benjamini–Schramm convergence. In particular, this implies that infinite sequences of finite
d
-regular Ramanujan–Schreier graphs have essentially large girth.
Miklós Abért. Yair Glasner. Bálint Virág. "Kesten’s theorem for invariant random subgroups." Duke Math. J. 163 (3) 465 - 488, 15 February 2014. https://doi.org/10.1215/00127094-2410064
Secondary: 05C81 , 35P20 , 53C24
Miklós Abért, Yair Glasner, Bálint Virág "Kesten’s theorem for invariant random subgroups," Duke Mathematical Journal, Duke Math. J. 163(3), 465-488, (15 February 2014)
|
AddConversionRule - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Skew Polynomials : OreTools : Converters Subpackage : AddConversionRule
OreTools[Converters]
FromOrePolyToLinearEquation
convert an OrePoly structure to the corresponding linear functional equation
FromLinearEquationToOrePoly
convert a linear functional equation to the corresponding OrePoly structure
AddConversionRule
add conversion rules to FromLinearEquationToOrePoly
FromOrePolyToPoly
convert an OrePoly structure to the corresponding polynomial
FromPolyToOrePoly
convert a polynomial to the corresponding OrePoly structure
FromOrePolyToLinearEquation (Ore, f, A)
FromLinearEquationToOrePoly (expr, f, A)
AddConversionRule (CaseName, ConvertProc)
FromOrePolyToPoly(Ore, x)
FromPolyToOrePoly(P, x)
name; dependent variable
expression; linear functional equation
name; labels the new conversion case
ConvertProc
procedure; algorithm for converting a linear functional equation to an Ore polynomial
name; noncommutative indeterminate
The FromOrePolyToLinearEquation(Ore, f, A) calling sequence converts the Ore polynomial Ore to the corresponding linear functional equation.
The FromLinearEquationToOrePoly(expr, f, A) calling sequence converts the linear functional equation expr to the corresponding Ore polynomial. The Ore polynomial is returned as an OrePoly structure.
The FromLinearEquationToOrePoly command currently supports only the shift and differential cases.
The AddConversionRule(CaseName, ConvertProc) command adds a new conversion rule from a linear equation to an Ore polynomial to be used by the FromLinearEquationToOrePoly command.
The ConvertProc procedure must accept three arguments: linear functional equation, name of dependent variable, and Ore algebra; and return an Ore polynomial as an OrePoly structure.
The FromOrePolyToPoly(Ore, x) calling sequence converts the Ore polynomial Ore to the corresponding polynomial.
The FromPolyToOrePoly(P, x) calling sequence converts the polynomial P to the corresponding Ore polynomial. The Ore polynomial is returned as an OrePoly structure.
\mathrm{with}\left(\mathrm{OreTools}\right):
\mathrm{with}\left(\mathrm{OreTools}[\mathrm{Converters}]\right):
A≔\mathrm{SetOreRing}\left(x,'\mathrm{differential}'\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{UnivariateOreRing}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{differential}}\right)
L≔\mathrm{OrePoly}\left(1,-x,{x}^{2}+Cx\right)
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{OrePoly}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)
\mathrm{eq}≔\mathrm{FromOrePolyToLinearEquation}\left(L,f,A\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)
\mathrm{FromLinearEquationToOrePoly}\left(\mathrm{eq},f,A\right)
\textcolor[rgb]{0,0,1}{\mathrm{OrePoly}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)
To convert a linear difference equation to an OrePoly structure, first define the difference polynomial ring.
A := SetOreRing(n, 'difference', 'sigma' = proc(p, x) eval(p, x=x+1) end,
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{UnivariateOreRing}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{difference}}\right)
Consider the following difference operator.
L≔\mathrm{OrePoly}\left(\frac{{n}^{4}}{n-1},{n}^{4}-{n}^{3},\frac{n-1}{n+2}\right)
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{OrePoly}}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{4}}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\right)
Convert L to a linear difference equation.
\mathrm{eq}≔\mathrm{FromOrePolyToLinearEquation}\left(L,s,A\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{3}}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)\right)\textcolor[rgb]{0,0,1}{+}\frac{\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)\right)}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}
The FromLinearEquationToOrePoly function cannot convert eq to the corresponding OrePoly structure.
\mathrm{FromLinearEquationToOrePoly}\left(\mathrm{eq},s,A\right)
Error, (in OreTools:-Converters:-FromLinearEquationToOrePoly) unable to handle the difference case
You must define the rule for the difference case.
difference_case := proc(eq, func, A)
local reqn, info, fcn, E, receq, req, func_set, n, place, i, var;
var := OreTools[Properties][GetVariable](A);
# Do some argument checking and processing
reqn := LREtools['REcreate'](eq, func(var), {});
info := eval(op(4, reqn));
if info['linear']=false then
error "only linear equations are handled"
n := info['vars'];
if not (nops([n])=1 and type(n, name)) then
error "only single equations are handled"
fcn := op(2, reqn)[1];
# locate n in arguments to fcn
for place to nops(fcn) while op(place, fcn) <> n do end do;
# Transform the equation from a difference equation to a
# difference operator
req := args[1];
func_set := indets(req, specfunc(anything, info['functions']));
receq := collect( subs( map( (x, y, z, i) -> x = (z+1)^(op(i, x)-y),
func_set, n, 'E', place), req), 'E', Normalizer);
'OrePoly'(seq(coeff(receq, 'E', i), i=0..degree(receq, 'E')))
Add this rule to OreTools, and then use the FromOrePolyToLinearEquation function again.
\mathrm{AddConversionRule}\left('\mathrm{difference}',\mathrm{difference_case}\right)
\mathrm{FromLinearEquationToOrePoly}\left(\mathrm{eq},s,A\right)
\textcolor[rgb]{0,0,1}{\mathrm{OrePoly}}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{4}}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\right)
\mathrm{Ore}≔\mathrm{OrePoly}\left(n,n-1,\frac{1}{n}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Ore}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{OrePoly}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}}\right)
q≔\mathrm{FromOrePolyToPoly}\left(\mathrm{Ore},S\right)
\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{S}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{n}}
\mathrm{FromPolyToOrePoly}\left(q,S\right)
\textcolor[rgb]{0,0,1}{\mathrm{OrePoly}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}}\right)
|
Pressure Welding of Thin Sheet Metals: Experimental Investigations and Analytical Modeling | J. Manuf. Sci. Eng. | ASME Digital Collection
Sasawat Mahabunphachai,
Sasawat Mahabunphachai
Department of Mechanical Engineering, NSFI/UCR Center for Precision Forming,
, Richmond, VA 23284; Department of Mechanical Engineering,
Muammer Koç,
, Richmond, VA 23284
Mahabunphachai, S., Koç, M., and Ni, J. (July 7, 2009). "Pressure Welding of Thin Sheet Metals: Experimental Investigations and Analytical Modeling." ASME. J. Manuf. Sci. Eng. August 2009; 131(4): 041003. https://doi.org/10.1115/1.3160597
Emerging applications, such as fuel cell, fuel processor, heat exchanger, microreactors, etc., require joining of thin metallic plates in confined places with small dimensions and minimal damage to the surrounding areas. In this study, the feasibility and modeling of pressure welding (solid state bonding) process are investigated, specifically for bonding of thin sheet metals. The effects of material type (e.g., copper, nickel, and stainless steel) and initial plate thickness
(51–254 μm)
as well as process conditions (e.g., welding pressure and temperature,
25–300°C
) on the minimum welding pressure and the final bond strength are experimentally studied. A pressure welding apparatus was developed for testing of different materials and process conditions. Based on the experimental results, the effects of material and process conditions on the final bond quality are characterized. At room temperature, copper and nickel blanks were successfully bonded, while stainless steel blanks could only be joined at elevated temperature levels (
150°C
300°C
). The material type (i.e., strength) and thickness were shown to have significant impact on the welding pressure; in that more pressure is required to bond the blanks with higher strength or thinner. To reduce the required welding pressure, the process can be carried out at elevated temperature levels. In this study, the bond strength of the welded blanks was characterized with uniaxial testing. The tensile test results showed that the bond strength could be increased by increasing the welding pressure or temperature. However, the increase in bond strength by increasing the welding pressure was shown to have an optimal point, after which the bond strength would decrease with further increase in pressure. This critical pressure value was found to be dependent on the material and process conditions. In addition, bond formation mechanisms for different materials were studied through microscopic analyses. The microscopy images of the weld spots showed that for a successful bonding to take place, the contaminant layers at the surfaces must be removed or broken to allow the virgin metal underneath to be extruded through. The metallic bonds only form at these locations where both surfaces are free of contaminant layers. Finally, a model for bond strength prediction in pressure welding was developed and validated. This model includes the sheet thickness parameter, which is shown to be a critical factor in bonding thin sheet metals with the sheet thickness in the range of a few hundred micrometers.
copper, nickel, plates (structures), sheet materials, stainless steel, tensile strength, tensile testing, welding, pressure welding, cold welding, solid state welding, microwelding, fuel cell plate
Bond strength, Pressure, Welding, Modeling, Sheet metal, Temperature, Blanks
Part 1 Cold Pressure Welding
Lehrheuer
Proceedings of the International Manufacturing Science and Engineering Conference (MSEC)
, Atlanta, GA, Oct. 15–17.
Wire Ind.
Lukaschkin
Interface Surface Behaviour in the Upsetting of Sandwich Metal Sheets
Interfacial Bonding State on Different Metals Ag, Ni in Cold Pressure Welding
Cold Welding: Part 1. Characteristics, Bonding Mechanisms, Bond Strength
Industrial Practice in Cold Pressure Welding
Br. Weld. J.
Pendrous
Cold Roll and Indent Welding of Some Metals
Met. Technol. (London)
Materials Joined by New Cold Welding Process
Weld. J. (Miami, FL, U.S.)
A Bonding Map for Cu and Al Plates by Pressure Welding at Cold and Warm Temperatures
Pressure Welding by Rolling
Interfacial Conditions and Bond Strength in Cold Pressure Welding by Rolling
Pressure Welding by Rolling at Elevated Temperatures
Madaah-Hosseini
Cold Roll Bonding of 5754-Aluminum Strips
Investigation of Size Effects on Material Behavior of Thin Sheet Metals Using Hydraulic Bulge Testing At Micro/Meso-Scales
Short-Time Tensile Properties of Type 316 Stainless Steel at Very High Temperatures
|
Section 100.6 (04YX): Higher diagonals—The Stacks project
Section 100.6: Higher diagonals (cite)
100.6 Higher diagonals
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. In this situation it makes sense to consider not only the diagonal
\[ \Delta _ f : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} \]
but also the diagonal of the diagonal, i.e., the morphism
\[ \Delta _{\Delta _ f} : \mathcal{X} \longrightarrow \mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})} \mathcal{X} \]
Because of this we sometimes use the following terminology. We denote $\Delta _{f, 0} = f$ the zeroth diagonal, we denote $\Delta _{f, 1} = \Delta _ f$ the first diagonal, and we denote $\Delta _{f, 2} = \Delta _{\Delta _ f}$ the second diagonal. Note that $\Delta _{f, 1}$ is representable by algebraic spaces and locally of finite type, see Lemma 100.3.3. Hence $\Delta _{f, 2}$ is representable, a monomorphism, locally of finite type, separated, and locally quasi-finite, see Lemma 100.3.4.
We can describe the second diagonal using the relative inertia stack. Namely, the fibre product $\mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})} \mathcal{X}$ is equivalent to the relative inertia stack $\mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ by Categories, Lemma 4.34.1. Moreover, via this identification the second diagonal becomes the neutral section
\[ \Delta _{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \]
of the relative inertia stack. By analogy with what happens for groupoids in algebraic spaces (Groupoids in Spaces, Lemma 77.29.2) we have the following equivalences.
Lemma 100.6.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
$\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is separated,
$\Delta _{f, 1} = \Delta _ f : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is separated, and
$\Delta _{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is a closed immersion.
$\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is quasi-separated,
$\Delta _{f, 1} = \Delta _ f : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is quasi-separated, and
$\Delta _{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is a quasi-compact.
$\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is locally separated,
$\Delta _{f, 1} = \Delta _ f : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is locally separated, and
$\Delta _{f, 2} = e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is an immersion.
$\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is unramified,
$f$ is DM.
$\mathcal{I}_{\mathcal{X}/\mathcal{Y}} \to \mathcal{X}$ is locally quasi-finite,
$f$ is quasi-DM.
Proof. Proof of (1), (2), and (3). Choose an algebraic space $U$ and a surjective smooth morphism $U \to \mathcal{X}$. Then $G = U \times _\mathcal {X} \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is an algebraic space over $U$ (Lemma 100.5.1). In fact, $G$ is a group algebraic space over $U$ by the group law on relative inertia constructed in Remark 100.5.2. Moreover, $G \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is surjective and smooth as a base change of $U \to \mathcal{X}$. Finally, the base change of $e : \mathcal{X} \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ by $G \to \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is the identity $U \to G$ of $G/U$. Thus the equivalence of (a) and (c) follows from Groupoids in Spaces, Lemma 77.6.1. Since $\Delta _{f, 2}$ is the diagonal of $\Delta _ f$ we have (b) $\Leftrightarrow $ (c) by definition.
Proof of (4) and (5). Recall that (4)(b) means $\Delta _ f$ is unramified and (5)(b) means that $\Delta _ f$ is locally quasi-finite. Choose a scheme $Z$ and a morphism $a : Z \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. Then $a = (x_1, x_2, \alpha )$ where $x_ i : Z \to \mathcal{X}$ and $\alpha : f \circ x_1 \to f \circ x_2$ is a $2$-morphism. Recall that
\[ \vcenter { \xymatrix{ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2) \ar[d] \ar[r] & Z \ar[d] \\ \mathcal{X} \ar[r]^{\Delta _ f} & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2) \ar[d] \ar[r] & Z \ar[d]^{x_2} \\ \mathcal{I}_{\mathcal{X}/\mathcal{Y}} \ar[r] & \mathcal{X} } } \]
are cartesian squares. By Lemma 100.5.4 the algebraic space $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2)$ is a pseudo torsor for $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ over $Z$. Thus the equivalences in (4) and (5) follow from Groupoids in Spaces, Lemma 77.9.5. $\square$
Lemma 100.6.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent:
the morphism $f$ is representable by algebraic spaces,
the second diagonal of $f$ is an isomorphism,
the group stack $ \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ is trivial over $\mathcal X$, and
for a scheme $T$ and a morphism $x : T \to \mathcal{X}$ the kernel of $\mathit{Isom}_\mathcal {X}(x, x) \to \mathit{Isom}_\mathcal {Y}(f(x), f(x))$ is trivial.
Proof. We first prove the equivalence of (1) and (2). Namely, $f$ is representable by algebraic spaces if and only if $f$ is faithful, see Algebraic Stacks, Lemma 93.15.2. On the other hand, $f$ is faithful if and only if for every object $x$ of $\mathcal{X}$ over a scheme $T$ the functor $f$ induces an injection $\mathit{Isom}_\mathcal {X}(x, x) \to \mathit{Isom}_\mathcal {Y}(f(x), f(x))$, which happens if and only if the kernel $K$ is trivial, which happens if and only if $e : T \to K$ is an isomorphism for every $x : T \to \mathcal{X}$. Since $K = T \times _{x, \mathcal{X}} \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ as discussed above, this proves the equivalence of (1) and (2). To prove the equivalence of (2) and (3), by the discussion above, it suffices to note that a group stack is trivial if and only if its identity section is an isomorphism. Finally, the equivalence of (3) and (4) follows from the definitions: in the proof of Lemma 100.5.1 we have seen that the kernel in (4) corresponds to the fibre product $T \times _{x, \mathcal{X}} \mathcal{I}_{\mathcal{X}/\mathcal{Y}}$ over $T$. $\square$
This lemma leads to the following hierarchy for morphisms of algebraic stacks.
Lemma 100.6.3. A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is
a monomorphism if and only if $\Delta _{f, 1}$ is an isomorphism, and
representable by algebraic spaces if and only if $\Delta _{f, 1}$ is a monomorphism.
Moreover, the second diagonal $\Delta _{f, 2}$ is always a monomorphism.
Proof. Recall from Properties of Stacks, Lemma 99.8.4 that a morphism of algebraic stacks is a monomorphism if and only if its diagonal is an isomorphism of stacks. Thus Lemma 100.6.2 can be rephrased as saying that a morphism is representable by algebraic spaces if the diagonal is a monomorphism. In particular, it shows that condition (3) of Lemma 100.3.4 is actually an if and only if, i.e., a morphism of algebraic stacks is representable by algebraic spaces if and only if its diagonal is a monomorphism. $\square$
Lemma 100.6.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then
$\Delta _{f, 1}$ separated $\Leftrightarrow $ $\Delta _{f, 2}$ closed immersion $\Leftrightarrow $ $\Delta _{f, 2}$ proper $\Leftrightarrow $ $\Delta _{f, 2}$ universally closed,
$\Delta _{f, 1}$ quasi-separated $\Leftrightarrow $ $\Delta _{f, 2}$ finite type $\Leftrightarrow $ $\Delta _{f, 2}$ quasi-compact, and
$\Delta _{f, 1}$ locally separated $\Leftrightarrow $ $\Delta _{f, 2}$ immersion.
Proof. Follows from Lemmas 100.3.5, 100.3.6, and 100.3.7 applied to $\Delta _{f, 1}$. $\square$
The following lemma is kind of cute and it may suggest a generalization of these conditions to higher algebraic stacks.
$f$ is separated if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are universally closed, and
$f$ is quasi-separated if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are quasi-compact.
$f$ is quasi-DM if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are locally quasi-finite.
$f$ is DM if and only if $\Delta _{f, 1}$ and $\Delta _{f, 2}$ are unramified.
Proof. Proof of (1). Assume that $\Delta _{f, 2}$ and $\Delta _{f, 1}$ are universally closed. Then $\Delta _{f, 1}$ is separated and universally closed by Lemma 100.6.4. By Morphisms of Spaces, Lemma 66.9.7 and Algebraic Stacks, Lemma 93.10.9 we see that $\Delta _{f, 1}$ is quasi-compact. Hence it is quasi-compact, separated, universally closed and locally of finite type (by Lemma 100.3.3) so proper. This proves “$\Leftarrow $” of (1). The proof of the implication in the other direction is omitted.
Proof of (2). This follows immediately from Lemma 100.6.4.
Proof of (3). This follows from the fact that $\Delta _{f, 2}$ is always locally quasi-finite by Lemma 100.3.4 applied to $\Delta _ f = \Delta _{f, 1}$.
Proof of (4). This follows from the fact that $\Delta _{f, 2}$ is always unramified as Lemma 100.3.4 applied to $\Delta _ f = \Delta _{f, 1}$ shows that $\Delta _{f, 2}$ is locally of finite type and a monomorphism. See More on Morphisms of Spaces, Lemma 75.14.8. $\square$
Lemma 100.6.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a separated (resp. quasi-separated, resp. quasi-DM, resp. DM) morphism of algebraic stacks. Then
given algebraic spaces $T_ i$, $i = 1, 2$ and morphisms $x_ i : T_ i \to \mathcal{X}$, with $y_ i = f \circ x_ i$ the morphism
\[ T_1 \times _{x_1, \mathcal{X}, x_2} T_2 \longrightarrow T_1 \times _{y_1, \mathcal{Y}, y_2} T_2 \]
is proper (resp. quasi-compact and quasi-separated, resp. locally quasi-finite, resp. unramified),
given an algebraic space $T$ and morphisms $x_ i : T \to \mathcal{X}$, $i = 1, 2$, with $y_ i = f \circ x_ i$ the morphism
\[ \mathit{Isom}_\mathcal {X}(x_1, x_2) \longrightarrow \mathit{Isom}_\mathcal {Y}(y_1, y_2) \]
is proper (resp. quasi-compact and quasi-separated, resp. locally quasi-finite, resp. unramified).
Proof. Proof of (1). Observe that the diagram
\[ \xymatrix{ T_1 \times _{x_1, \mathcal{X}, x_2} T_2 \ar[d] \ar[r] & T_1 \times _{y_1, \mathcal{Y}, y_2} T_2 \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } \]
is cartesian. Hence this follows from the fact that $f$ is separated (resp. quasi-separated, resp. quasi-DM, resp. DM) if and only if the diagonal is proper (resp. quasi-compact and quasi-separated, resp. locally quasi-finite, resp. unramified).
Proof of (2). This is true because
\[ \mathit{Isom}_\mathcal {X}(x_1, x_2) = (T \times _{x_1, \mathcal{X}, x_2} T) \times _{T \times T, \Delta _ T} T \]
hence the morphism in (2) is a base change of the morphism in (1). $\square$
Comment #1407 by Ariyan Javanpeykar on April 15, 2015 at 14:36
Would it be possible to split off (3) of Lemma 79.6.2? I would suggest finishing with (2) and adding an extra sentence "Moreover, the second diagonal [..] is always a monomorphism".
@#1407: Do you really think that this makes the statement easier to read? I thought it was fun how the numbering suggests a hierarchy of results...
Comment #1417 by jojo on April 16, 2015 at 14:19
I think Ariyan was trying to say that the 3) is not well phrased and doesn't belong in the list. I agree with him and his suggestion of correction.
Comment #1418 by Ariyan on April 16, 2015 at 15:27
I also think it is cool how the numbering suggests a hierarchy. To keep this hierarchy and make the Lemma well-phrased, would it be possble to add "and" after "is a monomorphism," in 79.6.2.(2)?
Honnestly I think your first suggestion was better Ariyan, I'm not sure if the second one would really make more sense than what is written right now. But maybe i'm missing something.
f:X\to Y
be a morphism of algebraic stacks (as considered throughout this tag.) Is the group stack
K\to X
defined as the kernel of
I_X\to I_Y
given any special attention elsewhere in the Stacks Project? It occurs in this Tag because the second diagonal
\Delta_{f,2}
can be identified with the identity section of
K\to X
. Would it be useful to record that the representability of
K\to X
being the trivial group (i.e., the second diagonal being an isomorphism as in Lemma 79.6.1)?
As requested in #1407 and #1417, I made the change as suggested. See here.
@#1431: Yes, we could add this to the first lemma. Feel free to send us an edited version of the chapter doing so.
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Talk:QB/a19ElectricPotentialField KE PE - Wikiversity
Talk:QB/a19ElectricPotentialField KE PE
{\displaystyle \Delta V_{AB}=V_{A}-V_{B}=-\int _{A}^{B}{\vec {E}}\cdot d{\vec {\ell }}}
{\displaystyle {\vec {E}}=-{\tfrac {\partial V}{\partial x}}{\hat {i}}-{\tfrac {\partial V}{\partial y}}{\hat {j}}-{\tfrac {\partial V}{\partial z}}{\hat {k}}=-{\vec {\nabla }}V}
{\displaystyle q\Delta V}
{\displaystyle U=qV}
{\displaystyle Power={\tfrac {\Delta U}{\Delta t}}={\tfrac {\Delta q}{\Delta t}}V=IV=e{\tfrac {\Delta N}{\Delta t}}}
{\displaystyle K={\tfrac {1}{2}}mv^{2}}
{\displaystyle V(r)=k{\tfrac {q}{r}}}
{\displaystyle V_{P}=k\sum _{1}^{N}{\frac {q_{i}}{r_{i}}}\to k\int {\frac {dq}{r}}}
{\displaystyle Q=CV}
{\displaystyle C=\varepsilon _{0}{\tfrac {A}{d}}}
{\displaystyle {\text{Series}}:\;{\tfrac {1}{C_{S}}}=\sum {\tfrac {1}{C_{i}}}.}
{\displaystyle {\text{ Parallel:}}\;C_{P}=\sum C_{i}.}
{\displaystyle u={\tfrac {1}{2}}QV={\tfrac {1}{2}}CV^{2}={\tfrac {1}{2C}}Q^{2}}
{\displaystyle u_{E}={\tfrac {1}{2}}\varepsilon _{0}E^{2}}
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Return to "QB/a19ElectricPotentialField KE PE" page.
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suman made a cardboard box in the form of a cube without a lid if the total surface area - Maths - Visualising Solid Shapes - 8741889 | Meritnation.com
Since the cardboard box is a cube.
Thus, its TSA = 6s2 where s stands for side of the cube.
Since, this cube is without lid so we remove the area of its lid, that is, s2 from its TSA.
Thus, the area to be considered is 5s2.
Also, we are given that surface area of the box without lid = 180 cm2
⇒5{s}^{2}=180\phantom{\rule{0ex}{0ex}}⇒{s}^{2}=36\phantom{\rule{0ex}{0ex}}⇒s=\sqrt{36}\phantom{\rule{0ex}{0ex}}⇒s=±6
We reject the negative value as length of the cubical box cannot be negative.
Thus, side of the box = 6 cm
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the deer population in a forest grows at 4% per annum if initially there were 45000 deers - Maths - Algebraic Expressions and Identities - 13345903 | Meritnation.com
the deer population in a forest grows at 4% per annum . if initially there were 45000 deers . find the number of deers after 2 years
Mamta Joshi answered this
it is given that there are 45000 deers
growth of dear per year =4%
so growth in first year = 4%of 45000 =
\frac{4}{100}×45000\phantom{\rule{0ex}{0ex}}4×450\phantom{\rule{0ex}{0ex}}1800
so total deer after first year= 45000+1800=46800
so growth after second year= 4%of 46800 = 1872
so total deer after second year =46800+1872=48672
Sankar S answered this
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Chordal graph - Wikipedia
(Redirected from Dirac's theorem on chordal graphs)
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs[1] or triangulated graphs.[2]
A cycle (black) with two chords (green). As for this part, the graph is chordal. However, removing one green edge would result in a non-chordal graph. Indeed, the other green edge with three black edges would form a cycle of length four with no chords.
Chordal graphs are a subset of the perfect graphs. They may be recognized in linear time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it.
1 Perfect elimination and efficient recognition
2 Maximal cliques and graph coloring
3 Minimal separators
4 Intersection graphs of subtrees
5 Relation to other graph classes
6 Chordal completions and treewidth
Perfect elimination and efficient recognitionEdit
A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering.[3]
Rose, Lueker & Tarjan (1976) (see also Habib et al. 2000) show that a perfect elimination ordering of a chordal graph may be found efficiently using an algorithm known as lexicographic breadth-first search. This algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices. The algorithm repeatedly chooses a vertex v from the earliest set in the sequence that contains previously unchosen vertices, and splits each set S of the sequence into two smaller subsets, the first consisting of the neighbors of v in S and the second consisting of the non-neighbors. When this splitting process has been performed for all vertices, the sequence of sets has one vertex per set, in the reverse of a perfect elimination ordering.
Since both this lexicographic breadth first search process and the process of testing whether an ordering is a perfect elimination ordering can be performed in linear time, it is possible to recognize chordal graphs in linear time. The graph sandwich problem on chordal graphs is NP-complete[4] whereas the probe graph problem on chordal graphs has polynomial-time complexity.[5]
The set of all perfect elimination orderings of a chordal graph can be modeled as the basic words of an antimatroid; Chandran et al. (2003) use this connection to antimatroids as part of an algorithm for efficiently listing all perfect elimination orderings of a given chordal graph.
Maximal cliques and graph coloringEdit
Another application of perfect elimination orderings is finding a maximum clique of a chordal graph in polynomial-time, while the same problem for general graphs is NP-complete. More generally, a chordal graph can have only linearly many maximal cliques, while non-chordal graphs may have exponentially many. To list all maximal cliques of a chordal graph, simply find a perfect elimination ordering, form a clique for each vertex v together with the neighbors of v that are later than v in the perfect elimination ordering, and test whether each of the resulting cliques is maximal.
The clique graphs of chordal graphs are the dually chordal graphs.[6]
The largest maximal clique is a maximum clique, and, as chordal graphs are perfect, the size of this clique equals the chromatic number of the chordal graph. Chordal graphs are perfectly orderable: an optimal coloring may be obtained by applying a greedy coloring algorithm to the vertices in the reverse of a perfect elimination ordering.[7]
The chromatic polynomial of a chordal graph is easy to compute. Find a perfect elimination ordering
{\displaystyle v_{1},v_{2},\ldots ,v_{n}.}
Let Ni equal the number of neighbors of vi that come after vi in that ordering. For instance, Nn = 0. The chromatic polynomial equals
{\displaystyle (x-N_{1})(x-N_{2})\cdots (x-N_{n}).}
(The last factor is simply x, so x divides the polynomial, as it should.) Clearly, this computation depends on chordality.[8]
Minimal separatorsEdit
In any graph, a vertex separator is a set of vertices the removal of which leaves the remaining graph disconnected; a separator is minimal if it has no proper subset that is also a separator. According to a theorem of Dirac (1961), chordal graphs are graphs in which each minimal separator is a clique; Dirac used this characterization to prove that chordal graphs are perfect.
The family of chordal graphs may be defined inductively as the graphs whose vertices can be divided into three nonempty subsets A, S, and B, such that A ∪ S and S ∪ B both form chordal induced subgraphs, S is a clique, and there are no edges from A to B. That is, they are the graphs that have a recursive decomposition by clique separators into smaller subgraphs. For this reason, chordal graphs have also sometimes been called decomposable graphs.[9]
Intersection graphs of subtreesEdit
A chordal graph with eight vertices, represented as the intersection graph of eight subtrees of a six-node tree.
An alternative characterization of chordal graphs, due to Gavril (1974), involves trees and their subtrees.
From a collection of subtrees of a tree, one can define a subtree graph, which is an intersection graph that has one vertex per subtree and an edge connecting any two subtrees that overlap in one or more nodes of the tree. Gavril showed that the subtree graphs are exactly the chordal graphs.
A representation of a chordal graph as an intersection of subtrees forms a tree decomposition of the graph, with treewidth equal to one less than the size of the largest clique in the graph; the tree decomposition of any graph G can be viewed in this way as a representation of G as a subgraph of a chordal graph. The tree decomposition of a graph is also the junction tree of the junction tree algorithm.
Relation to other graph classesEdit
Interval graphs are the intersection graphs of subtrees of path graphs, a special case of trees. Therefore, they are a subfamily of chordal graphs.
Split graphs are graphs that are both chordal and the complements of chordal graphs. Bender, Richmond & Wormald (1985) showed that, in the limit as n goes to infinity, the fraction of n-vertex chordal graphs that are split approaches one.
Ptolemaic graphs are graphs that are both chordal and distance hereditary. Quasi-threshold graphs are a subclass of Ptolemaic graphs that are both chordal and cographs. Block graphs are another subclass of Ptolemaic graphs in which every two maximal cliques have at most one vertex in common. A special type is windmill graphs, where the common vertex is the same for every pair of cliques.
Strongly chordal graphs are graphs that are chordal and contain no n-sun (for n ≥ 3) as an induced subgraph. Here an n-sun is an n-vertex chordal graph G together with a collection of n degree-two vertices, adjacent to the edges of a Hamiltonian cycle in G.
K-trees are chordal graphs in which all maximal cliques and all maximal clique separators have the same size.[10] Apollonian networks are chordal maximal planar graphs, or equivalently planar 3-trees.[10] Maximal outerplanar graphs are a subclass of 2-trees, and therefore are also chordal.
SuperclassesEdit
Chordal graphs are a subclass of the well known perfect graphs. Other superclasses of chordal graphs include weakly chordal graphs, cop-win graphs, odd-hole-free graphs, even-hole-free graphs, and Meyniel graphs. Chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see holes in graph theory).
Every chordal graph is a strangulated graph, a graph in which every peripheral cycle is a triangle, because peripheral cycles are a special case of induced cycles. Strangulated graphs are graphs that can be formed by clique-sums of chordal graphs and maximal planar graphs. Therefore, strangulated graphs include maximal planar graphs.[11]
Chordal completions and treewidthEdit
Main article: Chordal completion
If G is an arbitrary graph, a chordal completion of G (or minimum fill-in) is a chordal graph that contains G as a subgraph. The parameterized version of minimum fill-in is fixed parameter tractable, and moreover, is solvable in parameterized subexponential time.[12][13] The treewidth of G is one less than the number of vertices in a maximum clique of a chordal completion chosen to minimize this clique size. The k-trees are the graphs to which no additional edges can be added without increasing their treewidth to a number larger than k. Therefore, the k-trees are their own chordal completions, and form a subclass of the chordal graphs. Chordal completions can also be used to characterize several other related classes of graphs.[14]
^ Dirac (1961).
^ Berge (1967).
^ Fulkerson & Gross (1965).
^ Bodlaender, Fellows & Warnow (1992).
^ Berry, Golumbic & Lipshteyn (2007).
^ Szwarcfiter & Bornstein (1994).
^ Maffray (2003).
^ For instance, Agnarsson (2003), Remark 2.5, calls this method well known.
^ Peter Bartlett. "Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations" (PDF).
^ a b Patil (1986).
^ Seymour & Weaver (1984).
^ Kaplan, Shamir & Tarjan (1999).
^ Fomin & Villanger (2013).
^ Parra & Scheffler (1997).
Agnarsson, Geir (2003), "On chordal graphs and their chromatic polynomials", Mathematica Scandinavica, 93 (2): 240–246, doi:10.7146/math.scand.a-14421, MR 2009583 .
Bender, E. A.; Richmond, L. B.; Wormald, N. C. (1985), "Almost all chordal graphs split", J. Austral. Math. Soc., A, 38 (2): 214–221, doi:10.1017/S1446788700023077, MR 0770128 .
Berge, Claude (1967), "Some Classes of Perfect Graphs", in Harary, Frank (ed.), Graph Theory and Theoretical Physics, Academic Press, pp. 155–165, MR 0232694 .
Berry, Anne; Golumbic, Martin Charles; Lipshteyn, Marina (2007), "Recognizing chordal probe graphs and cycle-bicolorable graphs", SIAM Journal on Discrete Mathematics, 21 (3): 573–591, doi:10.1137/050637091 .
Bodlaender, H. L.; Fellows, M. R.; Warnow, T. J. (1992), "Two strikes against perfect phylogeny" (PDF), Proc. of 19th International Colloquium on Automata Languages and Programming, Lecture Notes in Computer Science, vol. 623, pp. 273–283, doi:10.1007/3-540-55719-9_80, hdl:1874/16653 .
Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), "Enumerating and characterizing the perfect elimination orderings of a chordal graph" (PDF), Theoretical Computer Science, 307 (2): 303–317, doi:10.1016/S0304-3975(03)00221-4 .
Dirac, G. A. (1961), "On rigid circuit graphs" (PDF), Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25 (1–2): 71–76, doi:10.1007/BF02992776, MR 0130190, S2CID 120608513 .
Fomin, Fedor V.; Villanger, Yngve (2013), "Subexponential Parameterized Algorithm for Minimum Fill-In", SIAM J. Comput., 42 (6): 2197–2216, arXiv:1104.2230, doi:10.1137/11085390X, S2CID 934546 .
Fulkerson, D. R.; Gross, O. A. (1965), "Incidence matrices and interval graphs", Pacific J. Math., 15 (3): 835–855, doi:10.2140/pjm.1965.15.835 .
Gavril, Fănică (1974), "The intersection graphs of subtrees in trees are exactly the chordal graphs", Journal of Combinatorial Theory, Series B, 16: 47–56, doi:10.1016/0095-8956(74)90094-X .
Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press .
Habib, Michel; McConnell, Ross; Paul, Christophe; Viennot, Laurent (2000), "Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition, and consecutive ones testing", Theoretical Computer Science, 234 (1–2): 59–84, doi:10.1016/S0304-3975(97)00241-7 .
Kaplan, Haim; Shamir, Ron; Tarjan, Robert (1999), "Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs", SIAM J. Comput., 28 (5): 1906–1922, doi:10.1137/S0097539796303044 .
Maffray, Frédéric (2003), "On the coloration of perfect graphs", in Reed, Bruce A.; Sales, Cláudia L. (eds.), Recent Advances in Algorithms and Combinatorics, CMS Books in Mathematics, vol. 11, Springer-Verlag, pp. 65–84, doi:10.1007/0-387-22444-0_3, ISBN 0-387-95434-1 .
Parra, Andreas; Scheffler, Petra (1997), "Characterizations and algorithmic applications of chordal graph embeddings", Discrete Applied Mathematics, 79 (1–3): 171–188, doi:10.1016/S0166-218X(97)00041-3, MR 1478250 .
Patil, H. P. (1986), "On the structure of k-trees", Journal of Combinatorics, Information and System Sciences, 11 (2–4): 57–64, MR 0966069 .
Rose, D.; Lueker, George; Tarjan, Robert E. (1976), "Algorithmic aspects of vertex elimination on graphs", SIAM Journal on Computing, 5 (2): 266–283, doi:10.1137/0205021, MR 0408312 .
Seymour, P. D.; Weaver, R. W. (1984), "A generalization of chordal graphs", Journal of Graph Theory, 8 (2): 241–251, doi:10.1002/jgt.3190080206, MR 0742878 .
Szwarcfiter, J.L.; Bornstein, C.F. (1994), "Clique graphs of chordal and path graphs", SIAM Journal on Discrete Mathematics, 7 (2): 331–336, doi:10.1137/s0895480191223191, hdl:11422/1497 .
Information System on Graph Class Inclusions: chordal graph
Weisstein, Eric W. "Chordal Graph". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Chordal_graph&oldid=1071172771#Minimal_separators"
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Dirac large numbers hypothesis - Wikipedia
The Dirac large numbers hypothesis (LNH) is an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. The ratios constitute very large, dimensionless numbers: some 40 orders of magnitude in the present cosmological epoch. According to Dirac's hypothesis, the apparent similarity of these ratios might not be a mere coincidence but instead could imply a cosmology with these unusual features:
The strength of gravity, as represented by the gravitational constant, is inversely proportional to the age of the universe:
{\displaystyle G\propto 1/t\,}
The mass of the universe is proportional to the square of the universe's age:
{\displaystyle M\propto t^{2}}
Physical constants are actually not constant. Their values depend on the age of the Universe.
2 Dirac's interpretation of the large number coincidences
3 Later developments and interpretations
3.1 Modern studies
LNH was Dirac's personal response to a set of large number 'coincidences' that had intrigued other theorists of his time. The 'coincidences' began with Hermann Weyl (1919),[1][2] who speculated that the observed radius of the universe, RU, might also be the hypothetical radius of a particle whose rest energy is equal to the gravitational self-energy of the electron:
{\displaystyle {\frac {R_{\text{U}}}{r_{\text{e}}}}\approx {\frac {r_{\text{H}}}{r_{\text{e}}}}\approx 10^{42},}
{\displaystyle r_{\text{e}}={\frac {e^{2}}{4\pi \epsilon _{0}m_{\text{e}}c^{2}}},}
{\displaystyle r_{\text{H}}={\frac {e^{2}}{4\pi \epsilon _{0}m_{\text{H}}c^{2}}},}
{\displaystyle m_{\text{H}}c^{2}={\frac {Gm_{\text{e}}^{2}}{r_{\text{e}}}}}
and re is the classical electron radius, me is the mass of the electron, mH denotes the mass of the hypothetical particle, and rH is its electrostatic radius.
The coincidence was further developed by Arthur Eddington (1931)[3] who related the above ratios to N, the estimated number of charged particles in the universe[clarification needed]:
{\displaystyle {\frac {e^{2}}{4\pi \epsilon _{0}Gm_{\text{e}}^{2}}}\approx {\sqrt {N}}\approx 10^{42}.}
In addition to the examples of Weyl and Eddington, Dirac was also influenced by the primeval-atom hypothesis of Georges Lemaître, who lectured on the topic in Cambridge in 1933. The notion of a varying-G cosmology first appears in the work of Edward Arthur Milne a few years before Dirac formulated LNH. Milne was inspired not by large number coincidences but by a dislike of Einstein's general theory of relativity.[4][5] For Milne, space was not a structured object but simply a system of reference in which relations such as this could accommodate Einstein's conclusions:
{\displaystyle G=\left({\frac {c^{3}}{M_{\text{U}}}}\right)t,}
where MU is the mass of the universe and t is the age of the universe. According to this relation, G increases over time.
Dirac's interpretation of the large number coincidencesEdit
The Weyl and Eddington ratios above can be rephrased in a variety of ways, as for instance in the context of time:
{\displaystyle {\frac {ct}{r_{\text{e}}}}\approx 10^{40},}
where t is the age of the universe,
{\displaystyle c}
is the speed of light and re is the classical electron radius. Hence, in units where c = 1 and re = 1, the age of the universe is about 1040 units of time. This is the same order of magnitude as the ratio of the electrical to the gravitational forces between a proton and an electron:
{\displaystyle {\frac {e^{2}}{4\pi \epsilon _{0}Gm_{\text{p}}m_{\text{e}}}}\approx 10^{40}.}
Hence, interpreting the charg{\displaystyle e}
of the electron, the masses
{\displaystyle m_{\text{p}}}
{\displaystyle m_{\text{e}}}
of the proton and electron, and the permittivity factor
{\displaystyle 4\pi \epsilon _{0}}
in atomic units (equal to 1), the value of the gravitational constant is approximately 10−40. Dirac interpreted this to mean that
{\displaystyle G}
varies with time as
{\displaystyle G\approx 1/t}
. Although George Gamow noted that such a temporal variation does not necessarily follow from Dirac's assumptions,[6] a corresponding change of G has not been found.[7] According to general relativity, however, G is constant, otherwise the law of conserved energy is violated. Dirac met this difficulty by introducing into the Einstein field equations a gauge function β that describes the structure of spacetime in terms of a ratio of gravitational and electromagnetic units. He also provided alternative scenarios for the continuous creation of matter, one of the other significant issues in LNH:
'additive' creation (new matter is created uniformly throughout space) and
'multiplicative' creation (new matter is created where there are already concentrations of mass).
Later developments and interpretationsEdit
Dirac's theory has inspired and continues to inspire a significant body of scientific literature in a variety of disciplines. In the context of geophysics, for instance, Edward Teller seemed to raise a serious objection to LNH in 1948[8] when he argued that variations in the strength of gravity are not consistent with paleontological data. However, George Gamow demonstrated in 1962[9] how a simple revision of the parameters (in this case, the age of the Solar System) can invalidate Teller's conclusions. The debate is further complicated by the choice of LNH cosmologies: In 1978, G. Blake[10] argued that paleontological data is consistent with the 'multiplicative' scenario but not the 'additive' scenario. Arguments both for and against LNH are also made from astrophysical considerations. For example, D. Falik[11] argued that LNH is inconsistent with experimental results for microwave background radiation whereas Canuto and Hsieh[12][13] argued that it is consistent. One argument that has created significant controversy was put forward by Robert Dicke in 1961. Known as the anthropic coincidence or fine-tuned universe, it simply states that the large numbers in LNH are a necessary coincidence for intelligent beings since they parametrize fusion of hydrogen in stars and hence carbon-based life would not arise otherwise.
Various authors have introduced new sets of numbers into the original 'coincidence' considered by Dirac and his contemporaries, thus broadening or even departing from Dirac's own conclusions. Jordan (1947)[14] noted that the mass ratio for a typical star (specifically, a star of the Chandrasekhar mass, itself a constant of nature, approx. 1.44 solar masses) and an electron approximates to 1060, an interesting variation on the 1040 and 1080 that are typically associated with Dirac and Eddington respectively. (The physics defining the Chandrasekhar mass produces a ratio that is the -3/2 power of the gravitational fine-structure constant, 10−40.)
Modern studiesEdit
Several authors have recently identified and pondered the significance of yet another large number, approximately 120 orders of magnitude. This is for example the ratio of the theoretical and observational estimates of the energy density of the vacuum, which Nottale (1993)[15] and Matthews (1997)[16] associated in an LNH context with a scaling law for the cosmological constant. Carl Friedrich von Weizsäcker identified 10120 with the ratio of the universe's volume to the volume of a typical nucleon bounded by its Compton wavelength, and he identified this ratio with the sum of elementary events or bits of information in the universe.[17] Valev (2019) [18] found equation connecting cosmological parameters (for example density of the universe) and Planck units (for example Planck density). This ratio of densities, and other ratios (using four fundamental constants - speed of light in vacuum c, Newtonian constant of gravity G, reduced Planck constant ℏ, and Hubble constant H) computes to an exact number, 32.8 x 10120. This provides evidence of the Dirac large numbers hypothesis by connecting the macro-world and the micro-world.
Time-variation of physical constants
^ H. Weyl (1917). "Zur Gravitationstheorie". Annalen der Physik (in German). 359 (18): 117–145. Bibcode:1917AnP...359..117W. doi:10.1002/andp.19173591804.
^ H. Weyl (1919). "Eine neue Erweiterung der Relativitätstheorie". Annalen der Physik. 364 (10): 101–133. Bibcode:1919AnP...364..101W. doi:10.1002/andp.19193641002.
^ A. Eddington (1931). "Preliminary Note on the Masses of the Electron, the Proton, and the Universe". Proceedings of the Cambridge Philosophical Society. 27 (1): 15–19. Bibcode:1931PCPS...27...15E. doi:10.1017/S0305004100009269.
^ E. A. Milne (1935). Relativity, Gravity and World Structure. Oxford University Press.
^ H. Kragh (1996). Cosmology and Controversy: The historical development of two theories of the universe. Princeton University Press. pp. 61–62. ISBN 978-0-691-02623-7.
^ H. Kragh (1990). Dirac: A Scientific Biography. Cambridge University Press. p. 177. ISBN 978-0-521-38089-8.
^ J. P.Uzan (2003). "The fundamental constants and their variation, Observational status and theoretical motivations". Reviews of Modern Physics. 75 (2): 403. arXiv:hep-ph/0205340. Bibcode:2003RvMP...75..403U. doi:10.1103/RevModPhys.75.403. S2CID 118684485.
^ E. Teller (1948). "On the change of physical constants". Physical Review. 73 (7): 801–802. Bibcode:1948PhRv...73..801T. doi:10.1103/PhysRev.73.801.
^ G. Gamow (1962). Gravity. Doubleday. pp. 138–141. LCCN 62008840.
^ G. Blake (1978). "The Large Numbers Hypothesis and the rotation of the Earth". Monthly Notices of the Royal Astronomical Society. 185 (2): 399–408. Bibcode:1978MNRAS.185..399B. doi:10.1093/mnras/185.2.399.
^ D. Falik (1979). "Primordial Nucleosynthesis and Dirac's Large Numbers Hypothesis". The Astrophysical Journal. 231: L1. Bibcode:1979ApJ...231L...1F. doi:10.1086/182993.
^ V. Canuto, S. Hsieh (1978). "The 3 K blackbody radiation, Dirac's Large Numbers Hypothesis, and scale-covariant cosmology". The Astrophysical Journal. 224: 302. Bibcode:1978ApJ...224..302C. doi:10.1086/156378.
^ V. Canuto, S. Hsieh (1980). "Primordial nucleosynthesis and Dirac's large numbers hypothesis". The Astrophysical Journal. 239: L91. Bibcode:1980ApJ...239L..91C. doi:10.1086/183299.
^ P. Jordan (1947). "Die Herkunft der Sterne". Astronomische Nachrichten. 275 (10–12): 191. Bibcode:1947dhds.book.....J. doi:10.1002/asna.19472751012.
^ L. Nottale. "Mach's Principle, Dirac's Large Numbers and the Cosmological Constant Problem" (PDF).
^ R. Matthews (1998). "Dirac's coincidences sixty years on". Astronomy & Geophysics. 39 (6): 19–20. doi:10.1093/astrog/39.6.6.19.
^ H. Lyre (2003). "C. F. Weizsäcker's Reconstruction of Physics: Yesterday, Today and Tomorrow". arXiv:quant-ph/0309183.
^ D. Valev (2019). "Evidence of Dirac large numbers hypothesis" (PDF). Proceedings of the Romanian Academy. 20 (+4): 361–368.
P. A. M. Dirac (1938). "A New Basis for Cosmology". Proceedings of the Royal Society of London A. 165 (921): 199–208. Bibcode:1938RSPSA.165..199D. doi:10.1098/rspa.1938.0053.
P. A. M. Dirac (1937). "The Cosmological Constants". Nature. 139 (3512): 323. Bibcode:1937Natur.139..323D. doi:10.1038/139323a0. S2CID 4106534.
P. A. M. Dirac (1974). "Cosmological Models and the Large Numbers Hypothesis". Proceedings of the Royal Society of London A. 338 (1615): 439–446. Bibcode:1974RSPSA.338..439D. doi:10.1098/rspa.1974.0095. S2CID 122802355.
G. A. Mena Marugan; S. Carneiro (2002). "Holography and the large number hypothesis". Physical Review D. 65 (8): 087303. arXiv:gr-qc/0111034. Bibcode:2002PhRvD..65h7303M. doi:10.1103/PhysRevD.65.087303. S2CID 119452710.
C.-G. Shao; J. Shen; B. Wang; R.-K. Su (2006). "Dirac Cosmology and the Acceleration of the Contemporary Universe". Classical and Quantum Gravity. 23 (11): 3707–3720. arXiv:gr-qc/0508030. Bibcode:2006CQGra..23.3707S. doi:10.1088/0264-9381/23/11/003. S2CID 119339090.
S. Ray; U. Mukhopadhyay; P. P. Ghosh (2007). "Large Number Hypothesis: A Review". arXiv:0705.1836 [gr-qc].
A. Unzicker (2009). "A Look at the Abandoned Contributions to Cosmology of Dirac, Sciama and Dicke". Annalen der Physik. 18 (1): 57–70. arXiv:0708.3518. Bibcode:2009AnP...521...57U. doi:10.1002/andp.200810335. S2CID 11248780.
Audio of Dirac talking about the large numbers hypothesis
Full transcript of Dirac's speech.
Robert Matthews: Dirac's coincidences sixty years on
The Mysterious Eddington–Dirac Number
Retrieved from "https://en.wikipedia.org/w/index.php?title=Dirac_large_numbers_hypothesis&oldid=1073090233"
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Introducing box-plate beam-to-column moment connections | JVE Journals
A. Shishegaran1 , S. Rahimi2 , H. Darabi3
1Structural Engineering, School of Civil Engineering, Science and Technology of Iran, Tehran, Iran
2, 3Structural Engineering, School of Civil Engineering, Azad University of Noor, Noor, Iran
Nowadays, using high-ductility structures in the construction and use of significant buildings is highly appreciated. To use more ductile structures, effort has been made in this research to introduce box-plate beam-to-column connections. They underwent hysteretic loading and it was found from their moment-rotation curves that the bending capacity and ductility of the box-plate connection were more than ordinary rigid connection, and those of the latter were more than those of the normal typical one. It was also shown that stress concentration in box-plate connections disappears over the top and bottom flange plates.
Keywords: top/bottom flange plate rigid connection, box-plate rigid connection, ductility, stress concentration, connection energy dissipation.
Before North Ridge Earthquake in 1994, structures that had lateral resisting systems with rigid frames were considered as the most ductile and most earthquake resisting compared with other similar systems, but the “Earthquake” resulted in much damage in steel rigid structures that had welded beam-to-column connections. Numerous studies were done and showed that the reasons for the damage were different brittle fractures in the welded connections; such fractures prevent the connections’ inelastic behavior and hence reduce the structure ductility [1].
Chen et al. (2003) studied the longitudinal toothed connection and its bending capacity through laboratory tests and simulation methods and showed that it prevented brittle fractures (in the beam penetration part) and reduced stress concentration at the connection end and hence prevented the flange failure. They showed in their study that local stress concentration was reduced, plastic hinge was formed at a good distance from the column, the system showed appropriate ductility (without any brittle fractures) [2].
Tehranizadeh et al. (2012) studied the effects of the upper and lower plates on the bending rigidity of the restrained joints using three real-size beam-to-column connections all tested under hysteretic loading. The results, validated through simulations in the ABAQUS Software, revealed that the deformations were nearly similar in all three models. Accordingly, they suggested that it would be beneficial if the flange plates had the shortest lengths, welding materials were high-strength (resistant), and the upper and lower plates’ thicknesses for welding were the highest allowable [3].
In the present research, one novel rigid connection is introduced and compared, for their ductility and moment capacity, with typical restrained connections through their related moment-rotation diagrams drawn under similar regular dimensions, loading, and boundary conditions. [4, 5].
In this research, the typical and novel connections have been studied by the FEM under ultimate and hysteretic loading and their performances have been compared through their moment-rotation diagrams. This research uses weld and beam-to-column sections and relates them together with ties and friction interaction to do modeling in the ABAQUS by virtue of the papers validated by laboratory methods and the mentioned software [5, 6, 7, and 8].
For a better study of the finite element model and the modeling verification, first the beam-to-column connection investigated by Tehranizadeh et al. was modeled in the ABAQUS and then its finite element analyses results were compared with those of the laboratory tests. In short, the dimensions of the beam-column plates, connections, welds and specifications of the plates (Table 1), were used in the ABAQUS finite element software as inputs. The simulation models of the ordinary typical and box-plate connections are shown in Figs. 1 and 2 respectively [7].
Table 1. Dimensions of the plates used in the FE modeling and specifications of the plates used in the finite element model [3]
{\epsilon }_{y}
Stress (MPa)
{\sigma }_{y}
{\epsilon }_{u}
{\sigma }_{u}
Col. Continuity
Box-plate num1
Fig. 1. Simulation of an ordinary typical connection and boundary conditions in the ABAQUS Software [7]
Fig. 2. Simulation of the box-plate connection in the ABAQUS Software [7]
Fig. 3 shows Three different typical curves of Moment-rotation for fully restrained (FR), partially restrained (PR) and simple (S) connections.
Fig. 3. Three different typical curves of moment-rotation for fully restrained (FR), partially restrained (PR) and simple (S) connections [9, 10]
Fig. 4 shows Protocol load in this research. Fig. 5 shows their moment-rotation diagrams and Fig. 8 shows pushover curves and the bending capacity of model. Fig. 10 shows energy dissipation of the ordinary and four box-plate connections under the application of 22 similar loading cycles (of the displacement type), and. Also, connections with no brittle failure have rotated more than 4 radians. It is clears that Tangent of graph shows stiffness for models [7, 9-11].
Fig. 4. Displacement-time analyses of all the models [7]
Fig. 5. Moment-rotation curves of five models under 22 loading cycles [7]
Fig. 6 shows the stress at the end of the 22nd loading cycle in the ordinary and box-plate models respectively [7].
Fig. 7 shows the plastic strain in the 22nd loading cycle in the ordinary and box-plate connections respectively. In the first model (ordinary), the plastic strain at the junction of the upper and lower plates is 43.33 and 6.6 times those in the third models respectively [7].
Fig. 6. Triaxial stress in five models: a) ordinary model, b) box plate connection thickness = 20 mm [7]
Fig. 7. Plastic strain in five models: a) ordinary model, b) box plate connection thickness = 20 mm [7]
Fig. 8 shows pushover curves of five models under 22 loading cycles. The integral of this graph is cumulative energy dissipation. It is clears that Tangent of graph pushover curve is stiffness for model. Fig. 9 shows limits of stiffness for simple connection and rigid connection. Energy Dissipations for models is calculated as integral of graph Fig. 8. The results this calculation shows in Fig. 10. [7, 9 and 10]
Fig. 8. Pushover curves of five models under 22 loading cycles
Fig. 9. Two different typical curves of moment-rotation for fully restrained (FR), partially restrained (PR) and simple (S) connections [9, 10]
Fig. 10. Cumulative energy dissipation of five models under 22 loading cycles
The modeling of the ordinary and box-plate connections under hysteretic loading showed that:
1) They all rotated more than 4 radians in the 22nd loading cycle meaning that according to the AISC Seismic Code they can be used in the Special Moment Frame systems.
2) The bending moment capacity of the box-plate model is 10.6 % more than that of the Ordinary model.
3) Energy dissipation in the box-plate model is 65 % more than that in the ordinary model.
4) In the box-plate model, the stress concentration in the upper and lower plates is less than those of the ordinary model and its value at the web center is 1.35 times that in the upper and lower plates.
5) Tangent Box plate connection shows that this model is semi-rigid connection.
Mirghaderi S. R., Dehghani R. Continuous beam-to-column rigid seismic connection. Journal of Constructional Steel Research, 2008, p. 1516-1529. [Search CrossRef]
Chen Cheng-Chi, Lee Jen-Ming, Lin Ming-Chi Behavior of Steel Moment Connections with a Single Flange Rib. Department of Civil Engineering, National Chiao Tung University, Taiwan, 2003, p. 1419-1428. [Search CrossRef]
Gholami M., Deylami A., Tehranizadeh M. Seismic performance of flange plate connections between steel beams and box columns. Journal of Constructional Steel Research, 2012, p. 38-48. [Search CrossRef]
Wang Wei, Zhang Yanyan, Chen Yiyi, Lu Zhihao Enhancement of ductility of steel moment connections with non-compact beam web. Journal of Constructional Steel Research, 2012, p. 114-123. [Search CrossRef]
Azhari M., Mirghaderi R. Design of Steel Structures. Based on the AISC Code, First Editing, Vol. 1, 2005. [Search CrossRef]
ABAQUS Ver13.4. [Search CrossRef]
Kaufmann E. J. Dynamic Tension Tests of Simulated Moment Resisting Frame Weld Joints, Steel Tips, Structural Steel Education Council, AISC, Chicago, 1997. [Search CrossRef]
Specification for Structural Steel Buildings. AISC 360, American Institute of Steel Construction, Chicago, Illinois, 2005. [Search CrossRef]
State of the Art Repot on Connection Performance. Prepared by the SAC Joint Venture for the Federal Emergency Management Agency, FEMA-355D, Washington, DC, 2000. [Search CrossRef]
An innovative model for predicting the displacement and rotation of column-tree moment connection under fire
Mohammad Ali Naghsh, Aydin Shishegaran, Behnam Karami, Timon Rabczuk, Arshia Shishegaran, Hamed Taghavizadeh, Mehdi Moradi
Measurement of temperature and displacement with NiTi actuators under certain electrical conditions
Ersin Toptas, Mehmet Fatih Celebi, Sezgin Ersoy
Aydin Shishegaran, Mehdi Moradi, Mohammad Ali Naghsh, Behnam Karami, Arshia Shishegaran
Evaluation of a developed bypass viscous damper performance
Mahrad Fahiminia, Aydin Shishegaran
Computational predictions for estimating the maximum deflection of reinforced concrete panels subjected to the blast load
Aydin Shishegaran, Mohammad Reza Khalili, Behnam Karami, Timon Rabczuk, Arshia Shishegaran
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Compute track probabilities using the CGH algorithm - MATLAB toccgh - MathWorks Australia
toccgh
Tracker Operating Characteristic Curves
Compute Track Probabilities
Custom Gate Growth Sequence
Varying False-Alarm Probabilities
GateGrowthSequence
Common Gate History Algorithm
Compute track probabilities using the CGH algorithm
[pdt,pft,eft] = toccgh(pd,pfa)
[pdt,pft,eft] = toccgh(pd,pfa,Name,Value)
toccgh(___)
[pdt,pft,eft] = toccgh(pd,pfa) computes track probabilities using the Common Gate History Algorithm. The algorithm uses a 2-out-of-3 track confirmation logic, where 2 hits must be observed in 3 updates for a track to be confirmed.
[pdt,pft,eft] = toccgh(pd,pfa,Name,Value) specifies additional options using name-value arguments. Options include the confirmation logic, the gate size in bins, and the gate growth sequence.
toccgh(___) with no output arguments plots the tracker operating characteristic (TOC), which is the probability of target track, pdt, as a function of the probability of false track, pft.
The tracker operating characteristic (TOC) curve is a plot of the probability of a target track as a function of the probability of a false track. Plot the TOC curves for three different values of signal-to-noise ratio (SNR) assuming a 2/3 confirmation logic and use a one-dimensional constant-velocity Kalman filter to generate the tracker gate growth sequence.
Compute the probability of detection and the probability of false alarm for SNR values of 3, 6, and 9 dB. Assume a coherent receiver with a nonfluctuating target. Generate 20 probability-of-false-alarm values logarithmically equally spaced between
{10}^{-10}
{10}^{-3}
and calculate the corresponding probabilities of detection.
SNRdB = [3 6 9];
[pd,pfa] = rocsnr(SNRdB,'SignalType','NonfluctuatingCoherent', ...
'NumPoints',20,'MaxPfa',1e-3);
Compute and plot the TOC curves and the corresponding receiver operating characteristic (ROC) curves.
toccgh(pd,pfa)
Compute the probability of target track, the probability of false track, and the expected number of false tracks corresponding to a probability of detection of 0.9, a probability of false alarm of
{10}^{-6}
, and a 3-of-5 track confirmation logic.
logic = [3 5];
Use a modified version of the default one-dimensional constant-velocity Kalman filter to generate the tracker gate growth sequence. Specify an update time of 0.3 second and a maximum target acceleration of 20 meters per square second.
KFpars = {'UpdateTime',0.3,'MaxAcceleration',20};
Compute the probabilities and the expected number of false tracks.
[pdf,pft,eft] = toccgh(pd,pfa,'ConfirmationThreshold',logic,KFpars{:})
pdf = 0.9963
pft = 2.1555e-19
eft = 1
Use the common gate history algorithm to compute the probability of target track and the probability of track for a probability of detection of 0.5 and a probability of false alarm of
{10}^{-3}
. Use a custom gate growth sequence and a confirmation threshold of 3/4.
cp = [3 4];
gs = [21 39 95 125];
Compute the probabilities.
[pdf,pft] = toccgh(pd,pfa,'ConfirmationThreshold',cp, ...
'GateGrowthSequence',gs)
Investigate how receiver operating characteristic (ROC) and tracker operating characteristic (TOC) curves change with the probability of false alarm.
Compute probability-of-detection and signal-to-noise-ratio (SNR) values corresponding to probabilities of false alarm of
{10}^{-4}
{10}^{-6}
. Assume a coherent receiver with a nonfluctuating target. Plot the resulting ROC curves. Use larger markers to denote a larger SNR value.
pfa = [1e-4 1e-6];
[pd,SNRdB] = rocpfa(pfa,'SignalType','NonfluctuatingCoherent');
scatter(SNRdB,pd,max(SNRdB,1),'filled')
title('Receiver Operating Characteristic (ROC)')
ylabel('P_d')
title(legend('10^{-6}','10^{-4}'),'P_{fa}')
Compute the TOC curves using the probabilities of detection and probabilities of false alarm that you obtained. As the SNR increases, the probability of a false track in the presence of target detection increases. As the SNR decreases, the probability of target detection decreases, thereby increasing the probability of a false track.
[pct,pcf] = toccgh(pd.',pfa);
scatter(pcf,pct,max(SNRdB,1),'filled')
title('Tracker Operating Characteristic (TOC)')
xlabel('P_{FT}')
ylabel('P_{DT}')
Probability of detection, specified as a vector or a matrix of values in the range [0, 1].
If pd is a vector, then it must have the same number of elements as pfa
If pd is a matrix, then its number of rows must equal the number of elements of pfa. In that case, the number of columns of pd equals the length of the signal-to-noise (SNR) ratio input to rocsnr or output by rocpfa.
If you use rocpfa to obtain pd, you must transpose the output before using it as input to toccgh. If you use rocsnr to obtain pd, you do not have to transpose the output.
Example: [pd,pfa] = rocsnr(6) returns single-pulse detection probabilities and false-alarm probabilities for a coherent receiver with a nonfluctuating target and a signal-to-noise ratio of 6 dB.
Probability of false alarm per cell (bin), specified as a vector of values in the range [0, 1].
Use pfa values of 10–3 or smaller to satisfy the assumptions of the common gate history algorithm.
Example: 'UpdateTime',0.25,'MaximumAcceleration',8 specifies that the 1-D constant-velocity track Kalman filter used to compute the track gate growth has an update time of 0.25 second and a maximum acceleration of targets of interest of 8 meters per square second.
[2 3] (default) | two-element row vector of positive integers | positive integer scalar
Confirmation threshold, specified as a two-element row vector of positive integers or a scalar. The two-element vector [M N] corresponds to an M-out-of-N or M/N confirmation logic, a test that stipulates that an event must occur at least M times in N consecutive updates.
A track is confirmed if there are at least M detections in N updates.
A track is deleted if there are less than M detections in N updates.
If this argument is specified as a scalar, toccgh treats it as a two-element vector with identical elements. N cannot be larger than 50.
NumCells — Number of cells
16384 (default) | positive integer scalar
Number of cells, specified as a positive integer scalar. Use this argument to compute the expected number of false tracks.
NumTargets — Number of targets
Number of targets, specified as a positive integer scalar. Use this argument to compute the expected number of false tracks.
UpdateTime — Update time for Kalman filter
Update time for the default one-dimensional constant-velocity Kalman filter, specified as a positive scalar in seconds. This argument impacts the track gate growth.
MaxAcceleration — Maximum acceleration of targets of interest
10 (default) | nonnegative scalar in meters per square second
Maximum acceleration of targets of interest, specified as a nonnegative scalar in meters per square second. Use this input to tune the process noise in the default one-dimensional constant-velocity Kalman filter. This argument impacts the track gate growth.
Resolution — Range and range-rate resolution
[1 1] (default) | two-element row vector of positive values
Range and range-rate resolution, specified as a two-element row vector of positive values. The first element of 'Resolution' is the range resolution in meters. The second element of 'Resolution' is the range rate resolution in meters per second. This argument is used to convert the predicted tracker gate size to bins.
GateGrowthSequence — Tracker gate growth sequence
Tracker gate growth sequence, specified as a vector of positive integers. The values in the vector represent gate sizes in bins corresponding to N possible misses in N updates, where N is specified using 'ConfirmationThreshold'. If 'ConfirmationThreshold' is a two-element vector, then N is the second element of the vector.
If this argument is not specified, toccgh generates the tracker gate growth sequence using a one-dimensional constant-velocity Kalman filter implemented as a trackingKF object with these settings:
Update time — 0.5 second
Maximum target acceleration — 10 meters per square second
Range resolution — 1 meter
Range rate resolution — 1 meter per second
StateTransitionModel — [1 dt; 0 1], where dt is the update time
StateCovariance — [0 0; 0 0], which means the initial state is known perfectly
MeasurementNoise — 0
ProcessNoise — [dt^4/4 dt^3/2; dt^3/2 dt^2]*q, where dt is the update time, the tuning parameter q is amax^2*dt, and amax is the maximum acceleration. The tuning parameter is given in Equation 1.5.2-5 of [2].
To compute the gate sizes, the algorithm:
Uses the predict function to compute the predicted state error covariance matrix.
Calculates the area of the error ellipse as π times the product of the square roots of the eigenvalues of the covariance matrix.
Divides the area of the error ellipse by the bin area to express the gate size in bins. The bin area is the product of the range resolution and the range rate resolution.
If this argument is specified, then the 'UpdateTime', 'MaxAcceleration', and 'Resolution' arguments are ignored.
Example: [21 39 95 125 155 259 301] specifies a tracker grate growth sequence that occurs on some radar applications.
pdt — Probability of true target track in presence of false alarms
Probability of true target track in the presence of false alarms, returned as a matrix. pdt has the same size as pd.
pft — Probability of false track in presence of targets
Probability of false alarm track in the presence of targets, returned as a matrix. pft has the same size as pd.
eft — Expected number of false tracks
Expected number of false tracks, returned as a matrix of the same size as pd. toccgh computes the expected number of tracks using
{E}_{\text{ft}}={P}_{\text{ft,nt}}{N}_{\text{c}}+{P}_{\text{ft}}{N}_{\text{t}},
where Pft,nt is the probability of false track in the absence of targets, Nc is the number of resolution cells specified in 'NumCells', Pft is the probability of false track in the presence of targets, and Nt is the number of targets specified in 'NumTargets'.
The common gate history (CGH) algorithm was developed by Bar-Shalom and collaborators and published in [1]. For more information about the CGH algorithm, see Assessing Performance with the Tracker Operating Characteristic.
The algorithm proceeds under these assumptions:
A track is one of these:
Detections from targets only
Detections from targets and from false alarms
The probability of more than one false alarm in a gate is low, which is true when the probability of false alarm Pfa is low (Pfa ≤ 10–3).
The location of a target in a gate obeys a uniform spatial distribution.
The algorithm sequentially generates the gate history vector ω = [ωl, ωlt, λ], where:
ωl is the number of time steps since the last detection, either of a target or of a false alarm.
ωlt is the number of time steps since the last detection of a target.
λ is the number of detections.
The state vector evolves as a Markov chain by means of these steps:
The algorithm initially creates a track. Only two events can initialize a track:
A target detection
There are only four types of events that continue a track:
A1 — No detection
Events of Type 1 occur with probability
P\left\{{A}_{1}\right\}=\left(1-\frac{g\left({\omega }_{l}\right)}{g\left({\omega }_{lt}\right)}{P}_{\text{d}}\right){\left(1-{P}_{\text{fa}}\right)}^{g\left({\omega }_{l}\right)}
where Pd is the probability of detection specified using pd, Pfa is the probability of false alarm specified using pfa, g(ωl) is the gate size at step ωl, and g(ωlt) is the gate size at step ωlt.
To reduce Pd to a lower effective value, toccgh weights it with the ratio
\frac{g\left({\omega }_{l}\right)}{g\left({\omega }_{lt}\right)}=\frac{\text{Actual gate size}}{\text{Size of gate taking into account the time elapsed since the last target detection}},
which assumes a uniform spatial distribution of the location of a target in a gate. The gate sizes are specified using 'GateGrowthSequence'.
Events of Type 1 update the gate history vector as [ωl, ωlt, λ] ➔ [ωl + 1, ωlt + 1, λ].
A2 — Target detection
P\left\{{A}_{2}\right\}=\frac{g\left({\omega }_{l}\right)}{g\left({\omega }_{lt}\right)}{P}_{\text{d}}{\left(1-{P}_{\text{fa}}\right)}^{g\left({\omega }_{l}\right)}
and update the gate history vector as [ωl, ωlt, λ] ➔ [1, 1, λ + 1].
A3 — False alarm
P\left\{{A}_{3}\right\}=\left(1-{\left(1-{P}_{\text{fa}}\right)}^{g\left({\omega }_{l}\right)}\right)\left(1-\frac{g\left({\omega }_{l}\right)}{g\left({\omega }_{lt}\right)}{P}_{\text{d}}\right)
and update the gate history vector as [ωl, ωlt, λ] ➔ [1, ωlt + 1, λ + 1].
A4 — Target detection and false alarm
P\left\{{A}_{4}\right\}=\left(1-{\left(1-{P}_{\text{fa}}\right)}^{g\left({\omega }_{l}\right)}\right)\left(\frac{g\left({\omega }_{l}\right)}{g\left({\omega }_{lt}\right)}{P}_{\text{d}}\right)
and cause the track to split into a false track and a true track:
As,2a — Continue with A3, updating [ωl, ωlt, λ] ➔ [1, ωlt + 1, λ + 1].
As,2b — Continue with A2, updating [ωl, ωlt, λ] ➔ [1, 1, λ + 1].
At each step, the algorithm multiplies each track probability by the probability of the event that continues the track.
The procedure then lumps together the tracks that have a common gate history vector ω by adding their probabilities:
Tracks continued with A4 are lumped with tracks that continue with A3 (one false alarm only).
Tracks continued with A4 are lumped with tracks that continue with A2 (target detection only).
This step controls the number of track states within the Markov chain.
At the end, the algorithm computes and assigns the final probabilities:
A target track is a sequence of detections that satisfies the M/N confirmation logic and contains at least one detection from a target. To compute the probability of target track:
Determine the sequences that satisfy the confirmation logic under the assumption As,2b that A4 yields A2.
Separately store these probabilities.
To compute the probability of false track:
Compute the probability of target track under the assumption As,2a that A4 yields A3.
Subtract this probability from the probability of all detection sequences that satisfy the confirmation logic.
[1] Bar‐Shalom, Yaakov, Leon J. Campo, and Peter B. Luh. "From Receiver Operating Characteristic to System Operating Characteristic: Evaluation of a Track Formation System." IEEE® Transactions on Automatic Control 35, no. 2 (February 1990): 172–79. https://doi.org/10.1109/9.45173.
[2] Bar-Shalom, Yaakov, Peter K. Willett, and Xin Tian. Tracking and Data Fusion: A Handbook of Algorithms. Storrs, CT: YBS Publishing, 2011.
rocpfa | rocsnr
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Not to be confused with Bhāskara I.
Find sources: "Bhāskara II" – news · newspapers · books · scholar · JSTOR (November 2015) (Learn how and when to remove this template message)
c. 1114 AD
Vijjadavida, Maharashtra
(Identified as Patan near Chalisgaon in present-day Khandesh[1] or Jalgaon[2][3][4]in present-day Utter Maharashtra)
Algebra, Calculus, Arithmetic, Trigonometry
Siddhānta Shiromani (Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya), Karaṇa-Kautūhala
Bhaskara's proof of the Pythagorean Theorem.
1 Date, place and family
2 The Siddhānta-Śiromani
2.1 Līlāvatī
2.2 Bijaganita
2.3 Grahaganita
6.1 "Behold!"
Date, place and familyEdit
The Siddhānta-ŚiromaniEdit
LīlāvatīEdit
BijaganitaEdit
The second section Bījagaṇita(Algebra) has 213 verses.[12] It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method.[12] In particular, he also solved the
{\displaystyle 61x^{2}+1=y^{2}}
case that was to elude Fermat and his European contemporaries centuries later.[12]
GrahaganitaEdit
{\displaystyle \sin y'-\sin y\approx (y'-y)\cos y}
{\displaystyle y'}
{\displaystyle y}
, or in modern notation:[18]
{\displaystyle {\frac {d}{dy}}\sin y=\cos y}
In his words:[18]
A proof of the Pythagorean theorem by calculating the same area in two different ways and then cancelling out terms to get a2 + b2 = c2.[19]
In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.[20]
Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century.
The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhaskara II.[21]
Solutions of Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.[20]
Solved quadratic equations with more than one unknown, and found negative and irrational solutions.[citation needed]
Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.[22]
Conceived differential calculus, after discovering an approximation of the derivative and differential coefficient.
Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.
In Siddhanta-Śiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)
Indeterminate equations (Kuṭṭaka), integer solutions (first and second order). His contributions to this topic are particularly important,[citation needed] since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.
The 'unknown' (includes determining unknown quantities).
Kuṭṭaka (for solving indeterminate equations and Diophantine equations).
TrigonometryEdit
The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for
{\displaystyle \sin \left(a+b\right)}
{\displaystyle \sin \left(a-b\right)}
There is evidence of an early form of Rolle's theorem in his work. The modern formulation of Rolle's theorem states that if
{\displaystyle f\left(a\right)=f\left(b\right)=0}
{\displaystyle f'\left(x\right)=0}
{\displaystyle x}
{\displaystyle \ a<x<b}
He gave the result that if
{\displaystyle x\approx y}
{\displaystyle \sin(y)-\sin(x)\approx (y-x)\cos(y)}
, thereby finding the derivative of sine, although he never developed the notion of derivatives.[25]
He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.[citation needed] In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.
The three problems of diurnal rotation.(Diurnal motion is an astronomical term referring to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle.)
The Moon's crescent.
Ellipse calculations.[citation needed]
"Behold!"Edit
A number of institutes and colleges in India are named after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College of Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications and Geo-Informatics in Gandhinagar. He was a great mathematician.
^ Indian Journal of History of Science, Volume 35, National Institute of Sciences of India, 2000, p. 77
^ M. S. Mate, G. T. Kulkarni (1974), Studies in Indology and Medieval History: Prof. G. H. Khare Felicitation Volume, Joshi & Lokhande Prakashan, pp. 42-44
^ Proceedings, Indian History Congress, Volume 40, Indian History Congress, 1979, p. 71
^ Indian History and Epigraphy: Dr. G.S. Gai Felicitation Volume, Agam Kala Prakashan, 1990, p. 119
^ गणिती (Marathi term meaning Mathematicians) by Achyut Godbole and Dr. Thakurdesai, Manovikas, First Edition 23, December 2013. p. 34.
^ Mathematics in India by Kim Plofker, Princeton University Press, 2009, p. 182
^ Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara by Henry Colebrooke, Scholiasts of Bhascara p., xxvii
^ Sahni 2019, p. 50.
^ Chopra 1982, pp. 52–54.
^ Poulose 1991, p. 79.
^ a b c d e f g h i j k l m n S. Balachandra Rao (13 July 2014), ನವ ಜನ್ಮಶತಾಬ್ದಿಯ ಗಣಿತರ್ಷಿ ಭಾಸ್ಕರಾಚಾರ್ಯ, Vijayavani, p. 17 [unreliable source?]
^ Seal 1915, p. 80.
^ Goonatilake 1999, p. 134.
^ The Illustrated Weekly of India, Volume 95. Bennett, Coleman & Company, Limited, at the Times of India Press. 1974. p. 30. Deshasthas have contributed to mathematics and literature as well as to the cultural and religious heritage of India. Bhaskaracharaya was one of the greatest mathematicians of ancient India.
^ a b Pingree 1970, p. 299.
^ a b c d e Scientist (13 July 2014), ನವ ಜನ್ಮಶತಾಬ್ದಿಯ ಗಣಿತರ್ಷಿ ಭಾಸ್ಕರಾಚಾರ್ಯ, Vijayavani, p. 21 [unreliable source?]
^ Verses 128, 129 in Bijaganita Plofker 2007, pp. 476–477
^ a b Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
^ a b Stillwell1999, p. 74. sfn error: no target: CITEREFStillwell1999 (help)
^ Students& Britannica India. 1. A to C by Indu Ramchandani
^ a b c 50 Timeless Scientists von K.Krishna Murty
^ Shukla 1984, pp. 95–104.
^ Cooke 1997, pp. 213–215.
^ IERS EOP PC Useful constants. An SI day or mean solar day equals 86400 SI seconds. From the mean longitude referred to the mean ecliptic and the equinox J2000 given in Simon, J. L., et al., "Numerical Expressions for Precession Formulae and Mean Elements for the Moon and the Planets" Astronomy and Astrophysics 282 (1994), 663–683.[1]
^ Selin 2008, pp. 269–273.
^ Mazur 2005, pp. 19–20
^ a b Plofker 2007, p. 477
^ Bhaskara NASA 16 September 2017
^ "Anand Narayanan". IIST.
^ "Great Indian Mathematician - Bhaskaracharya". indiavideodotorg. 22 September 2015. Archived from the original on 12 December 2021.
Burton, David M. (2011), The History of Mathematics: An Introduction (7th ed.), McGraw Hill, ISBN 978-0-07-338315-6
Eves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Saunders College Publishing, ISBN 978-0-03-029558-4
Mazur, Joseph (2005), Euclid in the Rainforest, Plume, ISBN 978-0-452-28783-9
Sarkār, Benoy Kumar (1918), Hindu achievements in exact science: a study in the history of scientific development, Longmans, Green and co.
Colebrooke, Henry T. (1817), Arithmetic and mensuration of Brahmegupta and Bhaskara
White, Lynn Townsend (1978), "Tibet, India, and Malaya as Sources of Western Medieval Technology", Medieval religion and technology: collected essays, University of California Press, ISBN 978-0-520-03566-9
Selin, Helaine, ed. (2008), "Astronomical Instruments in India", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition), Springer Verlag Ny, ISBN 978-1-4020-4559-2
Shukla, Kripa Shankar (1984), "Use of Calculus in Hindu Mathematics", Indian Journal of History of Science, 19: 95–104
Pingree, David Edwin (1970), Census of the Exact Sciences in Sanskrit, vol. 146, American Philosophical Society, ISBN 9780871691460
Plofker, Kim (2007), "Mathematics in India", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 9780691114859
Plofker, Kim (2009), Mathematics in India, Princeton University Press, ISBN 9780691120676
Cooke, Roger (1997), "The Mathematics of the Hindus", The History of Mathematics: A Brief Course, Wiley-Interscience, pp. 213–215, ISBN 0-471-18082-3
Poulose, K. G. (1991), K. G. Poulose (ed.), Scientific heritage of India, mathematics, Ravivarma Samskr̥ta granthāvali, vol. 22, Govt. Sanskrit College (Tripunithura, India)
Chopra, Pran Nath (1982), Religions and communities of India, Vision Books, ISBN 978-0-85692-081-3
Goonatilake, Susantha (1999), Toward a global science: mining civilizational knowledge, Indiana University Press, ISBN 978-0-253-21182-8
Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2001), "Mathematics across cultures: the history of non-western mathematics", Science across cultures, Springer, 2, ISBN 978-1-4020-0260-1
Stillwell, John (2002), Mathematics and its history, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-95336-6
Sahni, Madhu (2019), Pedagogy Of Mathematics, Vikas Publishing House, ISBN 978-9353383275
O'Connor, John J.; Robertson, Edmund F., "Bhāskara II", MacTutor History of Mathematics archive, University of St Andrews University of St Andrews, 2000.
Pingree, David (1970–1980). "Bhāskara II". Dictionary of Scientific Biography. Vol. 2. New York: Charles Scribner's Sons. pp. 115–120. ISBN 978-0-684-10114-9.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bhāskara_II&oldid=1087158738"
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Simulation on metro railway induced vibration. Part II: effect of corrugated rail | JVE Journals
Hougui Zhang1 , Zhou Ren2 , Qiong Wu3 , Jie Yang4
1, 3, 4Beijing Municipal Institute of Labour Protection, Beijing 100054, China
2Beijing Jiaotong University, Beijing 100044, China
The deterioration of running surface, including out-of-round wheels and corrugated rails. The effect of out-of-round wheels on simulation of rail dynamic behavior was presented in a companion paper [1]. In this paper, the effect of corrugated rails was discussed. Considering a perfect round wheels moving on different irregularities rail surface, including perfect smooth rail, irregularities generated from a US PSD, measurement data from corrugated rails. Calculation result indicated that (1) the traditional used irregularity generated from US-PSD would not satisfy the requirement simulation on rolling noise and vibration generation in high frequency range; (2) corrugated rail plays the dominant role on the vibration which obviously related to its typical wavelength.
Simulation on rolling noise and vibration generation in high frequency range needs considering "acoustic roughness"
In low frequency range (<10 Hz), there is no obvious difference between US-PSD irregularity and perfect smooth rail
Corrugated rail plays the dominant role on the vibration (about 250 Hz) which obviously related to its typical wavelength (63mm) at speed of 60km/h
Keywords: railway induced vibration, corrugated rail, and acoustic roughness.
The deterioration of running surface, including out-of-round wheels and corrugated rails, as which typical wavelength less than 1000 mm, is also defined as ‘acoustic roughness’. In order to distinguish the contribution of running surface deterioration by wheels and rail respectively, out-of-round wheels moving on perfect smooth rail surface in different train speeds had already discussed in a companion paper [1].
Were different from out-of-round wheels that were commonly neglected in modeling, track irregularities play an important role on nearly all of the calculation models as the excitation element. When modeling the dynamic stability of train running, calculating the substructures subjected to the moving loads from railway trains [2-4], and so on, it would be sufficient using irregularities which was simplify generated from functions of power spectrum density (PSD) summarized from US or Germany railway main lines. However, it should be figured out that the wavelength of US-PSD was scoped from 1.524 m to 304.8 m, which highest effected frequency would be about 11Hz when the train’s speed at 60 km/h. Obviously, the common used track irregularities would not satisfy the requirement simulation on rolling noise and vibration generation, which interested frequency would be up to 80 Hz, 250 Hz [5, 6], and even 4000 Hz.
Rail corrugation is a periodic deterioration on running surface of rail (Fig. 2). Measurement data shows rail corrugation lead to 15 to 20 dB (A) pass-by noise increasing than normal condition. The enlarged vibration and boring squeal noise are complained by the resident living nearby, metro passengers and even the metro train drivers [7].
In this paper, the effect of rail roughness on the simulation modeling was discussed, considering a perfect round wheels moving on different irregularities rail surface, including perfect smooth rail, irregularities generated from a US PSD, measurement data from corrugated rails.
2. Irregularities input data
The SIMPACK-ABQUES simulation model was presented in the companion paper [1]. As the only alternating quantity, the input wheel-rail interaction irregularities were introduced in this section.
2.1. Traditional used irregularities
Generally, it would be sufficient using irregularities which was simplify generated from US-PSD. In this paper, the Level 6 irregularity was generated in default SIMPACK package, using Eq. (1) and Table 1 parameters:
{S}_{v}\left(\mathrm{\Omega }\right)=\frac{k{A}_{v}{{\mathrm{\Omega }}_{c}}^{2}}{{\mathrm{\Omega }}^{2}\left({\mathrm{\Omega }}^{2}+{{\mathrm{\Omega }}_{c}}^{2}\right)}.
And the final irregularities were shown in Fig. 1.
Table 1. US-PSD track irregularity spectrum parameters
{\mathrm{\Omega }}_{c}
rad/m
{\mathrm{\Omega }}_{s}
{A}_{a}
cm2∙rad/m
{A}_{v}
0.0
Fig. 1. Irregularities generated from US-PDF Level 6
2.2. Roughness from measured data for corrugated rail
For the corrugated rail, the roughness was measured using a corrugation analysis trolley (CAT) (Fig. 1). It is a hand operated device for measurement of longitudinal rail irregularities in the wavelength range from approximately 10 mm to 3,000 mm. Rail irregularities are determined by integrating the signal from a vertical accelerometer mounted on a hard steel ball which rolls on the rail. Another wheel with rubber coating is used to determine the longitudinal position of the trolley and to trigger the samples. The sampling distance was 1 mm. The data acquisition hardware was connected to a PC via a USB interface [8].
Fig. 2. Corrugated rail
Fig. 3. Corrugation analysis trolley
The roughness data was captured from a measurement conducted in the section of DTVI2 track system in Beijing, in accordance with the model parameter. Raw data (Fig. 4) were provided as the input irregularities. The 1/3 octave wavelength spectrum was shown in Fig. 5, indicted that the typical wavelength of the measured corrugated rail was 63 mm.
Fig. 4. Roughness of corrugated rail (Raw data)
Fig. 5. 1/3 octave central wavelength
The effect of rail roughness on the simulation modeling was discussed in this section, considering a train with perfect round wheels moved on different irregularities rail surface at the speed of 60 km/h, including perfect smooth rail, irregularities generated from a US PSD, measurement data from corrugated rails. As the rail vibration velocity always related to rolling noise, 1/3 octave band vibration velocity response were recorded and analyzed.
Fig. 6. Rail velocity vibration for different irregularities
From Fig. 6, it was clear that:
1) The curve of rail velocity vibration for using US PSD Lever 6 was very close to the irregularity of perfect smooth rail surface. It indicated the dynamic property of vehicle and track system plays the dominant role.
2) In low frequency range (less than 10 Hz), measurement irregularity data of corrugated rail was not made difference from perfect smooth rail;
3) In the frequency from 10 to 63 Hz, although the measurement irregularity data of corrugated rail raised the vibration velocity value, it still not alter the tendency of the curve that resulted by the dynamic property of vehicle and track system.
4) In high frequency that more than 63 Hz, the effect of corrugated rail would be not neglected any more, especially at the band of 250 Hz, corrugated rail plays the dominant role on the vibration which obviously related to the typical wavelength of 63 mm according to Eq. (2):
f=\frac{V}{\lambda }=\frac{60 \mathrm{k}\mathrm{m}/\mathrm{h}}{63 \mathrm{m}\mathrm{m}}=264 \mathrm{H}\mathrm{z}.
1) It was demonstrated that the irregularity generated from US-PSD would not satisfy the requirement simulation on rolling noise and vibration generation in high frequency range;
2) Corrugated rail plays the dominant role on the vibration which obviously related to its typical wavelength.
This work was under the support of Beijing Natural Science Foundation (No. 3184047), Beijing Academy of Science and Technology Funds (No.OTP-2018-002) and Beijing Public Finance innovation project in 2018(PXM2018-178304_000007).
Hougui Zhang, Zhou Ren, et al. Simulation on metro railway induced vibration. Part I: effect of out-of-round wheels. Vibroengineering Procedia, Vol. 29, https://doi.org/10.21595/vp.2019.21153, 2019. [Search CrossRef]
Thompson D. J. Railway noise and vibration: mechanisms, modeling and means of control. Elsevier, Oxford, 2008. [Search CrossRef]
Zhang H. G., Liu W. N., Liu W. F., et al. Study on the cause and treatment of rail corrugation for Beijing metro. Wear, Vol. 317, Issues 1-2, 2014, p. 120-128. [Publisher]
Grassie S. L. Rail corrugation: characteristics, causes and treatments. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, Vol. 223, 2009, p. 581-595. [Publisher]
Impact vibration behavior of railway vehicles: a state-of-the-art overview
Lin Jing, Kaiyun Wang, Wanming Zhai
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Numerical simulation of vortex induced vibrations on a circular cylinder at different Reynold’s number | JVE Journals
Shivam Yadav1 , Akshaj Kulshreshtha2 , Baij Nath Singh3
3Indian Institute of Technology Dhanbad, Dhanbad- 826004, Jharkhand, India
Copyright © 2019 Shivam Yadav, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This research investigates the effect of vortex induced vibrations on flow past a circular cylinder for two-dimensional unsteady incompressible flow at different Reynold’s number. The pressure bases steady solver is used for computation along with standard k-ε turbulence model. The change in the lift and drag coefficient with respect to increase in Reynolds number is studied and contours of vorticity are plotted. The pressure distribution on the fixed cylinder for different Reynolds number is also presented. It is found that drag coefficient reduces with the increasing Re and lift coefficient increases up to Reynold’s number 104. Moreover, the pressure difference on the fixed cylinder increases with the increasing Reynold’s number.
Keywords: vortex induced vibration (VIV), wake flow, standard k-ε turbulence model.
The flow past circular cylinder is gaining attention of researchers due to its primitive significance in the engineering applications. Steady flow past a circular cylinder gives the various flow regimes at different Reynolds numbers and coefficient of drag rely upon the Reynolds number [1].
In his research, Xiong elaborates that Vortex induced vibrations generates when a structure is placed normally to the direction of flow in a Newtonian fluid so that it can fluctuate due to separation in vortices shedding and both natural and shedding frequency becomes equal [2].
Vortex shedding can successfully lead to separation of delay, drag reduction, and reduction of vibration and noise. There are many features which results from the interactions between the external disturbance, and the flow around the circular cylinder.
Vortex shedding is an oscillating flow of a fluid like air, and water around the bluff bodies at Reynold’s number based on the characteristic length of the body. On the bluff bodies, the flow separation takes place to causes the pressure difference on different surface of the body. The vortices detach from the body periodically producing the Von-Karman street. Results computed from the experimental studies are most reliable, but due to the cost limitations, nowadays numerical methods are more preferred as they also provide reliable, and acceptable results [3].
Fluid passing through the model separates the flow at one or more than one sharp corner and makes the contours and as the Reynolds number increases the substantial difficulties starts occurring [4].
By using laminar separation bubble, and turbulent model, an increase in a Reynolds number we found that there is a rapid reduction in drag force and increases in lift force [5].
By doing experiment on a circular cylinder the investigation of vortex induced vibrations with Low damping ratio gives the idea that there is a change in velocity [6].
To do the computational simulation first order SAS-SST turbulent model has been used on a 2-D cylinder near different wake regions and found that shape of mean velocity is related to fluctuation in velocities [7].
In this paper, we consider the case of flow past a circular cylinder at different Reynolds number. Our research work includes the computational work. The simulations carried out for 2-D Circular Cylinder only. Thus, the present work deals with the lift and drag coefficient in comparison to increase in Reynolds number.
Two major governing equations for 2-D incompressible turbulent laminar flow we have used.
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.
Navier stokes equations are given in Eq. (2) and Eq. (3).
X
\frac{\partial \left(\rho u\right)}{\partial t}+\nabla \left(\rho uU\right)=-\frac{\partial p}{\partial x}+\frac{\partial \tau xx}{\partial x}+\frac{\partial \tau yx}{\partial y}.
Y
\frac{\partial \left(\rho v\right)}{\partial t}+\nabla \left(\rho vU\right)=-\frac{\partial p}{\partial y}+\frac{\partial \tau yx}{\partial x}+\frac{\partial \tau yy}{\partial y},
\rho
U
is the free stream velocity,
p
\mu
u
v
x
y
Re=\frac{\rho UD}{\mu }, { C}_{l}=\frac{{F}_{l}}{\frac{1}{2 }\rho A{U}^{2}}, { C}_{d}=\frac{{F}_{d}}{\frac{1}{2 }\rho A{U}^{2}}, { C}_{p}=\frac{p-{p}_{\infty }}{\frac{1}{2 }\rho {U}^{2}}.
Re
{ C}_{l}
{C}_{d}
are the lift and drag coefficient.
{C}_{p}
is the pressure coefficient and
{C}_{f}
is the skin friction coefficient.
D
is the diameter of cylinder,
{F}_{l}
{F}_{d}
are the lift and drag force respectively.
3. 3.Computational domain and meshing
The numerical computation of drag and lift coefficient for different Reynold’s number is studied using Ansys Fluent 19.1. The 2-D cylinder of diameter 5 mm is formed and computational domain is created around it having dimensions of 150 mm×70 mm. The fluid domain with boundary conditions is given in Fig. 1. The inlet is assigned as velocity inlet and outlet is having atmospheric pressure i.e. 101325 Pa and all other walls are having no slip condition. The structured 2-D quadrilateral mesh is used for the domain and inflation layers are formed around the cylinder. The mesh with inflation layers is shown in Fig. 2.
The steady pressure-based solver is used to study the aerodynamic effects around the cylinder. The turbulence model used is standard
k
\epsilon
and SIMPLE scheme is used for pressure and velocity coupling with second order upwind for momentum.
Fig. 1. Domain around the cylinder
Fig. 2. Meshing around the cylinder with inflation layers
The numerical simulation around cylinder is carried out by varying the velocity at inlet to get different Reynold’s number and to analyze the change in lift, drag and pressure coefficient with respect to
Re
. Table 1 shows the drag coefficient and lift coefficient values for different Reynold’s number. The plot for the averaged
{C}_{d}
{C}_{l}
along the increasing
Re
is shown in Fig. 3 and Fig. 4 respectively. It is observed that the drag coefficient decreases on increasing the
Re
while lift coefficient increases up to
Re=
104 and drops for higher
Re
Table 1. Coefficient of lift and drag coefficient for different Reynold’s number
{C}_{d}
Coefficient of lift (
{C}_{l}
Fig. 3. Coeffcient of drag vs Reynold’s number
Fig. 4. Coeffcient of lift vs Reynold’s number
Fig. 5 represents the coefficient of lift and pressure distribution on fixed cylinder for at
Re=
102 ,103 ,104 and 105. The
{C}_{l}
is plotted for the iteration ranging from 800 to 1800. The
{C}_{l}
value increasing as
Re
increases till 104 while decreases for further higher values which can be seen from the graph. The value of pressure coefficient is having overlapped curve for at
Re=
102 and the difference between pressure distribution increases along the increasing
Re
Fig. 5. The lift coefficient and pressure coefficient distribution along the fixed cylinder wall at different Reynold’s number
The vorticity contour is plotted in Fig. 6 to check the wake pattern on the cylinder for increasing
Re
. The range of vortex shedding is kept from –104 to 104. The oscillations around the cylinder increases with Reynold’s number forming von Karman street around the cylinder. It can be seen that separate shear layers are rolling on the upper and lower wall of fixed cylinder. The red color in the vorticity contour represents the clockwise rotation while blue color represents the counterclockwise rotation of vortices.
To understand the oscillations around the cylinder due to formation of vortex shedding the power spectral density of the lift coefficient is determined using Fast Fourier Transformation. The maximum frequency for the
Re=
105 using FFT is 0.018 Hz. Fig. 7 represents the power spectral distribution for the same.
Fig. 6. Velocity contours at different Reynolds numbers
Fig. 7. Power spectral distribution vs frequency
In this paper the numerical computation for the lift and drag coefficient with increasing number of Reynold’s number is represented. The drag coefficient reduces with the increasing
Re
and lift coefficient increases up to Reynold’s number 104. Similarly, the pressure difference on the fixed cylinder increases with the increasing Reynold’s number. The vorticity contour and power spectral distribution of lift coefficient is also presented in the results and discussion section.
Martinez R. C., Sweeney L. G., Finlay W. H. Aerodynamic forces and moment on a sphere or cylinder attached to a wall in a Blasius boundary layer. Engineering Applications of Computational Fluid Mechanics, Vol. 3, Issue 3, 2009, p. 289-295. [Publisher]
Xiong Y., Peng S., Zhang M., Yang D. Numerical study on the vortex-induced vibration of a circular cylinder in viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics, Vol. 272, 2019, p. 104170. [Publisher]
Singha S., Sinhamahapatra K. Flow past a circular cylinder between parallel walls at low Reynolds numbers. Ocean Engineering, Vol. 37, Issues 8-9, 2010, p. 757-769. [Publisher]
Rajani B. N., Kandasamy A., Majumdar S. Numerical simulation of laminar flow past a circular cylinder. Applied Mathematical Modelling, Vol. 33, Issue 3, 2009, p. 1228-1247. [Publisher]
Chopra G., Mittal S. Numerical simulations of flow past a circular cylinder. Journal of Physics: Conference Series, Vol. 822, 2017, p. 012019. [Search CrossRef]
Brika D., Laneville A. Vortex-induced vibrations of a long flexible circular cylinder. Journal of Fluid Mechanics, Vol. 250, 1993, p. 1-481. [Publisher]
Shim Y. M., Sharma R., Richards P. Numerical study of the flow over a circular cylinder in the near wake at Reynolds number 3900. 39th AIAA Fluid Dynamics Conference, 2009. [Publisher]
Anuj Yadav, N. K. Singh
|
Role of slanted reinforcement on bending capacity SS beams | JVE Journals
Mohammad Reza Ghasemi1 , Aydin Shishegaran2
1Structural Engineering, School of Civil Engineering, Sistan and Baluchestan University, Zahedan, Iran
2Environmental Engineering, School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran
The wide application range of simply supported beams in building construction, has always caused an attraction to somehow increase their bending capacity with high ductility. In this research, for the very purpose, the reinforcement bars under compression are bent at 45° from 1/3 of the beam length from the two ends and led to the tension zone. A sealed rubber tube of diameter twice that of the reinforcement bar covers the slanted part to separate it from the beam’s concrete. This will in fact reduce the stress intensity created in the bars above and below the neutral plane and increase the beam’s bending capacity considerably through making the two tensile and compressive forces acting opposite to each other. Actually, the proposed system can be specified by applying a superposition of the sum of the effects of the compressive stresses of the reinforcement bars above the beam’s 1/3 ends plus the sum of the effects of the tensile stresses created at 1/3 of the beam midpoint. The compressive stress created in the upper part tends to pass through the slanted part and reach the tensile part, and an opposite act for the tensile stress created in the lower part. Therefore, it is obvious that a compressive force found by the solution of the first superposition equation is applied at the middle 1/3 of the lower part and causes up to 25 % increase in the beam bending capacity.
Keywords: bending capacity increase, sealed rubber tube, stress transfer, slanted reinforcement bar, ductility.
There are many techniques for increasing bending capacity. A post-tensioning system for reinforcing and creating a compressive force in the tensile member will cause a reduction in the ultimate tensile stress. some methods have been proposed to reinforce the tensile section to enhance the beam’s bending capacity. In this study use a new reinforcement, an auxiliary member, or a primary compressive force applied on the tensile section to enhance the bending capacity [1-3].
2. Literary systems for increasing bending capacity
Figeys et al. (2008) used the finite element method and studied a case in the laboratory where a post-tensioned FRP member stuck under a concrete beam. They compared two similar beams one simple and one reinforced with a tensioned FRP plate and concluded that the latter increased the beam bending capacity by about 40 %. Easy execution (compared with pretensioned cables), capability of being implemented after the beam has been constructed, and easier maintenance are the merits of such a system. Challal (1991) investigated fiber reinforced bars (FRB) and showed that they reduced cracks and increased the bending strength. Jerom and Ross (1999) showed that FRP-reinforced beams gained almost 30-70 % more bending strength, but lost about 40 % of their ductility [1, 4].
3. Proposed system for increasing bending capacity
Fig. 1 shows that in an ordinary beam, the compressive and tensile stresses are created respectively above and below the neutral plane. In Fig. 2 that shows the new beam model, it is quite clear that the slanted reinforcement separated from the beam concrete can be a place where the compressive stress above can be transferred to the tensile reinforcement below. Fig. 2 that uses the superposition method, shows that the stress created in the compression zone and its transferring to the tensile zone can somewhat reduce the tensile stress created in the mid-beam tensile section; the reverse occurs too for the tensile stress created in the tensile zone. This resembles the case of pre-stressing which causes an increase in the beam bending capacity. In short, this method reduces the tensile stress of the beam’s mid-section and increases its bending capacity through transferring the compressive force created in the upper part of the concrete beam to its lower middle one-third where the tensile stress has its highest value.
Fig. 1. Ordinary beam models. It shows number of compressive and tension reinforcement bars
Fig. 2. Superposition of the stress created in the bent reinforcement bar
Earlier studies show that some methods have been proposed to reinforce the tensile section to enhance the beam’s bending capacity. These studies use a new reinforcement, an auxiliary member, or a primary compressive force applied on the tensile section to enhance the bending capacity. The common point between the present and earlier researches is that this research too tries to present a new method of applying a compressive stress in the tensile zone to strengthen the tensile section.
To study accuracy and behavior of the proposed theory, use was made of the finite element (with the ABAQUS Software) and laboratory test methods. The beam specimens were built in IUST, Tehran, in Sept. 2014 and tested two months later in BHRC. The
p
∆
diagrams were then drawn for two beam specimens with concentrated loads in the mid-span (Figs. 6-7).
In simulation, use was made of the damaged plastic concrete and specifications of the 30 MPa concrete for modeling in the ABAQUS Software. For reinforcement bars, both the elastic and plastic states were considered; the specifications used in the above software are shown in Fig. 3. The results are related to the
{\mathrm{\Phi }}_{20}
bars extensometer tests performed at the laboratory of the School of Mechanics, IUST. In the steel-concrete interaction modeling, the embedded rigid element model was used and only the slanted part was excluded [5-10].
The ordinary and new laboratory specimens were made similar to their finite element counterparts, but before the concrete placement, the slanted bar was covered with a rubber tube and the two ends were sealed with glue to prevent the concrete to enter the tube (as shown in Fig. 4). Meanwhile, to compare the laboratory results and check their accuracy, two beam specimens were made for each model. The reinforced bars were of the A-III type and underwent the extensometer test [7].
Fig. 3. Steel bar stress-strain diagram
Fig. 4. Reinforcements in the ordinary and new models (proposed models)
Next, the cubical specimens were tested in the BHRC structure laboratory under similar loading and boundary conditions; they were crushed on the test day and day 28 of the concrete age with a compressive strength of about 30 MPa and both the ordinary and new models were tested on day 30. A 100-ton jack was placed in the mid-span on a 20×20 cm metal plate; all the above mentioned arrangements are shown in Fig. 5. Up to the ultimate bending capacity in the elastic range, loading was static (100 kg/2 sec) after which, in the plastic range, it was changed to the displacement type (0.05 mm/2 sec) [7].
Fig. 5. Beam below a 100-ton jack and location of the deflect-meter and test setup
When tests started, the beam mid-span load-displacement information was given to the data logger so as to compare the results (shown as
p
\mathrm{\Delta }
curves) with those found by ABAQUS. Comparisons of the results (beams with beams and laboratory tests with ABAQUS) confirm the accuracy of the proposed theory and modeling, and provide, for other researchers, the possibility of modeling, concluding, and validating this theory [7, 9].
One specimen from each finite element model and two from the test models were studied and the
p
\mathrm{\Delta }
curves were drawn to determine each model’s bending capacity and ductility. The bending capacity for the elastic, elastoplastic, and plastic (ultimate limit) states is shown by
p
\mathrm{\Delta }
curves [6].
p
\mathrm{\Delta }
results and both the finite element (ABAQUS) and laboratory tests static analyses show that the ultimate bending capacity of the new model beam is 25 % more than that of the ordinary beam. This is clear in Figs. 6 and 7 which compare the
p
\mathrm{\Delta }
results of the two beam models; therefore, the conformity of the
p
\mathrm{\Delta }
results of the laboratory tests and finite element findings show the accuracy of the ABAQUS model before and up to the ultimate bending capacity. The
p
\mathrm{\Delta }
results of the new model’s laboratory tests show that after 25 % increase, the bending capacity starts decreasing due to the area reduction (where steel and concrete separate) and beam shear capacity reduction in this part due to excess loading. Figs. 8 show crack formation in the new model’s ultimate limit and failure time [9].
Fig. 6. Comparison of the modified results of the ordinary and new models specimens in the ABAQUS [9]
Fig. 7. Ordinary and new models’ laboratory tests results’
p
\mathrm{\Delta }
Fig. 8. New model’s cracks (ultimate shear capacity)
Although pre-tensioning with cables and post-tensioning with the help of FRP plates stuck to the beam are quite efficient and safe systems, the stress transfer can, through more studies, be another way of increasing the bending and load bearing capacities in structural engineering. We suggest that other researchers study and do researches on this structural engineering-related subject. In general, the results of our research and the suggestions we have for other colleagues are as follows:
1) To increase the bending capacity of a simply supported beam through transferring the compressive stress of the compressive reinforcement to the tensile reinforcement, there is a weak point around the rubber tube. To eliminate this, we suggest increasing stirrups and the beam cross sectional area in this part.
2) Stress transfer from compressive to tensile reinforcement can be an initiation for the subject of “bending capacity increase in concrete beams”. In general, compressive stress transfer to the tensile part from any beam section possible can result in an increase in the bending capacity. It is suggested that in simply supported beams the rubber tube (hose) be placed at the turning point and the investigations and tests be repeated.
3) To optimize the proposed system, it is suggested that more research be done on such topics as the bars yielding, point and angle of bending, and so on. Most probably, if the compression bar is bent towards the beam’s tensile section at its yielding point, the bending capacity increase will be more, but this needs more research.
Figeys W., Verstrynge T. E., Brosens T. K., Van Schepdael L., Dereymaeker J., Van Gemert D., Schueremans L. Feasibility of a novel system of prestressing externally bonded. Materials and Structures, Vol. 44, Issue 9, 2011, p. 1655-1669. [Search CrossRef]
Loo Y. C., Chowdhury S. H. Reinforced and Prestressed Concrete. Vol. 1, 2010. [Search CrossRef]
Khalou A. R. Design of Prestressed Concrete Structures. Vol. 1, 2003. [Search CrossRef]
Oehlers D. J., Liu I., Seracino R. A. Generic design approach for EB and NSM longitudinally plated RC beams. Constructions and Building Materials, Vol. 21, Issue 4, 2007, p. 697-708. [Search CrossRef]
Massart T. J., Vantomme J., Bouillard F., Iribarren B. S. Progressive Collapse Simulation of Reinforced Concrete Structures; Influence of Design and Material Parameters and Investigation of Strain Rate Effects. Ph.D. Thesis, Royal Military Academy Polytechnical Faculty, Universite Libre de Bruxelles Faculty of Applied Sciences, Bruxelles, 2010. [Search CrossRef]
Jankowiak T., Lodygowski O. Identification of Parameters of Concrete Damage Plasticity Constitutive Model Foundation of Civil and Environmental Engineering. Pozan, Poland, 2005. [Search CrossRef]
Mostofinezhadm D. Reinforced Concrete Structures. Based on ACI318-05, Vol. 1, 2007. [Search CrossRef]
Schreppers G. J. Embedded Reinforcements. TNO DIANA, Netherland, 2011, p. 11-15. [Search CrossRef]
ABAQUS CAE/6.13-4 Version. [Search CrossRef]
ABAQUS Theory Manual. Version 6.3, Hibbitt Karlson and Sorensen, Inc., 2002. [Search CrossRef]
Flexural behavior of high-strength, steel-reinforced, and prestressed concrete beams
Qing Jiang, Hanqin Wang, Xun Chong, Yulong Feng, Xianguo Ye
Nazim Abdul Nariman, Martin Husek, Ilham Ibrahim Mohammad, Kaywan Othman Ahmed, Diyako Dilshad, Ibrahim Khidr
Journal of Engineering and Thermal Sciences
Optimum Design of Flexural Strength and Stiffness for Reinforced Concrete Beams Using Machine Learning
Nazim Nariman, Khader Hamdia, Ayad Ramadan, Hamed Sadaghian
Damage modeling of ballistic penetration and impact behavior of concrete panel under low and high velocities
Chahmi Oucif, J.S. Kalyana Rama, K. Shankar Ram, Farid Abed
Experimental study on acceleration response laws of shallow-buried bias tunnels with a small distance
Feifei Wang, Qingyang Ren, Yang Peng, Yong Zhang, Ping Zou, Wanjie Hu, Zeng Ma
Mitigation of the blast load effects on a building structure using newly composite structural configurations
Ahmed K. Taha, M.S. Zahran, Zhengguo Gao
Seismic responses of concrete rectangular liquid storage structure with large height-width ratio
Wei Jing, Shangshang Xing, Yu Song
Performance of fixed beam without interacting bars
Aydin Shishegaran, Behnam Karami, Timon Rabczuk, Arshia Shishegaran, Mohammad Ali Naghsh, Mohammreza Mohammad Khani
Performance of a novel bent-up bars system not interacting with concrete
Aydin Shishegaran, Mohammad Reza Ghasemi, Hesam Varaee
|
(Redirected from Statistics - Below)
{\displaystyle {\hat {Q}}_{1}(p)}
{\displaystyle x_{1}}
{\displaystyle 0.95\times x_{i}+0.05\times x_{i+1}}
{\displaystyle x_{i+1}}
{\displaystyle {\hat {Q}}_{6}(p)}
{\displaystyle {\hat {Q}}_{1}(p)}
{\displaystyle x_{1}}
{\displaystyle 0.75\times x_{i}+0.25\times x_{i+1}}
{\displaystyle x_{i+1}}
{\displaystyle {\hat {Q}}_{6}(p)}
{\displaystyle {\hat {Q}}_{1}(p)}
{\displaystyle x_{1}}
{\displaystyle 0.25\times x_{i}+0.75\times x_{i+1}}
{\displaystyle x_{i+1}}
{\displaystyle {\hat {Q}}_{6}(p)}
{\displaystyle {\hat {Q}}_{1}(p)}
{\displaystyle x_{1}}
{\displaystyle 0.05\times x_{i}+0.95\times x_{i+1}}
{\displaystyle x_{i+1}}
{\displaystyle {\hat {Q}}_{6}(p)}
{\displaystyle {\hat {Q}}_{1}(p)}
{\displaystyle x_{1}}
{\displaystyle 0.5\times x_{i}+0.5\times x_{i+1}}
{\displaystyle x_{i+1}}
{\displaystyle {\hat {Q}}_{6}(p)}
{\displaystyle p}
{\displaystyle \sigma _{p}={\sqrt {\frac {p(1-p)}{\sum _{i=1}^{n}w_{i}-b}}}}
{\displaystyle b=1}
{\displaystyle w_{i}=1}
{\displaystyle w_{i}}
{\displaystyle \sigma _{\bar {x}}={\sqrt {{\frac {1}{(\sum _{i=1}^{n}w_{i})(\sum _{i=1}^{n}w_{i}-b)}}\sum _{i=1}^{n}w_{i}(x_{i}-{\frac {1}{\sum _{i=1}^{n}w_{i}}}\sum _{i=1}^{n}w_{i}x_{i}}})^{2}}
{\displaystyle b=1}
{\displaystyle w_{i}=1}
{\displaystyle w_{i}}
Retrieved from ‘https://wiki.q-researchsoftware.com/index.php?title=Statistics&oldid=58345’
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On the Spectrum and Spectral Norms of -Circulant Matrices with Generalized -Horadam Numbers Entries
Lele Liu, "On the Spectrum and Spectral Norms of -Circulant Matrices with Generalized -Horadam Numbers Entries", International Journal of Computational Mathematics, vol. 2014, Article ID 795175, 6 pages, 2014. https://doi.org/10.1155/2014/795175
Lele Liu1
Academic Editor: Chengpeng Bi
This work is concerned with the spectrum and spectral norms of -circulant matrices with generalized -Horadam numbers entries. By using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them. In addition, a sufficient condition for an -circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral norms of normal -circulant matrix with generalized -Horadam numbers, which generalize and improve the known results.
There is no doubt that the -circulant matrices have been one of the most interesting research areas in computation mathematics. It is well known that these matrices have a wide range of applications in signal processing, digital image disposal, coding theory, linear forecast, and design of self-regress.
There are many works concerning estimates for spectral norms of -circulant matrices with special entries. For example, Solak [1] established lower and upper bounds for the spectral norms of circulant matrices with Fibonacci and Lucas numbers entries. subsequently, Ipek [2] investigated some improved estimations for spectral norms of these matrices. Bani-Domi and Kittaneh [3] established two general norm equalities for circulant and skew circulant operator matrices. Shen and Cen [4] gave the bounds of the spectral norms of -circulant matrices whose entries are Fibonacci and Lucas numbers. In [5] they defined -circulant matrices involving -Lucas and -Fibonacci numbers and also investigated the upper and lower bounds for the spectral norms of these matrices.
Recently, Yazlik and Taskara [6] define a generalization of the special second-order sequences such as Fibonacci, Lucas, -Fibonacci, -Lucas, generalized -Fibonacci and -Lucas, Horadam, Pell, Jacobsthal, and Jacobsthal-Lucas sequences. For any integer number , the generalized -Horadam sequence is defined by the following recursive relation: where and are scaler-value polynomials, . The following are some particular cases.(i)If , and , , the -Fibonacci sequence is obtained: (ii)If , and , , the -Lucas sequence is obtained: (iii)If , and , , the Fibonacci sequence is obtained: (iv)If , and , , the Lucas sequence is obtained: (v)If , and , , the Jacobsthal sequence is obtained:
In [7], the authors present new upper and lower bounds for the spectral norm of an -circulant matrix , and they study the spectral norm of circulant matrix with generalized -Horadam numbers in [8]. In this paper, we first give an explicit formula for the eigenvalues of -circulant matrix with generalized -Horadam numbers entries using different methods in [7]. Afterwards, we present a sufficient condition for an -circulant matrix to be normal. Based on the results, the precise value for spectral norms of normal -circulant matrix whose entries are generalized -Horadam numbers is obtained, which generalize and improve the main results in [1, 2, 4, 5].
In this section, we present some known lemmas and results that will be used in the following study.
Definition 1. For any given , the -circulant matrix , denoted by , is of the form It is obvious that the matrix turns into a classical circulant matrix for .
Lemma 2 (see [9]). Let be an -circulant matrix; then the eigenvalues of are given by where is the th root of unity.
Let us take any matrix of order ; it is well known that the spectral norm of matrix is where is the conjugate transpose of and is the eigenvalue of .
For a normal matrix (i.e., ), we have the following lemma.
Lemma 3 (see [10]). Let be a normal matrix with eigenvalues . Then the spectral norm of is
Lemma 4 (see [11], Abel transformation). Suppose that and are two sequences, and ; then
3. Spectrum of -Circulant Matrix with Generalized -Horadam Numbers
We start this section by giving the following lemma.
Lemma 5. Suppose that is a generalized -Horadam sequence defined in (1). The following conclusions hold.(1)If , then (2)If , then
Proof. (1) According to (1), we have Changing the summation index in (14), we have By direct calculation, together with recursive relation (1), one can obtain that Therefore we immediately obtain (12) from .(2)Suppose that ; we first illustrate that . Let ; then . Combining (1) and , one can obtain that which shows that is a constant sequence, and therefore Evaluating summation from to , we have Changing the summation index in (19) gives Therefore In view of assumptions and , we know that . Thus we obtain (13) from (21).
Theorem 6. Let be an -circulant matrix with eigenvalues ; then for the following hold.(1)If , then (2)If , then
Proof. According to Lemma 2, we have Using Abel transformation (Lemma 4), we have (1)In the light of (12) and (25), one can obtain that It is clear that Substituting (27) into (26), we obtain that Therefore we have We immediately obtain formula (22) from (29).(2)Taking into account (13) and (25), we have It follows that Therefore we obtain (23). This concludes the proof.
4. Spectral Norms of Normal -Circulant Matrices
In this section, we consider the spectral norms of normal -circulant matrix whose entries are generalized -Horadam numbers. Our results generalize and improve the results in [1, 2, 4, 5]. The following lemma can be found in [9], and we give a concise proof.
Lemma 7. Let be an -circulant matrix. If , then is normal matrix.
Proof. It is well known that If , then
That is, . According to (32), we obtain that Therefore , which shows that is normal.
According to Theorem 6 and Lemma 7, we have the following theorem.
Theorem 8. Suppose that is an -circulant matrix. If and , then the spectral norm of is
The following theorem simplifies and generalizes the results of Theorem 2.2 in [12].
Theorem 9. Let be a circulant matrix; then
Proof. Suppose that ; it follows from Lemma 7 that is normal. Notice that It follows from Lemma 3 that . According to Theorem 6, if and , we obtain that Similarly, if , it follows that This completes the proof.
Taking into account formulae (4)–(6), we have the following corollary.
Corollary 10. Let be a circulant matrix; then
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r
-circulant matrices with the Fibonacci and Lucas numbers,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2891–2897, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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k
-Horadam sequence,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 36–41, 2012. View at: Publisher Site | Google Scholar | MathSciNet
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Copyright © 2014 Lele Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Constraints on Neutrino Masses from Baryon Acoustic Oscillation Measurements
Constraints on Neutrino Masses from Baryon Acoustic Oscillation Measurements ()
Universidad San Francisco de Quito, Quito, Ecuador.
From 21 independent Baryon Acoustic Oscillation (BAO) measurements we obtain the following sum of masses of active Dirac or Majorana neutrinos: , where and . This result may be combined with independent measurements that constrain the parameters Σmv, h, and Ωbh2 . For and , we obtain at 95% confidence.
Neutrino Mass, Baryon Accoustic Oscillations, Cosmology
Hoeneisen, B. (2018) Constraints on Neutrino Masses from Baryon Acoustic Oscillation Measurements. International Journal of Astronomy and Astrophysics, 8, 1-5. doi: 10.4236/ijaa.2018.81001.
We extend the analysis presented in “Study of baryon acoustic oscillations with SDSS DR13 data and measurements of
{\Omega }_{k}
{\Omega }_{\text{DE}}\left(a\right)
” [1] to include neutrino masses. The present analysis has three steps: 1) we calculate the distance of propagation
{r}_{s}
c/{H}_{0}
, referred to the present time, of sound waves in the photon-electron-baryon plasma until decoupling by numerical integration of Equation (16) and Equation (17) of Ref. [1] ; 2) we fit the Friedmann equation of evolution of the universe to 21 independent Baryon Acoustic Oscillation (BAO) distance measurements listed in [1] used as uncalibrated standard rulers and obtain the length d of these rulers, in units of
c/{H}_{0}
, referred to the present time; and 3) we set
{r}_{s}=d
to constrain the sum of neutrino masses
\sum \text{ }\text{ }{m}_{\nu }
. c is the speed of light, and
{H}_{0}\equiv 100h\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}
is the present day Hubble expansion parameter.
2. Constraints on Neutrino Masses
The main body of this article assumes: 1) flat space, i.e.
{\Omega }_{k}=0
, and 2) constant dark energy density relative to the critical density, i.e.
{\Omega }_{\text{DE}}
independent of the expansion parameter a. These constraints are in agreement with all observations to date [1] [2] . Results without these constraints are presented in Appendix 1. Results with partial data sets are presented in Appendix 2.
To be specific we consider three active neutrino flavors with three eigenstates with nearly the same mass
{m}_{\nu }
\sum \text{ }{m}_{\nu }=3{m}_{\nu }
. This is a useful scenario to consider since our current limits on
{m}_{\nu }^{2}
are much larger than the mass-squared-differences
\Delta {m}^{2}
\Delta {m}_{21}^{2}
obtained from neutrino oscillations [2] . These neutrinos become non-relativistic at a neutrino temperature
{T}_{\nu }={m}_{\nu }/3.15
or a photon temperature
T={m}_{\nu }{\left(11/4\right)}^{1/3}/3.15
. The corresponding expansion parameter is
{a}_{\nu }={T}_{0}/T=5.28\times {10}^{-4}\left(1\text{\hspace{0.17em}}\text{eV}/{m}_{\nu }\right)
The matter density relative to the present critical density is
{\Omega }_{m}/{a}^{3}
a>{a}_{\nu }
{\Omega }_{m}
includes the density
{\Omega }_{\nu }={h}^{-2}\sum \text{ }{m}_{\nu }/94\text{\hspace{0.17em}}\text{eV}
of Dirac or Majorana neutrinos that are non-relativistic today. Note that for Dirac neutrinos we are considering the scenario in which right-handed neutrinos and left-handed anti-neutrinos are sterile and never achieved thermal equilibrium. Our results can be amended for other specific scenarios. For
a<{a}_{\nu }
we take the matter density to be
\left({\Omega }_{m}-{\Omega }_{\nu }\right)/{a}^{3}
. The radiation density is
{\Omega }_{\gamma }{N}_{\text{eq}}/\left(2{a}^{4}\right)
a<{a}_{\nu }
{N}_{\text{eq}}=3.36
for three flavors of Dirac (mostly) left-handed neutrinos and right-handed anti-neutrinos. We also take
{N}_{\text{eq}}=3.36
for three active flavors of Majorana left-handed and right-handed neutrinos. For
a>{a}_{\nu }
, we take the radiation density to be
\left({\Omega }_{\gamma }{N}_{eq}/2-{a}_{\nu }{\Omega }_{\nu }\right)/{a}^{4}={\Omega }_{\gamma }/{a}^{4}
. The present density of photons relative to the critical density is
{\Omega }_{\gamma }=2.473\times {10}^{-5}{h}^{-2}
The data used to obtain d are 18 independent BAO distance measurements with Sloan Digital Sky Survey (SDSS) data release DR13 galaxies in the redshift range
z=0.1
to 0.7 [3] [4] [5] summarized in Table 3 of [1] , two BAO distance measurements in the Lyman-alpha forest (Lyα) at
z=2.36
(cross-correlation [6] ) and
z=2.34
(auto-correlation [7] ) summarized in Section 6 of [1] , and the Cosmic Microwave Background (CMB) correlation angle
{\theta }_{\text{MC}}=0.010410\pm 0.000005
[2] [8] , used as an uncalibrated standard ruler. Note that the correlation angle
{\theta }_{\text{MC}}
is also determined by BAO. These 21 independent BAO measurements and full details of the fitting method are presented in [1] .
As a reference we take
h=0.678\pm 0.009,\text{\hspace{1em}}{\Omega }_{b}{h}^{2}=0.02226\pm 0.00023
(at 68% confidence) from “Planck TT + low P + lensing” data (that does not contain BAO information) [2] .
{\Omega }_{b}
is the present density of baryons relative to the critical density.
Due to correlations and non-linearities we obtain our final result (Equation (9) below) with a global fit. The following equations are included to illustrate the dependence of
{r}_{s}
and d on the cosmological parameters h,
{\Omega }_{b}{h}^{2}
\sum \text{ }{m}_{\nu }
in limited ranges of interest. Integrating the comoving sound speed of the photon-baryon-electron plasma until
{a}_{\text{dec}}=1/\left(1+{z}_{\text{dec}}\right)
{z}_{\text{dec}}=1089.9\pm 0.4
[2] we obtain
{r}_{s}\approx 0.0339\times A\times {\left(\frac{0.28}{{\Omega }_{m}}\right)}^{0.24}
A\approx 0.990+0.007\cdot \delta h-0.001\cdot \delta b+0.020\cdot \frac{\sum \text{ }{m}_{\nu }}{1\text{\hspace{0.17em}}\text{eV}},
\delta h\equiv \left(h-0.678\right)/0.009,
\delta b\equiv \left({\Omega }_{b}{h}^{2}-0.02226\right)/0.00023.
To obtain d we minimize the
{\chi }^{2}
with 21 terms, corresponding to the 21 BAO observables, with respect to
{\Omega }_{\text{DE}}
and d, and obtain
{\Omega }_{\text{DE}}=0.718\pm 0.003
d\approx 0.0340\pm \mathrm{0.0002,}
{\chi }^{2}
per degree of freedom 19.8/19, and correlation coefficient 0.989 between
{\Omega }_{\text{DE}}
and d (this high correlation coefficient is due to the high precision of
{\theta }_{\text{MC}}
). Setting
{r}_{s}=d
{\displaystyle \sum }\text{\hspace{0.05em}}{m}_{\nu }\approx 0.73-0.35\cdot \delta h+0.05\cdot \delta b\pm 0.15\text{\hspace{0.17em}}\text{eV}\mathrm{.}
A more precise result is obtained with a global fit by minimizing the
{\chi }^{2}
with 21 terms varying
{\Omega }_{\text{DE}}
\sum \text{ }{m}_{\nu }
directly. We obtain
{\Omega }_{\text{DE}}=0.7175\pm 0.0023
{\displaystyle \sum }\text{\hspace{0.05em}}{m}_{\nu }=0.711-0.335\cdot \delta h+0.050\cdot \delta b\pm 0.063\text{\hspace{0.17em}}\text{eV},
{\chi }^{2}/\text{d}\text{.f}\text{.}=19.9/19
, and correlation coefficient 0.924 between
{\Omega }_{\text{DE}}
\sum \text{ }{m}_{\nu }
. This is our main result. Equation (9) is obtained from BAO measurements alone, and is written in a way that can be combined with independent constraints on the cosmological parameters
\sum \text{ }{m}_{\nu }
h
{\Omega }_{b}{h}^{2}
, such as measurements of the power spectrum of density fluctuations
P\left(k\right)
, the CMB, and direct measurements of the Hubble parameter.
\delta h=\pm 1
\delta b=\pm 1
we obtain the following upper bound on the mass of active neutrinos
{m}_{\nu }=\frac{1}{3}\sum \text{ }{m}_{\nu }
{m}_{\nu }<0.43\text{\hspace{0.17em}}\text{eV}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}95%\text{\hspace{0.17em}}\text{confidence}\text{.}
Appendix 1. Removing constraints
{\Omega }_{k}
and keeping
{\Omega }_{\text{DE}}
constant we obtain
{\Omega }_{k}=-0.003\pm 0.006
{\Omega }_{\text{DE}}+2.2{\Omega }_{k}=0.719\pm 0.003
{\displaystyle \sum }\text{\hspace{0.05em}}{m}_{\nu }=0.623-0.334\cdot \delta h+0.050\cdot \delta b\pm 0.191\text{\hspace{0.17em}}\text{eV}\mathrm{,}
{\chi }^{2}/\text{d}\text{.f}\text{.}=19.6/18
{\Omega }_{k}=0
{\Omega }_{\text{DE}}\left(a\right)={\Omega }_{\text{DE}}\cdot \left\{1+{w}_{a}\cdot \left(1-a\right)\right\}
{\Omega }_{\text{DE}}=0.716\pm 0.004
{w}_{a}=0.064\pm 0.148
{\displaystyle \sum }\text{\hspace{0.05em}}{m}_{\nu }=0.603-0.349\cdot \delta h+0.052\cdot \delta b\pm 0.257\text{\hspace{0.17em}}\text{eV},
{\chi }^{2}/\text{d}\text{.f}\text{.}=19.7/18
{\Omega }_{k}
{\Omega }_{\text{DE}}\left(a\right)={\Omega }_{\text{DE}}\cdot \left\{1+{w}_{a}\cdot \left(1-a\right)\right\}
{\Omega }_{k}=-0.008\pm 0.004
{\Omega }_{\text{DE}}+2.2{\Omega }_{k}=0.718\pm 0.004
{w}_{a}=0.227\pm 0.069
0<{\displaystyle \sum }\text{\hspace{0.05em}}{m}_{\nu }=-0.388-0.350\cdot \delta h+0.050\cdot \delta b\pm 0.830\text{\hspace{0.17em}}\text{eV},
{\chi }^{2}/\text{d}\text{.f}\text{.}=17.8/17
Appendix 2. Removing data.
In this Appendix we apply the constraints
{\Omega }_{k}=0
{\Omega }_{\text{DE}}
constant. Removing the measurement of
{\theta }_{\text{MC}}
{\Omega }_{\text{DE}}=0.722\pm 0.011
{\displaystyle \sum }\text{\hspace{0.05em}}{m}_{\nu }=0.579-0.333\cdot \delta h+0.049\cdot \delta b\pm 0.285\text{\hspace{0.17em}}\text{eV},
{\chi }^{2}/\text{d}\text{.f}\text{.}=19.7/18
Removing the measurement of
{\theta }_{\text{MC}}
and the two Lya measurements we obtain
{\Omega }_{\text{DE}}=0.716\pm 0.014
{\displaystyle \sum }\text{\hspace{0.05em}}{m}_{\nu }=0.743-0.330\cdot \delta h+0.049\cdot \delta b\pm 0.366\text{\hspace{0.17em}}\text{eV},
{\chi }^{2}/\text{d}\text{.f}\text{.}=11.2/16
Keeping only the measurement of
{\theta }_{\text{MC}}
we need to fix
{\Omega }_{\text{DE}}
in order to get zero degrees of freedom and have a unique solution. The best way to fix
{\Omega }_{\text{DE}}
is with BAO measurements, and that is the purpose of the present study.
[1] Hoeneisen, B. (2017) Study of Baryon Acoustic Oscillations with SDSS DR13 Data and Measurements of and . International Journal of Astronomy and Astrophysics, 7, 11-27.
[2] Patrignani, C., et al. (2016) Review of Particle Physics. Chinese Physics C, 40, Article ID: 100001.
[3] Albareti, F.D., et al. (2016) The Thirteenth Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the SDSS-IV Survey Mapping Nearby Galaxies at Apache Point Observatory. SDSS Collaboration, arXiv:1608.02013.
[4] Dawson, K.S., Schlegel, D.J., Ahn, C.P., et al. (2013) The Baryon Oscillation Spectroscopic Survey of SDSS-III. Astronomical Journal, 145, 10.
[5] Dawson, K.S., Kneib, J.-P., Percival, W.J., et al. (2016) The SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Overview and Early Data. Astronomical Journal, 151, 44.
[6] Font-Ribera, A., et al. (2014) Quasar-Lyman α Forest Cross-Correlation from BOSS DR11: Baryon Acoustic Oscillations. arXiv:1311.1767.
[7] Delubac, T., et al. (2014) Baryon Acoustic Oscillations in the Lyα Forest of BOSS DR11 Quasars. arXiv:1404.1801v2.
[8] Planck Collaboration (2015) Results XIII, Astron. & Astrophys. Submitted, arXiv:1502.01589v2.
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Simulating Interest Rates - MATLAB & Simulink - MathWorks Australia
Ensuring Positive Interest Rates
All simulation methods require that you specify a time grid by specifying the number of periods (NPeriods). You can also optionally specify a scalar or vector of strictly positive time increments (DeltaTime) and intermediate time steps (NSteps). These parameters, along with an initial sample time associated with the object (StartTime), uniquely determine the sequence of times at which the state vector is sampled. Thus, simulation methods allow you to traverse the time grid from beginning to end (that is, from left to right).
In contrast, interpolation methods allow you to traverse the time grid in any order, allowing both forward and backward movements in time. They allow you to specify a vector of interpolation times whose elements do not have to be unique.
Many references define the Brownian Bridge as a conditional simulation combined with a scheme for traversing the time grid, effectively merging two distinct algorithms. In contrast, the interpolation method offered here provides additional flexibility by intentionally separating the algorithms. In this method for moving about a time grid, you perform an initial Monte Carlo simulation to sample the state at the terminal time, and then successively sample intermediate states by stochastic interpolation. The first few samples determine the overall behavior of the paths, while later samples progressively refine the structure. Such algorithms are often called variance reduction techniques. This algorithm is simple when the number of interpolation times is a power of 2. In this case, each interpolation falls midway between two known states, refining the interpolation using a method like bisection. This example highlights the flexibility of refined interpolation by implementing this power-of-two algorithm.
Load the data. Load a historical data set of three-month Euribor rates, observed daily, and corresponding trading dates spanning the time interval from February 7, 2001 through April 24, 2006:
plot(dates, 100 * Dataset.EB3M)
datetick('x'), xlabel('Date'), ylabel('Daily Yield (%)')
title('3-Month Euribor as a Daily Effective Yield')
Fit a model to the data. Now fit a simple univariate Vasicek model to the daily equivalent yields of the three-month Euribor data:
d{X}_{t}=S\left(L-{X}_{t}\right)dt+\sigma d{W}_{t}
Given initial conditions, the distribution of the short rate at some time T in the future is Gaussian with mean:
E\left({X}_{T}\right)={X}_{0}{e}^{-ST}+L\left(1-{e}^{-ST}\right)
Var\left({X}_{T}\right)={\sigma }^{2}\left(1-{e}^{-ST}\right)/2S
To calibrate this simple short-rate model, rewrite it in more familiar regression format:
{y}_{t}=\alpha +\beta {x}_{t}+{\epsilon }_{t}
{y}_{t}=d{X}_{t},\alpha =SLdt,\beta =-Sdt
perform an ordinary linear regression where the model volatility is proportional to the standard error of the residuals:
\sigma =\sqrt{Var\left({\epsilon }_{t}\right)/dt}
yields = Dataset.EB3M;
regressors = [ones(length(yields) - 1, 1) yields(1:end-1)];
[coefficients, intervals, residuals] = ...
regress(diff(yields), regressors);
dt = 1; % time increment = 1 day
speed = -coefficients(2)/dt;
level = -coefficients(1)/coefficients(2);
sigma = std(residuals)/sqrt(dt);
Create an object and set its initial StartState. Create an hwv object with StartState set to the most recently observed short rate:
obj = hwv(speed, level, sigma, 'StartState', yields(end))
StartState: 7.70408e-05
Sigma: 4.77637e-07
Level: 6.00424e-05
Speed: 0.00228854
Simulate the fitted model. Assume, for example, that you simulate the fitted model over 64 (26) trading days, using a refined Brownian bridge with the power-of-two algorithm instead of the usual beginning-to-end Monte Carlo simulation approach. Furthermore, assume that the initial time and state coincide with those of the last available observation of the historical data, and that the terminal state is the expected value of the Vasicek model 64 days into the future. In this case, you can assess the behavior of various paths that all share the same initial and terminal states, perhaps to support pricing path-dependent interest rate options over a three-month interval.
Create a vector of interpolation times to traverse the time grid by moving both forward and backward in time. Specifically, the first interpolation time is set to the most recent short-rate observation time, the second interpolation time is set to the terminal time, and subsequent interpolation times successively sample intermediate states:
T = 64;
times = (1:T)';
t = NaN(length(times) + 1, 1);
t(1) = obj.StartTime;
t(2) = T;
delta = T;
jMax = 1;
for k = 1:log2(T)
i = delta / 2;
for j = 1:jMax
t(iCount) = times(i);
i = i + delta;
iCount = iCount + 1;
jMax = 2 * jMax;
Plot the interpolation times. Examine the sequence of interpolation times generated by this algorithm:
stem(1:length(t), t, 'filled')
xlabel('Index'), ylabel('Interpolation Time (Days)')
title ('Sampling Scheme for the Power-of-Two Algorithm')
The first few samples are widely separated in time and determine the course structure of the paths. Later samples are closely spaced and progressively refine the detailed structure.
Initialize the time series grid. Now that you have generated the sequence of interpolation times, initialize a course time series grid to begin the interpolation. The sampling process begins at the last observed time and state taken from the historical short-rate series, and ends 64 days into the future at the expected value of the Vasicek model derived from the calibrated parameters:
average = obj.StartState * exp(-speed * T) + level * ...
(1 - exp(-speed * T));
X = [obj.StartState ; average];
Generate five sample paths. Generate five sample paths, setting the Refine input flag to TRUE to insert each new interpolated state into the time series grid as it becomes available. Perform interpolation on a trial-by-trial basis. Because the input time series X has five trials (where each page of the three-dimensional time series represents an independent trial), the interpolated output series Y also has five pages:
nTrials = 5;
Y = obj.interpolate(t, X(:,:,ones(nTrials,1)), ...
'Times',[obj.StartTime T], 'Refine', true);
Plot the resulting sample paths. Because the interpolation times do not monotonically increase, sort the times and reorder the corresponding short rates:
[t,i] = sort(t);
Y = squeeze(Y);
Y = Y(i,:);
plot(t, 100 * Y), hold('on')
plot(t([1 end]), 100 * Y([1 end],1),'. black','MarkerSize',20)
xlabel('Interpolation Time (Days into the Future)')
ylabel('Yield (%)'), hold('off')
title ('Euribor Yields from Brownian Bridge Interpolation')
The short rates in this plot represent alternative sample paths that share the same initial and terminal values. They illustrate a special, though simplistic, case of a broader sampling technique known as stratified sampling. For a more sophisticated example of stratified sampling, see Stratified Sampling.
Although this simple example simulated a univariate Vasicek interest rate model, it applies to problems of any dimensionality.
Brownian-bridge methods also apply more general variance-reduction techniques. For more information, see Stratified Sampling.
All simulation and interpolation methods allow you to specify a sequence of functions, or background processes, to evaluate at the end of every sample time period. This period includes any intermediate time steps determined by the optional NSteps input, as discussed in Optimizing Accuracy: About Solution Precision and Error. These functions are specified as callable functions of time and state, and must return an updated state vector Xt:
{X}_{t}=f\left(t,{X}_{t}\right)
You must specify multiple processing functions as a cell array of functions. These functions are invoked in the order in which they appear in the cell array.
Processing functions are not required to use time (t) or state (Xt). They are also not required to update or change the input state vector. In fact, simulation and interpolation methods have no knowledge of any implementation details, and in this respect, they only adhere to a published interface.
Such processing functions provide a powerful modeling tool that can solve various problems. Such functions allow you to, for example, specify boundary conditions, accumulate statistics, plot graphs, and price path-dependent options.
Except for Brownian motion (BM) models, the individual components of the simulated state vector typically represent variables whose real-world counterparts are inherently positive quantities, such as asset prices or interest rates. However, by default, most of the simulation and interpolation methods provided here model the transition between successive sample times as a scaled (possibly multivariate) Gaussian draw. So, when approximating a continuous-time process in discrete time, the state vector may not remain positive. The only exception is simBySolution for gbm objects and simBySolution for hwv objects, a logarithmic transform of separable geometric Brownian motion models. Moreover, by default, none of the simulation and interpolation methods make adjustments to the state vector. Therefore, you are responsible for ensuring that all components of the state vector remain positive as appropriate.
Fortunately, specifying nonnegative states ensures a simple end-of-period processing adjustment. Although this adjustment is widely applicable, it is revealing when applied to a univariate cir square-root diffusion model:
d{X}_{t}=0.25\left(0.1-{X}_{t}\right)dt+0.2{X}_{t}^{\frac{1}{2}}d{W}_{t}=S\left(L-{X}_{t}\right)dt+\sigma {X}_{t}^{\frac{1}{2}}d{W}_{t}
Perhaps the primary appeal of univariate cir models where:
2SL\ge {\sigma }^{2}
is that the short rate remains positive. However, the positivity of short rates only holds for the underlying continuous-time model.
Simulate daily short rates of the cir model. To illustrate the latter statement, simulate daily short rates of the cir model, using cir, over one calendar year (approximately 250 trading days):
rng(14151617,'twister')
obj = cir(0.25,@(t,X)0.1,0.2,'StartState',0.02);
[X,T] = simByEuler(obj,250,'DeltaTime',1/250,'nTrials',5);
sprintf('%0.4f\t%0.4f+i%0.4f\n',[T(195:205)';...
real(X(195:205,1,4))'; imag(X(195:205,1,4))'])
'0.7760 0.0003+i0.0000
0.7800 0.0004+i0.0000
0.7880 -0.0000+i0.0000
Interest rates can become negative if the resulting paths are simulated in discrete time. Moreover, since cir models incorporate a square root diffusion term, the short rates might even become complex.
Repeat the simulation with a processing function. Repeat the simulation, this time specifying a processing function that takes the absolute magnitude of the short rate at the end of each period. You can access the processing function by time and state (t, Xt), but it only uses the state vector of short rates Xt:
[Y,T] = simByEuler(obj,250,'DeltaTime',1/250,...
'nTrials',5,'Processes',@(t,X)abs(X));
Compare the adjusted and non-adjusted paths. Graphically compare the magnitude of the unadjusted path (with negative and complex numbers!) to the corresponding path kept positive by using an end-of-period processing function over the time span of interest:
plot(T,100*abs(X(:,1,4)),'red',T,100*Y(:,1,4),'blue')
axis([0.75 1 0 0.4])
xlabel('Time (Years)'), ylabel('Short Rate (%)')
title('Univariate CIR Short Rates')
legend({'Negative/Complex Rates' 'Positive Rates'}, ...
You can use this method to obtain more accurate SDE solutions. For more information, see Performance Considerations.
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Context Panel - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 2018 : Context Panel
Clickable Math: Intelligent Context Panel
The new Context Panel on the right-hand side of the Maple interface is an important expansion of Maple’s Clickable Math capabilities. The Context Panel replaces and enhances the context-sensitive menus, which were accessed through the right-click menu, and offers a more discoverable way of exploring Maple's functionality.
When you click on a mathematical expression, Maple analyzes your expression and then presents you with a list of the most relevant operations and tools. Options could include solving for x, plotting your expression, finding the determinant, converting from one unit to another, applying a Fourier transform, integrating with respect to t, changing the numeric formatting in your result, calculating the average of your data, and much more, all depending on what makes sense. Simply selecting one of these options performs the operation. In addition, you can also use the context panel to customize the appearance of plots, change the properties of tables, and more. No knowledge of Maple syntax or Maple commands is required.
Context Panel operations are automatically documented, so you have a record of all your steps, from the original equation to the final result.
{a}^{3}{x}^{2} + 2 x b + 12
\stackrel{\text{differentiate w.r.t. x}}{\to }
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}
\stackrel{\text{solve for x}}{\to }
[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{b}}{{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{3}}}]]
\frac{{ⅆ}^{2}}{{ⅆ}^{\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}}{t}^{2}}f\left(t\right)+\frac{ⅆ}{ⅆ t}f\left(t\right)=\mathrm{sin}\left(t\right)
\frac{{s}^{}}{{s}^{2}+2 s+1}
\left[\begin{array}{ccc}27& 99& 92\\ 8& 29& -31\\ 69& 44& 67\end{array}\right]
\mathrm{AudioTools}:-\mathrm{Read}\left(\mathrm{FileTools}:-\mathrm{JoinPath}\left(\left[\mathrm{kernelopts}\left(\mathrm{datadir}\right), "audio","MapleSimMono11025.wav"\right]\right)\right)
\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{1 .. 49664}}\textcolor[rgb]{0,0,1}{\mathrm{Array}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type:}}{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Storage:}}\textcolor[rgb]{0,0,1}{\mathrm{rectangular}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Order:}}\textcolor[rgb]{0,0,1}{\mathrm{C_order}}\end{array}\right]
If you click the result below, you'll see tools for converting units and applying numeric formatting.
2.9873\cdot {10}^{3} ⟦\frac{J}{\mathrm{kg} K}⟧+3.7521\cdot {10}^{3} ⟦\frac{J}{\mathrm{kg} K}⟧
\textcolor[rgb]{0,0,1}{6.739400000}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{\mathrm{kJ}}}{\textcolor[rgb]{0,0,1}{\mathrm{kg}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{K}}⟧
This enhances Maple 2018's new tools for numerically solving and optimizing equations with units.
The Context Panel shows interactive popup options for certain types of expressions. Click the expression below to apply different trig identities.
\mathrm{sin}\left(2\mathrm{\theta }\right)
In addition, more operations are now available:
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Bayes' Theorem and Natural Frequencies - Blog | Julio Marquez
Bayes' Theorem and Natural Frequencies
Probability, risk and uncertainty are concepts that must be understood not only in the geosciences, but in life.
Probability, risk, and uncertainty are concepts that must be understood not only in the geosciences but in life. The way we process information to make decisions relies on many times in the reduction of quantitative uncertainty. The more information we gather, the better the improvement of our prediction capabilities. This information is expressed as discrete probabilities or as probability density functions. In this post I intend to clarify through an example the concepts of joint and conditional probability and their relationship with Bayes Theorem. Two approaches will be presented to help you get the ideas: a probabilistic one and a frequentist one.
Before we see the example let's review the definitions
Joint Probability (
P(X,Y)
P(X \cap Y)
): a statistical measure where the likelihood of two events occurring together and at the same point in time are calculated.
P(X \cap Y)=P(X).P(Y)
P(X \cap Y)=P(X).P(Y/X)
Conditional Probability (
P(X | Y)
): the probability that an event will occur, when another event is known to occur or have occurred.
P(X | Y) = P(X \cap Y)/P(Y); P(Y) > 0
A joint probability is a measure of the intersection of two conditions, whereas conditional probability is the joint probability of two conditions normalized by the probability of the conditional event. So far, ¡so theoretic and boring!
Let's get to the example. This example was taken from a New York Times article by Steven Strogatz and has to do with the estimation of risk and the limits of certainty.
We have to estimate the probability that a woman with a positive mammogram in a very low-risk group, with no symptoms or family history of breast cancer has breast cancer.
Once we started the problem, we need information, don't we? Here it is in the form of discrete probabilities or probabilistic facts:
The probability that a woman in this group has breast cancer is 0.8% (yes, less than 1 %).
In a group of women with breast cancer, the mammogram positive detection has a certain probability of 90% (ninety percent of the time, the mammogram gets it right).
In a group of healthy women, the probability of mammograms giving a positive detection is 7% (These are the false positives of the mammogram test, it's not perfect! Everything has its uncertainty).
That's it! If a woman gets a positive mammogram, what is the probability that she has breast cancer?.
This is the probabilistic approach promised in the introduction. The idea is to take advantage of Bayes theorem, which is a simple yet powerful equation relating to the conditional probabilities of two events. The theorem looks like this:
P(X | Y)=P(Y | X).P(X)/P(Y)
How do we adapt this theorem to our problem? That's the part that freaks everybody out! You feel like taking an exam, start sweating and don't know how to start (I have been there, I am there almost all of the time!).
First we have to write the given facts in this fancy “statistical” language of conditional probabilities. So:
P(C+) = 0.8\%
(Probability of having breast cancer in this low-risk group).
P(+m | C+) = 90\%
(Probability of positive mammogram, given that a woman has breast cancer).
P(+m | C-) = 7\%
(Probability of positive mammogram, given that a woman is healthy–has no cancer).
We want to know
P(C+ | +m)
, which is the probability of having breast cancer, given a positive mammogram.
I hope you are getting the differences here, especially between these two:
P(+m | C+)
P(C+ | +m)
. Both conditional probabilities ask, so to speak, different questions.
P(+m | C+)
: What is the probability of having a positive mammogram if a patient has breast cancer already?
P(C+ | +m)
: What is the probability of having breast cancer if a patient has a positive mammogram?
What is awesome about Bayes´ theorem is how those two different questions relate to each other.
I need a little visual help here. So I prepared a diagram showing the relationships above:
The diagram is not to scale but illustrates pretty well the dependencies and meaning of the probabilities. Conditional probabilities depend on a previous event and Joint probabilities are like areas that measure the uncertainty of an outcome. The bigger the area, the more probable that the outcome becomes.
Bayes' theorem boils down to this:
P(+C | m+)=P(+m | C+).P(+C)/P(m+)
We already have the probabilities of the numerator (initial facts) and the denominator, which measures the probability of a positive mammogram as an outcome, is the sum or union of the two blue areas (the joint probabilities of getting a positive mammogram together, either having breast cancer or not). So the final formula is:
P(+C | m+)=P(+m | C+).P(+C)/(P(C+).P(m+ | C+)+(1-P(C+)).P(+m | C-))
By doing the substitution,
P(+C | m+)
is around 9.39%. Problem solved!
This probabilistic approach has an incredible ability to be forgotten by students and practitioners alike because most of the time we see equations and theorems as magical objects. This “magical objects” were developed or conceived to concentrate the mathematical relationships between variables in a compact way, but not as substitutes for logic and common sense. Let's see now how logical thinking and frequencies can help us solve the same problem.
Here we have to write the initial facts as frequencies or number of events in a population. Let's do it:
8 out of 1000 women in the low-risk group have breast cancer.
9 out of 10 mammograms get it right with women with breast cancer.
In 7 women out of 100 healthy ones, the mammogram gets it wrong (7 false positives in 100 trials on healthy women).
In a population of 1000 women in this low-risk group, 8 will have breast cancer (Figure 2a). We need to know now how many of these healthy and non-healthy women will have a positive mammogram. In the group of 8 women there is a 90% percent chance of getting a positive mammogram, so 7 out of those 8 will have a positive mammogram (
P(+m | C+)=8*9/10
or around 7 women). In the remaining 992 healthy women, 7 percent of them will have a positive mammogram, this means (
P(+m | C-)=992*7/100
or around 69). How many women will have a positive mammogram, either healthy or not? The answer is 7+69 or 76 (the union or sum of the joint probabilities or the blue circle in Figure 2b).
If we want to know
P(+C | +m)
, we have to compare the number of women with cancer that gets positively tested (
P(+C,+m)
P(C+** \cap **+m)
) against the whole population of positive detections and false positives (the crazy denominator of the Bayesian approach).
P(+C | +m)=7/76
(Seven women out of seventy-six will have breast cancer and positive mammogram). That is 9.21% since we have rounded up to integers.
Both approaches do the same thing but depending on your way of thinking, one will be easier than the other. What is important to understand is the concept of uncertainty and how it surrounds everything, from medical tests like mammograms, HIV-tests, and even DNA recognition to the weather and games of chance. I hope you have gotten a better insight into conditional and joint probability after going through the example.
If you are interested in uncertainty I recommend the book of Gerd Gigerenzer “Reckoning with Risk: Learning to Live with Uncertainty” and as an extra recommendation this one from Nate Silver “The Signal and the Noise: Why So Many Predictions Fail-but Some Don't”. The article by Steven Strogatz can be read here: http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/.
Originally published at latraza.wordpress.com on September 18, 2017.
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Noise level of reverse motion gearing of gearbox | JVE Journals
Tomáš Oudrnický1 , Elias Tomeh2
1, 2Technical University in Liberec, Liberec, Czech Republic
Quality requirements in the automotive industry are currently at an extremely high level; therefore, it is necessary to make various complex measurements, testing or observation of all important components of a vehicle whose integral part is a driving gear – a combustion engine and a gearbox. These components produce vibrations and noise, which means that it is produced by every current user of a passenger car. Particularly in a small-city-cars segment where such a big emphasis is not put on sound absorption of the cab for the crew. When taking into consideration this class of vehicles, it is important to prevent vibrations formation itself. These vibrations not only cause the unpleasant noise, but they also affect the durability of the surrounding components.
Keywords: noise level, vibrations, gearbox, gearing.
1. Sources of noise of cogged gearboxes
Each automobile gearbox which uses gearings for changing the gear ratio produces a certain noise level (Fig. 1). These days, the noise level of a gearbox is one of the important aspects of evaluation of the whole vehicle. It is a parameter which can be subjectively distinguished by every customer without any expertise. The noise level generally belongs among the most commonly occurring negative impacts on the product quality, not only in engineering. Vibration indication of an automobile gearbox is closely connected with the quality of the designed construction and the production accuracy.
The noise can be described quantitatively or qualitatively. Quantitative variables are those that can be measured (such as intensity, frequency, rhythm, etc.) and are defined by numeric values. Qualitative variables include subjective evaluation and are defined by different standards and other prescribed methodologies [1, 2].
Fig. 1. Sources of noise in a gearbox [2]
1.1. Noise from the cogwheels grip
The main general reason of vibrations excitation from the gearing is the dynamic force which can be variable against its amplitude or place of action. The grip of the cogwheels does not consist only of the rolling itself, but also of spin. Therefore, the other frequent reason which was attached a great importance to in the past is vibrations excitation because of the existence of sliding speed. The vibrations from the grip are excited owing to the frequent changes of the sense of this sliding speed. Friction force dynamically changes when the grip is taking place. At first, it increases in one direction and after the rolling point is overcome – the shear force changes its sense – its size grows again. This feature is significant especially when taking into consideration the gear with a direct gearing.
In today's automobile gearboxes, gear sets with angular cogs which are at constant grip are exclusively used. When taking into consideration these gear sets, the vibrations which are produced owing to the influences of impacts from the cogs clearance of the free wheel are excited. By clearance it is meant a wheel placed on the shaft in a pivoted way. These impacts are also accompanied by impacts from the drive-axle assembly, so called “Disturbing noise” where the shaft is the subject to torsional oscillation. Therefore, flywheels are used to decrease the influences of these torsional oscillations [1, 2].
Parametric gearing excitation is directly influenced by the size of the coefficient of the grip duration ε. The advantage is that if this coefficient has an integral value (for example 2), a great change of grip rigidity will not appear because exactly two pairs of cogs will be in the grip all the time.
When taking into consideration the gearing of the automobile gearbox, a great attention should be paid to the constant gear – gearing – which connects the driven shaft of the gearbox with the differential gear. It occurs because this gear is loaded with torque whenever there is any change of gears. Therefore, its quality and service life should equate with it. The other gears which are currently without any load are affected only by passive resistances.
So, called tractive cog side and backward cog side are distinguished on each cogwheel. It means the side of the cog which is in gear when the vehicle is in traction, respectively when braking by the engine. Both sides can have differently defined parameters of gearing (for example
{f}_{h\alpha }
{f}_{h\beta }
{C}_{\alpha }
{C}_{\beta }
) because of the optimization of the starting conditions and decreasing the noise level of given gear both in traction and when braking by the engine.
The noise which appears when the cogwheels are in the grip is a significant source of noise in the whole gearbox. The gearing has a broad impact on the overall noise level of the gearbox. The following causes are presented in the production: tool geometry → tool adjustment in the machine → product (cogwheel) geometry → gearbox noise. When discovering that the noise level of the gearbox is above the set limit, the reverse process is analysed. For a correct reverse analysis, it is necessary to work correctly with the measured data. It is especially necessary to observe the tool, machine, material melting and hardening furnace which are currently being used. Today’s automobile gearboxes use cogwheels with angular gearing which in practice cause approximately 5 dB lower noise than a direct gearing.
1.2. Other sources of vibrations in a gearbox
Among other parts which can cause noise most often belong different parts connected with changing gears (synchronizer rings, clutches, shift forks etc.)
Shifting mechanism belongs among the indirect sources of vibrations. If the shift fork stays in contact with a cogwheel, the vibrations will be transferred from the wheel into the whole shifting mechanism. Shafts transferring the torque are subjects to strain not only on torsion but considerably also on bending – their distortion, deflection and badly secured parallelism are also very often a reason of vibrations occurrence.
All those sources of noise transfer the created vibrations into the gearbox housing. Because a gearbox is a closed acoustic system, the gearbox noise is spread mostly by vibrations of the housing surface or by vibrations of connected components into the surroundings. Therefore, it also has a significant influence on the overall noise level of the gearbox. The value of the gearbox housing oscillation itself should not correspond with the discreet components of the gearing frequency, bearings or other parts which can be sources of dynamic excitation [1, 2].
2. Manifestations of noise level causes in vibration spectra
Each source of noise in a gearbox has its specific manifestation in so called vibration spectra. For identifying the cause and the source of origin, different methods of technical diagnostics are used.
The spectrum can possibly be displayed as related to the individual shafts in a gearbox (SK-1, SK-2) or to the differential gear (SK-3), which enables closer determination of the source of the adverse noise. The “MIX” spectrum of the overall vibrations displays the sum of the three previously mentioned channels. Next, the spectra are marked as “S” (engine in acceleration) or “B” (engine braking).
2.1. Manifestations of gearbox defects in vibration spectra
Cogwheels produce combined spectrum of vibrations. Above the continuous part of the spectrum, individual discreet (harmonic) frequency components are highlighted. These components correspond with the frequency of grip of individual cogs – so called gear mesh frequency (
{f}_{Z}
). In the Fig. 2, it is possible to see manifestations of basic defects of cogwheels in vibration spectra.
Fig. 2. Typical examples of noise from a cogwheel [3]
Involute (or generally gearing) wearing becomes evident by increasing the first (
{f}_{Z}
) and the second harmonic gear mesh frequency (2×
{f}_{Z}
). Higher orders of the harmonic frequency are caused by an advanced stage of damage or by another specific defect.
Anti-friction bearings become evident – in vibration spectra – by so called bearing frequency (
{f}_{L}
). Its size depends on the geometry – bearing proportions, size of operating clearance and number of rolling bodies. We distinguish four bearing frequencies: outer ring frequency, inner ring frequency, cage frequency and rolling elements frequency.
Other sources of vibrations have their specific manifestations. For example, misalignment of shafts becomes evident by sidebands around gear mesh frequencies of gearing. Sideband frequencies are distant from
{f}_{Z}
from each other about the value of a rotor frequency (
{f}_{R}
) which is given by shaft frequency.
In the automobile gearbox there are more gearings which can – in spectra – also become evident, for example reverse motion gear set (which is the case of the gearbox).
3. Noise level of reverse motion
The reverse motion in a gearbox is realised by use of three cogwheels system. These cogwheels are equipped with straight conical cogs. From the motor, the torque is transferred through gearing on the drive shaft into the idle wheel of the reverse motion which is placed on an independent, stationary shaft. This gearing is in a constant grip. The last cogwheel of the reverse motion mechanics is a wheel which – when changing the gear – transfers the torque into the driven shaft of the gearbox and then through the driving system. This gearing, as the only one in a gearbox, is not in a constant grip.
The reverse gear is not provided with synchronization and the last mentioned gearing on the driven shaft is a part of the synchronizer sleeve of the fifth gear.
The interesting thing about this gear system is that the idle wheel of the reverse motion (green) is wider so that it could at the same time be in grip with two different cogwheels. Along its width there are two different segments of conical gearing (Fig. 3). The first segment of this wheel, with the angle
\beta =
0°, is for the grip with the driving gearing on a shaft (blue). The second segment, with the angle
\beta =
1°, is for the grip with the driven gearing on the clutch (red). This geometry results in pulling the wheel into the grip so that the “jumping-out” of the speed is avoided. This compound gearing is made by milling by use of an only tool [4].
Fig. 3. Reverse motion gear [5]
3.1. Comparison of a silent and a noisy gearbox
For verifying the noise level of gearing, two tests were performed. During both tests, the same shaft and the same sliding clutch were used. Only the idle wheel was changed.
The green curve shows a silent gearing and the blue one shows a noisy gearing. The red curve shows the boundary values of the noise level.
Fig. 4. Reverse motion gear set [1]
3.2. Difference on idle wheel
Fig. 5. Comparison of a trace after rolling on an idle wheel
Fig. 6. Comparison of surface deviation
Fig. 7. Comparison of surface roughness
a) i.O Rz radial/axial
b) n.i.O Rz radial/axial
The idle wheel is quite a complex product because it consists of two different segments. Considering that, the wheel inclines to have production problems.
The protocol on gearing measurement was all right in both cases. It did not show any differences. Therefore, the analysis of other geometric proportions for discovering the cause of the increased noise level was performed.
Fig. 8. Comparison of front runout value against its average
It appears from the metrological analysis of the idle wheel that even the smallest geometric deviations on the cogwheel can have a negative impact on the production of vibrations of gearing.
The noise level was characterised by a warble tone, which, in my opinion, happens because of the increased value of the front runout on the NIO part or also because of the increased value of roughness
{R}_{z}
After changing the cogwheel, the harmonic frequencies which correspond with the given gear set were decreased, the amplitude modulation was taken off and also the overall noise level was decreased. The manifestation corresponds with the presumption in accordance with the Fig. 2.
Oudrnický T. Analysis of Causes of Noise in the Gearbox. Bachelor Thesis, Technical University in Liberec, 2014. [Search CrossRef]
Tomeh E. Vibration and Noise of Automobile Gear-Box in Connection with Identified Defects on Machine Tools. Technical university of Liberec, Liberec, 2008. [Search CrossRef]
Documents of Škoda Auto a.s., Mladá Boleslav, (Internal Data Source). [Search CrossRef]
Škoda Service, 5-speed Manual Transmission 0CF and Automated Five-Speed ASG Transmission. Workshop Teaching Aid No. 93, 2012. [Search CrossRef]
Tomeh E. Identify the sources of vibration and noise on cars gearbox by spectral analysis. 54th International Conference of Machine Design Departments, 2013. [Search CrossRef]
Oudrnický T. Constructional Modification of Gearbox for Noise Reduce. Diploma Thesis. Technical University of Liberec, Liberec, 2016. [Search CrossRef]
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Spur or planetary active differential gear - Simulink - MathWorks India
Active differential type
Clutch 1 to differential case gear ratio, Ns1
Three-gang gear inertia, Jgc
Axle 1 planetary carrier to axle gear ratio, Np1
Axle 1 sun gear inertia, Js1
Axle 1 carrier inertia, Jc1
Axle 1 ring inertia, Jr1
Effective applied pressure area
Friction coefficient vector, mu
Torque - slip speed matrix, TdPdw
Clutch pressure vector, pT
Spur or planetary active differential gear
The Active Differential block implements an active differential to account for the power transfer from the transmission to the axles. The block models the active differential as an open differential coupled to either a spur or planetary differential gear set. The block uses external pressure signals to regulate the clutch pressure to either speed up or slow down each axle rotation.
Use the block in hardware-in-the-loop (HIL) and optimization workflows to dynamically couple the driveshaft to the wheel axles when you want to direct the transmission torque to a specific axle. For detailed front wheel driving studies, use the block to couple the driveshaft to universal joints. The block is suitable to use in system-level closed-loop control studies, for example, yaw stability and torque vectoring. All the parameters are tunable.
To specify the active differential, open the Active Differential parameters and specify Active differential type.
Spur gears, superposition clutches
Clutches are in superposition through a three-gang gear system and a differential case
Double planetary gears, stationary clutches
Clutches are fixed to the carrier and axles through double planetary gear sets
Use the Open Differential parameter Crown wheel (ring gear) located to specify the open differential location, either to the left or right of the center-line.
Depending on the available data, to specify the method to couple the different torques applied to the axles, use the Slip Coupling parameter Coupling type.
Torque modeled as a dry clutch with constant friction coefficients
Slip speed dependent torque data
Torque determined from a lookup table that is a function of slip-speed and clutch pressure
The Active Differential block does not include a controller or external clutch actuator dynamics. Use this information to control the input clutch pressure. The info bus contains the slip speeds at clutch 1, Δωcl1, and clutch 2, Δωcl2.
Input Axle Torque
Δωcl1
Input Clutch Pressure
Positive axle 1 torque
Increase clutch 1 pressure
Disengage clutch 1 and 2
The Active Differential block implements these equations to represent the mechanical dynamic response for the superposition and stationary clutch configurations. To determine the gear ratios, the block uses the clutch speed and the number of teeth for each gear pair. The allowable wheel speed difference (AWSD) limits the wheel speed difference for positive torque.
Superposition Clutches and Spur Gearing
Stationary Clutches and Planetary Gearing
{\stackrel{˙}{\omega }}_{d}\left({J}_{d}+{J}_{gs}\right)={T}_{d}-{\omega }_{d}{b}_{d}-{T}_{i}
{\stackrel{˙}{\omega }}_{d}\left( {J}_{d}+{J}_{s1}+{J}_{s2}\right)={T}_{d}-{\omega }_{d}{b}_{d}-{T}_{i}
{\stackrel{˙}{\omega }}_{1}{J}_{1}=\eta {T}_{1}-{\omega }_{1}{b}_{1}-{T}_{i1}
{\stackrel{˙}{\omega }}_{1}\left({J}_{1}+{J}_{r1}\right)={T}_{1}-{\omega }_{1}{b}_{1}-{T}_{i1}
{\stackrel{˙}{\omega }}_{2}{J}_{2}=\eta {T}_{2}-{\omega }_{2}{b}_{2}-{T}_{i2}
{\stackrel{˙}{\omega }}_{2}\left({J}_{axle2}+{J}_{r1}\right)={T}_{2}-{\omega }_{2}{b}_{2}-{T}_{i2}
\begin{array}{l}\frac{{\omega }_{cl1}}{{\omega }_{d}}={N}_{s1}=\frac{{z}_{1}{z}_{6}}{{z}_{4}{z}_{3}}\\ \frac{{\omega }_{cl2}}{{\omega }_{d}}={N}_{s2}=\frac{{z}_{1}{z}_{5}}{{z}_{4}{z}_{2}}\end{array}
\begin{array}{l}\frac{{\omega }_{cl1}}{{\omega }_{d}}={N}_{p1}=\frac{{z}_{1}{z}_{6}}{{z}_{4}{z}_{3}}\\ \frac{{\omega }_{cl2}}{{\omega }_{d}}={N}_{p2}=\frac{{z}_{1}{z}_{5}}{{z}_{4}{z}_{2}}\end{array}
Rigid Coupling Constraints
\begin{array}{l}{T}_{1}= \frac{N{T}_{i}}{2}-\frac{{N}_{s2}}{2}{T}_{cl2}+\frac{{N}_{s1}}{2}{T}_{cl1}\\ \\ {T}_{2}= \frac{N{T}_{i}}{2}+\left(1-\frac{{N}_{s2}}{2}\right){T}_{cl2}-\left(1-\frac{{N}_{s1}}{2}\right){T}_{cl1}\\ \\ {\omega }_{d=}=\frac{N}{2}\left({\omega }_{1}+{\omega }_{2}\right)\end{array}
\begin{array}{l}{T}_{1}= \frac{N{T}_{i}}{2}-\frac{{N}_{p2}}{\left({N}_{p2}-1\right)2}{T}_{cl2}+\frac{\left(2-{N}_{p1}\right)}{\left({N}_{p1}-1\right)2}{T}_{cl1}\\ \\ {T}_{2}= \frac{N{T}_{i}}{2}+\frac{\left(2-{N}_{p2}\right)}{\left({N}_{p2}-1\right)2}{T}_{cl2}-\frac{{N}_{p1}}{\left({N}_{p1}-1\right)2}{T}_{cl1}\\ \\ {\omega }_{d=}=\frac{N}{2}\left({\omega }_{1}+{\omega }_{2}\right)\end{array}
Allowable wheel speed difference (AWSD)
{\overline{\Delta \omega }}_{max}=\left({N}_{s2}-{N}_{s1}\right)\cdot 100%
{\overline{\Delta \omega }}_{max}=\left({N}_{p1,2}-1\right)\cdot 100%
These superposition clutch illustrations show the clutch configuration and schematic for torque transfer to the left wheel.
The illustrations show the stationary clutch configuration and schematic.
For both the ideal clutch and slip-speed configurations, the slip coupling is a function of the slip-speed and clutch pressure. The slip-speed depends on the slip velocity at each of the clutch interfaces.
\varpi =\left[\Delta {\omega }_{c1},\Delta {\omega }_{c2}\right]
The ideal clutch coupling model uses the axle slip speed, clutch pressure, and friction to calculate the clutch torque. The friction coefficient is a function of the slip speed.
{T}_{C}={F}_{T}{N}_{d}\mu \left(|\overline{\omega }|\right){R}_{eff}\mathrm{tanh}\left(4\overline{\omega }\right)
To calculate the total clutch force, the block uses the effective radius, clutch pressure, and clutch preload force.
{F}_{T}= {F}_{C}+{P}_{1,2}{\text{A}}_{eff}, {F}_{T}\ge 0
{R}_{eff}=\frac{2\left({R}_{o}{}^{3}-{R}_{i}{}^{3}\right)}{3\left({R}_{o}{}^{2}-{R}_{i}{}^{2}\right)}
Slip-Speed
To calculate the clutch torque, the slip speed coupling model uses torque data that is a function of slip speed and clutch pressure. The angular velocities of the axles determine the slip speed.
{T}_{C}={T}_{C}\left(\varpi , {P}_{1,2}\right)
Aeff Effective clutch pressure area
Axle 1 and 2 linear viscous damping, respectively
Fc, FT
Clutch preload force and total force, respectively
Carrier rotational inertia
Three-gang gear rotational inertia
Planetary carrier 1 and 2 rotational inertia, respectively
Jr1, Jr2
Planetary ring gear 1 and 2 rotational inertia, respectively
Planetary sun gear 1 and 2 rotational inertia, respectively
Axle 1 and 2 rotational inertia, respectively
Carrier-to-drive shaft gear ratio
Clutch 1 and 2 carrier-to-spur gear ratio, respectively
Planetary 1 and 2 carrier-to-axle gear ratio, respectively
Clutch 1 and 2 pressure, respectively
Annular disk inner and outer radius, respectively
Tcl1, Tcl2
Clutch 1 and 2 coupling torque, respectively
Axle 1 and 2 torque, respectively
Axle 1 and 2 internal resistance torque
ω1, ω2
Axle 1 and 2 angular velocity, respectively
Δωcl1, Δωcl2
Clutch 1 and 2 slip speed at interface, respectively
ωcl1, ωcl2
Clutch 1 and 2 angular velocity, respectively
Clutch coefficient of friction
Number of teeth on gear i
Prs1 — Clutch 1 pressure
Clutch 1 pressure, P1, in Pa.
DriveshftTrq — Driveshaft torque
Applied input torque, Td, typically from the engine driveshaft, in N·m.
Drive shaft angular velocity
Axle 1 angular velocity
CplngTrq1
Clutch 1 coupling torque
Clutch 2 coupling torque N·m
CplngSlipSpd1
Clutch 1 slip speed
CplngSlipSpd2 Clutch 2 slip speed rad/s
CplngPrs1
Clutch 1 input pressure
CplngPrs2 Clutch 2 input pressure Pa
DriveshftSpd — Angular velocity
Driveshaft angular velocity, ωd, in rad/s.
Axl1Spd — Angular velocity
Axle 1 angular velocity, ω1, in rad/s.
Active differential type — Differential
Spur gears, superposition clutches (default) | Double planetary gears, stationary clutches
Specify the type of active differential.
Clutch 1 to differential case gear ratio, Ns1 — Clutch 1-spur gear ratio
Clutch 1-to-carrier spur gear ratio, Ns1, dimensionless.
To enable the spur gear parameters, select Spur gears, superposition clutches for the Active differential type parameter.
Three-gang gear inertia, Jgc — Rotational inertia
Three-gang gear rotational inertia, Jgc, in kg·m^2.
Axle 1 planetary carrier to axle gear ratio, Np1 — Planetary 1 carrier gear ratio
Planetary 1 carrier-to-axle gear ratio, Np1, dimensionless.
To enable the planetary gear parameters, select Double planetary gears, stationary clutches for the Active differential type parameter.
Axle 1 sun gear inertia, Js1 — Planetary 1 sun gear inertia
Planetary 1 sun gear inertia, Js1, in kg·m^2.
Axle 1 carrier inertia, Jc1 — Planetary 1 carrier inertia
Planetary 1 carrier inertia, Jc1, in kg·m^2.
Axle 1 ring inertia, Jr1 — Planetary 1 ring gear inertia
Planetary 1 ring gear inertia, Jr1, kg·m^2.
Planetary 2 ring gear inertia, Jr2, in kg·m^2.
Specify the crown wheel connection to the drive shaft.
Carrier-to-drive shaft gear ratio, N.
Rotational inertia of the crown gear assembly, Jd, in kg·m^2. You can include the drive shaft inertia.
Ideal pre-loaded clutch (default) | Slip speed dependent torque data | Input torque dependent torque data
Torque modeled as a wet clutch with a constant velocity
Effective applied pressure area — Pressure area
Effective applied pressure area, in N/m^2.
To enable the clutch parameters, select Ideal pre-loaded clutch for the Coupling type parameter.
{R}_{eff}
{R}_{eff}=\frac{2\left({R}_{o}{}^{3}-{R}_{i}{}^{3}\right)}{3\left({R}_{o}{}^{2}-{R}_{i}{}^{2}\right)}
{R}_{o}
{R}_{i}
Friction coefficient vector, mu — Friction
[.16 0.13 0.115 0.11 0.105 0.1025 0.10125 .10125] (default) | vector
[0 10 20 40 60 80 100 500] (default) | vector
Torque - slip speed matrix, TdPdw — Clutch torque
[-1000,-500,-90,-50,-5,0,5,50,90,500,1000].*ones(11) (default) | matrix
Torque matrix, Tc, in N·m.
Clutch pressure vector, pT — Clutch pressure breakpoints
[0 1e3 5e3 7e3 1e4 2e4 5e4 1e5 5e5 1e6 5e6] (default) | vector
Clutch pressure breakpoints vector, P1,2, in Pa.
Slip speed vector, dwT — Slip speed breakpoints
[-500 -200, -175, -100, - 50, 0, 50, 100, 175, 200, 500] (default) | vector
Slip speed breakpoints vector, ω, in rad/s.
Open Differential | Limited Slip Differential
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Climate Change/Science/Atmospheric Balance - Wikibooks, open books for an open world
Climate Change/Science/Atmospheric Balance
One may wonder where exactly is the top of the atmosphere, and with good reason. We know that the atmosphere consists primarily of the gaseous envelop around Earth, and that pressure decreases with height, according to they hydrostatic approximation. Does the atmosphere end only when the pressure reaches a vanishingly small value? No, but there is not a good definition of the top of the atmosphere, and it changes with sub-discipline. For our purposes, we can usually take the top of the atmosphere, often abbreviated TOA, as somewhere in the low to mid-stratosphere, or even simply the tropopause. In this chapter, we can imagine it is the level at which the downward shortwave radiative flux is negligibly different from the solar constant and where there is negligible downward longwave flux (that due to the sun, which is small).
Now that we have an idea of what the TOA is, we can ask why it might be useful.
First consider conservation of energy in an equilibrium system. This could be a tank of water with a heating lamp above it all enclosed in a box. It could be a simple blackbody system, or any isolated system. Conservation of energy means that the total amount of energy does not change, which is equivalent to saying that any energy that is input to the system must be balanced by an outward flux of energy. In the case of Earth (as the "system"), this means that the energy coming in (the sunlight) must be balanced by outgoing radiation. We know the solar constant (
{\displaystyle 1367Wm^{-2}}
), so if we integrate over the Earth's surface, we know how much incoming energy there is. This incoming energy, sometimes called solar insolation or downward shortwave radiation, needs to be balanced. Why? Well, if it is not balanced by an equal loss of energy, then the temperature of the system must change (this is the 1st law of thermodynamics). Wien's law tells us that the wavelength of peak emission from a blackbody is inversely related to the temperature, and for normal Earth-like temperatures that puts the emission in the infrared part of the electromagnetic spectrum. Thankfully, this light is invisible to humans, and because the wavelength is longer than visible light (solar or shortwave), the terrestrial infrared radiation is often referred to as longwave radiation. The amount that is radiated to space (which differs from that emitted by the surface because of the greenhouse effect) is often called outgoing longwave radiation (OLR). The OLR (which is equal to the net longwave at TOA) balances the net shortwave at the top of the atmosphere when the system is in equilibrium.
Is the net shortwave radiation at TOA equal to the incoming shortwave? The answer is no. The net shortwave, which when averaged over suitable time and over the global, is the source of energy to the climate system, but not all the solar insolation is absorbed by the earth. Let's not beat around the bush. What happens to incoming solar radiation when it arrives in the atmosphere? There are really just three paths a photon (a "particle" of light) can take. First it can be absorbed, either in the atmosphere or at the surface. Absorption means that the energy associated with the photon is imparted to some atom or molecule, resulting in a higher energy level in that particle. Second, the photon can be reflected, which means that the path of the photon is reversed. More generally, we should say that the photon can be scattered, with some probability of being scattered back in the direction it came from, but we do not need to deal with scattering right here. Third, the photon can continue unimpeded, ultimately reaching the surface and being absorbed or reflected; while the photon is traveling through a medium without interacting, it is said to be transmitted. To study climate, one need not (usually) worry about individual photons, but the effects of the light in aggregate. Since we now know what can happen to each photon individually, we can sum over all the photons that make up the solar insolation such that
{\displaystyle F_{\downarrow }=A+R+T}
, where F is the total downward shortwave flux, A is the fraction of light absorbed by the atmosphere, R is the fraction reflected back to space before reaching the surface, and T is the light transmitted to the surface.
From our understanding of the downward shortwave flux, we can continue the analysis by considering the surface. The amount of light absorbed by the surface is not exactly equal to the transmitted light, T. Why? Well, the surface can be highly reflective. For example, snow and ice reflect up to 80% of incident light, while open ocean surfaces reflect almost none. The reflectivity of the surface is usually called the albedo, denoted
{\displaystyle \alpha }
, and is simply the fraction of incident light that gets reflected. Knowing that the surface has a given albedo, we can now say that the amount of light absorbed at the surface must be equal to
{\displaystyle F_{\mathrm {absorbed} }=(1-\alpha )^{-1}(F_{\downarrow }-A-R)}
. This says the absorbed light at the surface is equal to the transmitted insolation that is not reflected by the surface. Note that the albedo is constrained by definition to always be between 0 and 1, with typical global average of about 0.3.
It should also be noted that the shortwave light reflected by the surface does have a chance of being reflected (by clouds or particulate matter) or absorbed by atmospheric constituents. However, for most discussions of climate, and for the purposes here, we will neglect this process. Furthermore, we can (to a reasonable approximation) assume that the atmosphere is transparent to shortwave radiation, meaning there will be no absorption. This simplifies our previous expressions by eliminating the term A. To further simplify our notation, we can say that the total "planetary albedo" is the sum of the atmosphere albedo (later will will call this the cloud albedo) and the surface albedo,
{\displaystyle \alpha _{p}=\alpha _{s}+\alpha _{c}}
. These simplifications allow us to write the "net shortwave at TOA" as
{\displaystyle F_{net}=F_{\downarrow }-F_{\uparrow }=(1-\alpha _{p})F_{\downarrow }}
As mentioned above, the radiative flux from the surface acts approximately according to the Steffan-Boltzmann and Wien's laws of blackbody radiation. Even if we take the total energy from the solar constant spread over the full surface area of Earth, the emission must be in the infrared. The calculation is left as an exercise.
When we assume that the surface absorbs some fraction of the incident shortwave,
{\displaystyle (1-\alpha _{s})F_{sfc}}
, and that the temperature reaches equilibrium, the emitted flux is then
{\displaystyle Q_{sfc}=\sigma T_{sfc}^{4}}
. Of course, in reality there is some emissivity associated with different surface types, but we neglect that here. Once emitted, these photons face similar consequences as the down-welling shortwave radiation. The difference in the longwave is primarily that the atmosphere is much more opaque in the infrared than visible, so the absorption can not be neglected. Various atmospheric constituents absorb infrared energy, then emit at a wavelength commensurate with the temperature of that part of the atmosphere. This is the natural greenhouse effect, and the active gases are often referred to as greenhouse gases; primary among these are water vapor, carbon dioxide, and methane.
The consequences of the natural greenhouse effect are crucial for life on Earth. In the absence of an atmosphere, the longwave radiation emitted to space would be exactly equal to the shortwave absorbed, and the surface temperature would be a chilly 255 K. Because greenhouse gases absorb infrared radiation, they act to warm the planet. How? We can think of the effects in two ways. First, the gases are heated by the absorbed radiation, and then radiate isotropically (equally up and down), sending energy back toward the surface to act as an extra energy source. Second, the absorption and subsequent emission by greenhouse gases changes the effective emission temperature of Earth (as seen from space). This second effect is a useful way to understand the greenhouse effect, and can be easily applied to changing climates. As a thought experiment, consider all the absorption by greenhouse gases happening in a thin layer of the atmosphere, which can effectively be thought of as a thin shell around the Earth. From space, the emission from the planet will be coming from an elevated level, with a much colder temperature than the surface. Of course, that means that the flux from that surface will be less than the incoming flux of solar insolation. The only way the climate system can achieve equilibrium, which is required by conservation of energy, is for the lower levels to warm, emit more energy as longwave radiation, which in turns warms the atmosphere, and changes the effective emission height and temperature. This adjustment continues until the shortwave and longwave budgets are balanced at the top of the atmosphere.
The figure below shows a rough cartoon of the radiative balances in the atmosphere. So far, we have focused only on the clear-sky scenario (on the left of the cartoon). Later we will consider the modifications that arise in the presence of clouds, and we will also explore the implications of changing the atmospheric composition.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Climate_Change/Science/Atmospheric_Balance&oldid=3277639"
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Two-point_tensor Knowpia
Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates.[1] Examples include the deformation gradient and the first Piola–Kirchhoff stress tensor.
As with many applications of tensors, Einstein summation notation is frequently used. To clarify this notation, capital indices are often used to indicate reference coordinates and lowercase for present coordinates. Thus, a two-point tensor will have one capital and one lower-case index; for example, AjM.
Continuum mechanicsEdit
A conventional tensor can be viewed as a transformation of vectors in one coordinate system to other vectors in the same coordinate system. In contrast, a two-point tensor transforms vectors from one coordinate system to another. That is, a conventional tensor,
{\displaystyle \mathbf {Q} =Q_{pq}(\mathbf {e} _{p}\otimes \mathbf {e} _{q})}
actively transforms a vector u to a vector v such that
{\displaystyle \mathbf {v} =\mathbf {Q} \mathbf {u} }
where v and u are measured in the same space and their coordinates representation is with respect to the same basis (denoted by the "e").
In contrast, a two-point tensor, G will be written as
{\displaystyle \mathbf {G} =G_{pq}(\mathbf {e} _{p}\otimes \mathbf {E} _{q})}
and will transform a vector, U, in E system to a vector, v, in the e system as
{\displaystyle \mathbf {v} =\mathbf {GU} }
The transformation law for two-point tensorEdit
Suppose we have two coordinate systems one primed and another unprimed and a vectors' components transform between them as
{\displaystyle v'_{p}=Q_{pq}v_{q}}
For tensors suppose we then have
{\displaystyle T_{pq}(e_{p}\otimes e_{q})}
A tensor in the system
{\displaystyle e_{i}}
. In another system, let the same tensor be given by
{\displaystyle T'_{pq}(e'_{p}\otimes e'_{q})}
{\displaystyle T'_{ij}=Q_{ip}Q_{jr}T_{pr}}
{\displaystyle T'=QTQ^{\mathsf {T}}}
is the routine tensor transformation. But a two-point tensor between these systems is just
{\displaystyle F_{pq}(e'_{p}\otimes e_{q})}
which transforms as
{\displaystyle F'=QF}
The most mundane example of a two-point tensor is the transformation tensor, the Q in the above discussion. Note that
{\displaystyle v'_{p}=Q_{pq}u_{q}}
Now, writing out in full,
{\displaystyle u=u_{q}e_{q}}
{\displaystyle v=v'_{p}e'_{p}}
This then requires Q to be of the form
{\displaystyle Q_{pq}(e'_{p}\otimes e_{q})}
By definition of tensor product,
{\displaystyle (e'_{p}\otimes e_{q})e_{q}=(e_{q}.e_{q})e'_{p}=3e'_{p}}
{\displaystyle u_{p}e_{p}=(Q_{pq}(e'_{p}\otimes e_{q}))(v_{q}e_{q})}
{\displaystyle u_{p}e_{p}=Q_{pq}v_{q}(e'_{p}\otimes e_{q})e_{q}}
Incorporating (1), we have
{\displaystyle u_{p}e_{p}=Q_{pq}v_{q}e_{p}}
^ Humphrey, Jay D. Cardiovascular solid mechanics: cells, tissues, and organs. Springer Verlag, 2002.
Mathematical foundations of elasticity By Jerrold E. Marsden, Thomas J. R. Hughes
Two-point Tensors at iMechanica
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(Redirected from Compass-and-straightedge construction)
It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone, or by straightedge alone if given a single circle and its center.
1 Straightedge and compass tools
4 Much used straightedge and compass constructions
5 Constructible points
5.1 Constructible angles
5.2 Relation to complex arithmetic
6.4 Distance to an ellipse
8 Constructing a triangle from three given characteristic points or lengths
9 Restricted Constructions
9.1 Constructing with only ruler or only compass
10.1 Solid constructions
10.2 Angle trisection
10.5 Trisect a straight segment
Straightedge and compass tools[edit]
The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. The line drawn is infinitesimally thin point-width. It can only be used to draw a line segment between two points, with infinite precision to those points, or to extend an existing segment.
The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it. Circles can only be drawn starting from two given points: the centre and a point on the circle, and aligned to those points with infinite precision. The arc that is drawn is infinitesimally thin point-width. The compass may or may not collapse when it is not drawing a circle.
{\displaystyle \pi }
is a transcendental number, and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle.[3]: p. 47
The basic constructions[edit]
Much used straightedge and compass constructions[edit]
The most-used straightedge and compass constructions include:
Constructing the perpendicular bisector from a segment
Finding the midpoint of a segment.
Drawing a perpendicular line from a point to a line.
Mirroring a point in a line
Constructing a line through a point tangent to a circle
Constructing a circle through 3 noncollinear points
Drawing a line through a given point parallel to a given line.
Constructible points[edit]
Main article: Constructible number
Straightedge and compass constructions corresponding to algebraic operations
x=a·b (intercept theorem)
x=a/b (intercept theorem)
x=√a (Pythagorean theorem)
Constructible angles[edit]
{\displaystyle \cos {\left({\frac {2\pi }{17}}\right)}=-{\frac {1}{16}}\;+\;{\frac {1}{16}}{\sqrt {17}}\;+\;{\frac {1}{16}}{\sqrt {34-2{\sqrt {17}}}}\;+\;{\frac {1}{8}}{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}}
Relation to complex arithmetic[edit]
{\displaystyle \mathrm {Re} (z)={\frac {z+{\bar {z}}}{2}}\;}
{\displaystyle \mathrm {Im} (z)={\frac {z-{\bar {z}}}{2i}}\;}
{\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}.\;}
Impossible constructions[edit]
Doubling the cube[edit]
Main article: Doubling the cube
Angle trisection[edit]
Main article: Angle trisection
Distance to an ellipse[edit]
Alhazen's problem[edit]
Constructing regular polygons[edit]
3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272... (sequence A003401 in the OEIS)
Constructing a triangle from three given characteristic points or lengths[edit]
Restricted Constructions[edit]
Constructing with only ruler or only compass[edit]
Extended constructions[edit]
Solid constructions[edit]
7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97... (sequence A051913 in the OEIS)
11, 22, 23, 25, 29, 31, 33, 41, 43, 44, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 66, 67, 69, 71, 75, 77, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 98, 99, 100... (sequence A048136 in the OEIS)
Origami[edit]
Main article: Huzita–Hatori axioms
Markable rulers[edit]
Main article: Neusis construction
Archimedes, Nicomedes and Apollonius gave constructions involving the use of a markable ruler. This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects the two given lines, such that the distance between the points of intersection equals the given segment. This the Greeks called neusis ("inclination", "tendency" or "verging"), because the new line tends to the point. In this expanded scheme, we can trisect an arbitrary angle (see Archimedes' trisection) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible. Using a markable ruler, regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them.[20]
Trisect a straight segment[edit]
Trisection of a straight edge procedure.
Computation of binary digits[edit]
List of interactive geometry software, most of them show straightedge and compass constructions
Underwood Dudley, a mathematician who has made a sideline of collecting false straightedge and compass proofs.
^ Underwood Dudley (1983), "What To Do When the Trisector Comes" (PDF), The Mathematical Intelligencer, 5 (1): 20–25, doi:10.1007/bf03023502
^ Godfried Toussaint, "A new look at Euclid’s second proposition," The Mathematical Intelligencer, Vol. 15, No. 3, (1993), pp. 12-24.
^ a b c d e f g h i Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).
^ a b Wantzel, Pierre-Laurent (1837). "Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas" (PDF). Journal de Mathématiques Pures et Appliquées. 1. 2: 366–372. Retrieved 3 March 2014.
^ a b Kazarinoff, Nicholas D. (2003) [1970]. Ruler and the Round. Mineola, N.Y.: Dover. pp. 29–30. ISBN 978-0-486-42515-3.
^ Weisstein, Eric W. "Trigonometry Angles--Pi/17". MathWorld.
^ Stewart, Ian. Galois Theory. p. 75.
^ *Squaring the circle at MacTutor
^ Instructions for trisecting a 72˚ angle.
^ Azad, H., and Laradji, A., "Some impossible constructions in elementary geometry", Mathematical Gazette 88, November 2004, 548–551.
^ Neumann, Peter M. (1998), "Reflections on Reflection in a Spherical Mirror", American Mathematical Monthly, 105 (6): 523–528, doi:10.1080/00029890.1998.12004920, JSTOR 2589403, MR 1626185
^ Highfield, Roger (1 April 1997), "Don solves the last puzzle left by ancient Greeks", Electronic Telegraph, 676, archived from the original on November 23, 2004, retrieved 2008-09-24
^ Pascal Schreck, Pascal Mathis, Vesna Marinkoviċ, and Predrag Janičiċ. "Wernick's list: A final update", Forum Geometricorum 16, 2016, pp. 69–80. http://forumgeom.fau.edu/FG2016volume16/FG201610.pdf
^ Avron, Arnon (1990). "On strict strong constructibility with a compass alone". Journal of Geometry. 38 (1–2): 12–15. doi:10.1007/BF01222890.
^ T.L. Heath, "A History of Greek Mathematics, Volume I"
^ P. Hummel, "Solid constructions using ellipses", The Pi Mu Epsilon Journal, 11(8), 429 -- 435 (2003)
^ Gleason, Andrew: "Angle trisection, the heptagon, and the triskaidecagon", Amer. Math. Monthly 95 (1988), no. 3, 185-194.
^ Row, T. Sundara (1966). Geometric Exercises in Paper Folding. New York: Dover.
^ Conway, John H. and Richard Guy: The Book of Numbers
^ A. Baragar, "Constructions using a Twice-Notched Straightedge", The American Mathematical Monthly, 109 (2), 151 -- 164 (2002).
^ E. Benjamin, C. Snyder, "On the construction of the regular hendecagon by marked ruler and compass", Mathematical Proceedings of the Cambridge Philosophical Society, 156 (3), 409 -- 424 (2014).
^ Simon Plouffe (1998). "The Computation of Certain Numbers Using a Ruler and Compass". Journal of Integer Sequences. 1. ISSN 1530-7638.
Weisstein, Eric W. "Angle Trisection". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Straightedge_and_compass_construction&oldid=1074308751"
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New Microphone and Speaker Components
Meter, Rotary Gauge, and Volume Gauge Components
The new microphone and speaker components make it possible to capture and playback audio signals in a Maple worksheet. See an example of how both of these new components work in the new application: Interaural Time Delay.
With the advances in programmatic content generation, it is now possible to programmatically generate interactive embedded components. Several Maple commands have been updated to leverage this technology, including the AudioTools:-Preview command, which now returns a play button and a speaker component with the output option set to embed:
\mathrm{audiofile}≔\mathrm{AudioTools}:-\mathrm{Read}\left(\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right)\right):
\mathrm{AudioTools}:-\mathrm{Preview}\left(\mathrm{audiofile}, 0.0..0.1, \mathrm{color}="Orange",\mathrm{output}=\mathrm{embed}\right)
The check box component has been updated to support a selected image as well as including support for setting the following option programmatically:
groupname: the name of the component group
The look and feel for the code edit region has been improved with default settings, such as hidden border, hidden line numbers, and the vertical line to the left of the CER.
In addition, keyboard shortcuts to collapse or expand code edit regions have been added:
Alt+C : to collapse code edit region (Meta + Alt+C on Mac)
Alt+X : to expand code edit region (Meta + Alt + X on Mac)
The code edit region also has the following new option available in the Format menu:
Autoexecute property: control if the code edit region is executed on startup or not
The dial component has three new options which can be set programmatically or in the properties dialog:
startAngle: specify the initial value of the dial
angleRange: specify the range of the dial from 1 to 360 degrees
image: use a custom image for the background of the dial
The following option can now be set programmatically:
continuousupdate: indicates whether the component should update continuously during drag operations
The list box component now supports multiple selection. This can be configured programmatically or in the properties dialog:
selectmultiple: indicates whether selecting multiple items is permitted
selectedItemList: the set of all items which are currently selected
Choose one or more trigonometric functions to plot:
sin(x)cos(x)csc(x)sec(x)
The following option can now be set programmatically for meter, rotary gauge, and volume gauge components:
The following option for plot component can now be set programmatically:
clickdefault: indicates whether the default cursor action on a plot component is a click action
The following option got radio button component can now be set programmatically:
The slider component is now resizable:
The properties for pixelHeight and pixelWidth of the slider can be changed in the right click properties menu or programmatically set using the DocumentTools:-SetProperty command.
The following option for text area component can now be set programmatically:
wrapping: indicates whether content on one line of the text area which exceeds the horizontal space will automatically wrap to the next line
The toggle button image has been updated.
To use the legacy button image for toggle buttons, the image files can be found in the images directory of your Maple installation.
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The formal definition for the mode of a dataset is:
The most frequently occurring observation in the dataset. A dataset can have multiple modes if there is more than one value with the same maximum frequency.
While you may be able to find the mode of a small dataset by simply looking through it, if you have trouble, we recommend you follow these two steps:
24,\ 16,\ 12,\ 10,\ 12,\ 28,\ 38,\ 12,\ 28,\ 24
Let’s find the frequency of each number:
From the table, we can see that our mode is 12, the most frequent number in our dataset.
Determine the mode of the ages for the first ten authors in the Le Monde survey:
29, 49, 42, 43, 32, 38, 37, 41, 27, 27
Save the value to mode_age.
Determine the number of authors who were the age of the mode. Save the number to mode_count.
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Fatigue life estimation algorithm of structures with high frequency random vibration based on hybrid energy finite element method | JVE Journals
J. Zeng1 , H. B. Chen2 , Y. Y. Wang3
1, 2, 3Department of Modern Mechanics, University of Science and Technology of China, CAS Key Laboratory of Mechanical Behavior and Design of Materials, Hefei, China
3AVIC Chengdu Aircraft Design and Research Institute, Chengdu, China
A hybrid energy finite element method (Hybrid-EFEM) considering direct field is introduced and promoted to general coupling conditions. The equivalent stress amplitude based on energy density is derived. Then combined with zero order moment stress spectrum method, a high-frequency fatigue life prediction method based on Hybrid-EFEM is proposed. A fatigue life prediction example is carried out to verify the feasibility of the method.
Keywords: hybrid energy finite element method, zero order moment stress spectrum method, fatigue life prediction.
High-frequency random pressure may cause serious engineering structure fatigue damage [1], so, it is necessary to forecast the fatigue life of the structure under high frequency vibration. The classical frequency-domain fatigue life prediction like Rayleigh approximation [2], Dirlik’s formula [3] methods are based on the stress power spectral density. As such, the outcomes of the traditional high-frequency energy algorithms such as Statistical Energy Analysis (SEA) and EFEM, which refer to the average energy in the frequency band, cannot be applied directly to estimate the fatigue life. Wang et al. [4] researched the influence of the stress power spectral density (PSD) shape on fatigue life estimation. A large number of numerical simulations shows fatigue life is not sensitive to the shape when the zero-order moment of stress PSD remains unchanged. Thus, a Zero Order Moment Stress Spectrum Method is proposed, which only need the stress RMS in the band Compared with the traditional frequency domain method, and the feasibility of the method is verified by a large number of examples.
The zero-order moment of the stress PSD in Y. Y. Wang’s method is calculated by SEA. Since SEA is a statistical averaging method, which can only get the average energy of the substructure, so the maximum stress position on the substructure and its stress level cannot be predicted [5]. Therefore, the predicted life value is so large to bring the risk to the actual structural design. EFEM can overcome the shortcoming of SEA to some extent, but the predict result is still very rough due to many assumptions, one is plane wave assumption [6]. Smith [7] taken the direct circle wave field component into account to solve the high frequency point-loaded plate vibration, and then come up with the Hybrid-EFEM. The accuracy of simulated response near the exciting point has been greatly improved compared with traditional EFEM. P. Hardy [8] considered the energy flow on the boundary, then extended hybrid-EFEM to two board structures, which provides a new idea for the application of Hybrid-EFEM to general coupling conditions.
The bending wave energy accounted for the main role when the plate is under high-frequency transverse excitation. Therefore, this paper only focus on the bending wave, Hybrid-EFEM will be considered to extend to general coupling conditions, then combined with Zero Order Moment Stress Spectrum Method to predict the fatigue life of the actual structure with high frequency random vibration.
2. The theory of Hybrid-EFEM
The Hybrid-EFEM considers the energy in the plate to be composed of two parts: The direct field and the reverberant field. The direct field is composed of cylindrical waves coming from external sources. The reverberant field is due to reflection and transmission at subsystem junctions.
2.1. The direct field and reverberant field
Fig. 1 shows the wave field components in a plate: where
S
is the exciting position,
\stackrel{\to }{{q}_{d}}
\stackrel{\to }{{q}_{r}}
represent direct and reverberant power flow intensity.
\mathrm{\Omega }
represent domain,
\mathrm{\Gamma }
represent boundary.
Fig. 1. Wave field components in a plate
The total energy density and power flow intensity in the domain can be written as [7]:
e={e}_{d}+{e}_{r},
\stackrel{\to }{q}={\stackrel{\to }{q}}_{d}+{\stackrel{\to }{q}}_{r},
where, the subscript
d
represents the direct field, r represents the reverberant field.
The energy density of the direct field is:
e\left(r\right)=\left\{\begin{array}{ll}\frac{{\pi }_{in}}{2\pi r{c}_{g}}{e}^{-mr},& r>{r}_{0},\\ \frac{{\pi }_{in}}{2\pi {r}_{0}{c}_{g}}{e}^{-m{r}_{0}},& 0<r<{r}_{0},\end{array}\right\
represents the distance from the excitation point, and
{r}_{0}
{r}_{0}=1/2{\pi }^{2}/\lambda +m
The reverberant field can be written as [8]:
\left\{\begin{array}{l}-\frac{{{c}_{g}}^{2}}{\eta \omega }{\nabla }^{2}{e}_{r}\left(r\right)+\eta \omega {e}_{r}\left(r\right)=0,\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\forall r\in \mathrm{\Omega }\\mathrm{\Gamma }, \\ -\frac{{{c}_{g}}^{2}}{\eta \omega }{\nabla }^{2}{e}_{r}\left(r\right)\cdot \stackrel{\to }{n}={\stackrel{\to }{q}}_{r}\cdot \stackrel{\to }{n}, \forall r\in \mathrm{\Gamma }.\end{array}\right\
2.2. Coupling boundary
P. Hardy deduces the bending wave energy flow at the junction of two plates using the energy transfer coefficient and power flow balance [8]. According to P. Hardy’s thought, the Hybrid-EFEM, which only consider the role of bending wave can be extended to general coupling conditions. Considering the space limitations, here we do not present the detailed derivation.
2.3. An example of hybrid-EFEM
Consider the two square plates that are coupled as Fig. 2, and the material parameters and dimensions of the two plates are shown in the Table 1.
Fig. 2. L-type coupling plate
Table 1. Size and material parameter
Considering a unit force and
f=
1000 Hz,
\eta =
0.1. The result on the dotted line calculated by FEM, Hybrid-EFEM, EFEM and SEA is shown in Fig. 3.
It can be seen that compared with EFEM and SEA, Hybrid-EFEM can better simulate the responses especially places near the excitation point. This is of great significance to predict the location and stress level of the structure’s dangerous point.
Fig. 3. Comparison of bending energy results in four methods
3. Fatigue life prediction based on the zero order moment stress spectrum method
3.1. Zero order moment stress spectrum theory
Y. Y. Wang [4] analyzes the influence of the stress PSD shape on the fatigue damage and obtains that the fatigue damage is insensitive to the shape of the stress PSD, and only the stress RMS is enough for the fatigue life estimation, as for broad-band processes, the whole band can be divided into several partitions, and the stress RMS of each partition is used to estimate the fatigue damage.
3.2. Application of hybrid-EFEM in zero order moment stress spectrum method
From the Parseval theorem we can see: The zero order moment of a physical quantity’s PSD is equal to the mean square of the physical quantity. Thus, we can use the Hybrid-EFEM to determine the sturacture’s total energy in the 1/3 octave, then, A formulation for estimating the equivalent stress from the energy density is deduced like SEA [5]. Then, the zero order moment of the stress PSD in the octave band is obtained through Parseval theorem. Ultimately, the structural fatigue life prediction is carried out by combining the zero order moment stress spectrum method.
Considering the type of three coupling plate as shown in the Fig. 4. Which usually work as a support structure, it is significant to study its fatigue life. each plate is aluminum plate structure of 1 m×1 m length and 3 mm thickness, Poisson’s ratio is 0.32, The internal loss factor of each band is 0.0063. Assuming that the plate 2 is randomly excited in the center point. The force is shown in Fig. 5. Time domain force is translated into frequency domain by Fourier transform. The broad-band is divided into 10 1/3 octaves from 1 kHz to 8 kHz. The zero-order moment of the stress PSD in each octave is calculated respectively, and then the fatigue life is estimated.
Fig. 4. Three coupling plate structure
Fig. 5. Force – time domain curve
Fig. 6. Bending wave energy nephogram in center frequency
f=
a) EFEM
b) Hybrid-EFEM
The bending wave energy nephogram in center frequency
f=
1000 Hz calculated separately by EFEM and Hybrid-EFEM was shown in Fig. 6.
From Fig. 6 we can see the energy is symmetrical due to the symmetrical structure and force. And the dangerous point is the center of plate 2. Most of all, Fig. 6 shows the stress level of the dangerous point simulated by Hybrid-EFEM is much larger than that of EFEM.
Assumed the two coefficients of the S-N curve is
k=
c=
1.268×1013, The fatigue life calculated by the zero-order moment stress spectrum is shown in Table 2.
Table 2. Estimated fatigue life
Zero-moment Rayleigh
Zero-moment Wirsching
Zero-moment Dirlik
Fatigue life (h)
This paper presents a new high-frequency fatigue life prediction method. Firstly, a hybrid energy method considering the direct field is introduced and extended to general coupling structure. And the equivalent stress amplitude estimation method based on the energy density is deduced. Finally, combined with the zero-order moment stress spectrum method, the structural high-frequency vibration fatigue life is predicted. An example of two coupled plate shows that the Hybrid-EFEM’s accuracy compared with EFEM, and a practical fatigue life estimation example of three coupled plate proves the new high-frequency fatigue life prediction method is feasible and scientific.
This work is supported by the strategic Priority Research Program of the Chinese Academy of Sciences, Grant No. XDB22040502.
Barter J. W., Dolling D. S. Prediction of fluctuating pressure loads produced by shock-induced turbulent boundary layer separation. AIAA 96-2043, 1996. [Search CrossRef]
Bishop N. W. M. The Use of Frequency Domain Parameters to Predict Structural Fatigue. Ph.D. Thesis, University of Warwick, UK, 1988. [Search CrossRef]
Dirlik Dirlik T. T. Application of Computers in Fatigue Analysis. Ph.D. Thesis, University of Warwick, UK, 1985. [Search CrossRef]
Wang Y. Y., Chen H. B., Zhou H. W. A fatigue life estimation algorithm based on Statistical Energy Analysis in high-frequency random processes. International Journal of Fatigue, Vol. 83, 2015, p. 221-229. [Search CrossRef]
Lyon R. H. Statistical Energy Analysis of Dynamical Systems: Theory and Applications. MIT Press, Cambridge, 1997. [Search CrossRef]
Bouthier O. M., Bernhard R. J. Simple models of the energetics of transversely vibrating plates. Journal of Sound and Vibration, Vol. 182, Issue 1, 1995, p. 149-164. [Search CrossRef]
Smith M. J. A hybrid energy method for predicting high frequency vibrational response of point-loaded plates. Journal of Sound and Vibration, Vol. 202, Issue 3, 1997, p. 375-394. [Search CrossRef]
Hardy P., Ichchou M., Jezequel L., Trentin D. A hybrid local energy formulation for plates mid-frequency flexural vibrations. European Journal of Mechanics, A/Solids, Vol. 28, Issue 1, 2009, p. 121-130. [Search CrossRef]
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Can You Certify That a Solution Is Global? - MATLAB & Simulink - MathWorks Nordic
Check if a Solution Is a Local Solution with patternsearch
Identify a Bounded Region That Contains a Global Solution
Use MultiStart with More Start Points
How can you tell if you have located the global minimum of your objective function? The short answer is that you cannot; you have no guarantee that the result of a Global Optimization Toolbox solver is a global optimum. While all Global Optimization Toolbox solvers repeatedly attempt to locate a global solution, no solver employs an algorithm that can certify a solution as global.
However, you can use the strategies in this section for investigating solutions.
Before you can determine if a purported solution is a global minimum, first check that it is a local minimum. To do so, run patternsearch on the problem.
To convert the problem to use patternsearch instead of fmincon or fminunc, enter
problem.solver = 'patternsearch';
Also, change the start point to the solution you just found, and clear the options:
problem.x0 = x;
problem.options = [];
For example, Check Nearby Points shows the following:
problem = createOptimProblem('fmincon', ...
'objective',ffun,'x0',[1/2 1/3], ...
'lb',[0 -1],'ub',[1 1],'options',options);
[x,fval,exitflag] = fmincon(problem)
However, checking this purported solution with patternsearch shows that there is a better solution. Start patternsearch from the reported solution x:
% set the candidate solution x as the start point
[xp,fvalp,exitflagp] = patternsearch(problem)
fvalp =
exitflagp =
Suppose you have a smooth objective function in a bounded region. Given enough time and start points, MultiStart eventually locates a global solution.
Therefore, if you can bound the region where a global solution can exist, you can obtain some degree of assurance that MultiStart locates the global solution.
f={x}^{6}+{y}^{6}+\mathrm{sin}\left(x+y\right)\left({x}^{2}+{y}^{2}\right)-\mathrm{cos}\left(\frac{{x}^{2}}{1+{y}^{2}}\right)\left(2+{x}^{4}+{x}^{2}{y}^{2}+{y}^{4}\right).
The initial summands x6 + y6 force the function to become large and positive for large values of |x| or |y|. The components of the global minimum of the function must be within the bounds
–10 ≤ x,y ≤ 10,
since 106 is much larger than all the multiples of 104 that occur in the other summands of the function.
You can identify smaller bounds for this problem; for example, the global minimum is between –2 and 2. It is more important to identify reasonable bounds than it is to identify the best bounds.
To check whether there is a better solution to your problem, run MultiStart with additional start points. Use MultiStart instead of GlobalSearch for this task because GlobalSearch does not run the local solver from all start points.
For example, see Example: Searching for a Better Solution.
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EMS | Jochen Glück awarded Jaroslav and Barbara Zemánek Prize in functional analysis
Jochen Glück awarded Jaroslav and Barbara Zemánek Prize in functional analysis
The Jaroslav and Barbara Zemánek Prize in functional analysis with emphasis on operator theory for 2021 has been awarded to Jochen Glück (University of Passau, Germany).
The Jaroslav and Barbara Zemánek Prize in functional analysis with emphasis on operator theory for 2021 is awarded to Jochen Glück (University of Passau, Germany) for his work on one-parameter operator semigroups and their various applications, along with several outstanding contributions to general operator theory. The Jury emphasized his deep and elegant results regarding the existence of dilations of Banach space contractions and on the existence of spectral gaps for hyperbounded operators on
L^p
Following a generous donation of Zemánek's family, the annual Zemánek's Prize was founded by the Institute of Mathematics of the Polish Academy of Sciences (IM PAN) in March 2018, in order to encourage the research in functional analysis, operator theory and related topics. The Prize is established to promote young mathematicians, under 35 years of age, who made important contributions to the field.
The awarding ceremony will take place at IM PAN, Warsaw, in Fall 2021.
A more detailed information about the Prize can be found on the webpage https://www.impan.pl/en/events/awards/b-and-j-zemanek-prize.
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Create Markov-switching dynamic regression model - MATLAB - MathWorks í•œêµ
Create Fully Specified Univariate Model
Create Fully Specified Model for US GDP Rate
Create Partially Specified Univariate Model for Estimation
Create Fully Specified Multivariate Model
Create Fully Specified Model Containing Regression Component
Create Partially Specified Multivariate Model for Estimation
Specify Model Regression Component for Estimation
Create Model Specifying Equality Constraints for Estimation
Markov-Switching Dynamic Regression Model
Create Markov-switching dynamic regression model
The msVAR function returns an msVAR object that specifies the functional form of a Markov-switching dynamic regression model for the univariate or multivariate response process yt. The msVAR object also stores the parameter values of the model.
An msVAR object has two key components:
Switching mechanism among states, represented by a discrete-time Markov chain (dtmc object)
State-specific submodels, either autoregressive (ARX) or vector autoregression (VARX) models (arima or varm objects), which can contain exogenous regression components
The components completely specify the model structure. The Markov chain transition matrix and submodel parameters, such as the AR coefficients and innovation-distribution variance, are unknown and estimable unless you specify their values.
To estimate a model containing unknown parameter values, pass the model and data to estimate. To work with an estimated or fully specified msVAR object, pass it to an object function.
Alternatively, to create a threshold-switching dynamic regression model, which has a switching mechanism governed by threshold transitions and observations of a threshold variable, see threshold and tsVAR.
Mdl = msVAR(mc,mdl)
Mdl = msVAR(mc,mdl,'SeriesNames',seriesNames)
Mdl = msVAR(mc,mdl) creates a Markov-switching dynamic regression model Mdl (an msVAR object) that has the discrete-time Markov chain, switching mechanism among states mc and the state-specific, stable dynamic regression submodels mdl.
Mdl = msVAR(mc,mdl,'SeriesNames',seriesNames) optionally sets the SeriesNames property, which associates the names seriesNames to the time series of the model.
mc — Discrete-time Markov chain for switching mechanism among states
Discrete-time Markov chain for the switching mechanism among states, specified as a dtmc object.
The states represented in the rows and columns of the transition matrix mc.P correspond to the states represented in the submodel vector mdl.
msVAR processes and stores mc in the Switch property.
mdl — State-specific dynamic regression submodels
vector of arima objects | vector of varm objects
State-specific dynamic regression submodels, specified as a length mc.NumStates vector of model objects individually constructed by arima or varm. All submodels must be of the same type (arima or varm) and have the same number of series.
Unlike other model estimation tools, estimate does not infer the size of submodel regression coefficient arrays during estimation. Therefore, you must specify the Beta property of each submodel appropriately. For example, to include and estimate three predictors of the regression component of univariate submodel j, set mdl(j).Beta = NaN(3,1).
msVAR processes and stores mdl in the property Submodels.
You can set only the SeriesNames property when you create a model by using name-value argument syntax or by using dot notation. MATLAB® derives the values of all other properties from inputs mc and mdl.
For example, create a Markov-switching model for a 2-D response series, and then label the first and second series "GDP" and "CPI", respectively.
Mdl = msVAR(mc,mdl);
Mdl.SeriesNames = ["GDP" "CPI"];
NumSeries — Number of time series
Number of time series, specified as a positive integer. NumSeries specifies the dimensionality of the response variable and innovation in all submodels.
State labels, specified as a string vector of length NumStates.
SeriesNames — Series labels
string(1:numSeries) (default) | string vector | cell array of character vectors | numeric vector
Series labels, specified as a string vector, cell array of character vectors, or a numeric vector of length numSeries. msVAR stores the series labels as a string vector.
Switch — Discrete-time Markov chain for switching mechanism among states
Submodels — State-specific vector autoregression submodels
vector of varm objects
State-specific vector autoregression submodels, specified as a vector of varm objects of length NumStates.
msVAR removes unsupported submodel components.
For arima submodels, msVAR does not support the moving average (MA), differencing, and seasonal components. If any submodel is a composite conditional mean and variance model (for example, its Variance property is a garch object), msVAR issues an error.
For varm submodels, msVAR does not support the trend component.
msVAR converts submodels specified as arima objects to 1-D varm objects.
NaN-valued elements in either the properties of Switch or the submodels of Submodels indicate unknown, estimable parameters. Specified elements, except submodel innovation variances, indicate equality constraints on parameters in model estimation.
All unknown submodel parameters are state dependent.
Create a two-state Markov-switching dynamic regression model for a 1-D response process. Specify all parameter values (this example uses arbitrary values).
Create a two-state discrete-time Markov chain model that describes the regime switching mechanism. Label the regimes.
mc = dtmc(P,'StateNames',["Expansion" "Recession"])
StateNames: ["Expansion" "Recession"]
For each regime, use arima to create an AR model that describes the response process within the regime.
% AR coefficients
AR1 = [0.3 0.2]; % 2 lags
AR2 = 0.1; % 1 lag
% Innovations variances
% AR Submodels
mdl1 = arima('Constant',C1,'AR',AR1,...
'Variance',v1,'Description','Expansion State')
Description: "Expansion State"
'Variance',v2,'Description','Recession State')
Description: "Recession State"
mdl1 and mdl2 are fully specified arima objects.
Store the submodels in a vector with order corresponding to the regimes in mc.StateNames.
mdl = [mdl1; mdl2];
Use msVAR to create a Markov-switching dynamic regression model from the switching mechanism mc and the state-specific submodels mdl.
msVAR with properties:
SeriesNames: "1"
Switch: [1x1 dtmc]
Submodels: [2x1 varm]
Mdl.Submodels(1)
Covariance: 2
Mdl is a fully specified msVAR object representing a univariate two-state Markov-switching dynamic regression model. msVAR stores specified arima submodels as varm objects.
Because Mdl is fully specified, you can pass it to any msVAR object function for further analysis (see Object Functions). Or, you can specify that the parameters of Mdl are initial values for the estimation procedure (see estimate).
Consider a two-state Markov-switching dynamic regression model of the postwar US real GDP growth rate. The model has the parameter estimates presented in [1].
Create a discrete-time Markov chain model that describes the regime switching mechanism. Label the regimes.
P = [0.92 0.08; 0.26 0.74];
mc = dtmc(P,'StateNames',["Expansion" "Recession"]);
mc is a fully specified dtmc object.
Create separate AR(0) models (constant only) for the two regimes.
sigma = 3.34; % Homoscedastic models across states
mdl1 = arima('Constant',4.62,'Variance',sigma^2);
mdl2 = arima('Constant',-0.48,'Variance',sigma^2);
mdl = [mdl1 mdl2];
Create the Markov-switching dynamic regression model that describes the behavior of the US GDP growth rate.
Mdl is a fully specified msVAR object.
Consider fitting to data a two-state Markov-switching model for a 1-D response process.
Create a discrete-time Markov chain model for the switching mechanism. Specify a 2-by-2 matrix of NaN values for the transition matrix. This setting indicates that you want to estimate all transition probabilities. Label the states.
mc is a partially specified dtmc object. The transition matrix mc.P is completely unknown and estimable.
Create AR(1) and AR(2) models by using the shorthand syntax of arima. After you create each model, specify the model description by using dot notation.
mdl1.Description = "Expansion State"
mdl2.Description = "Recession State"
mdl1 and mdl2 are partially specified arima objects. NaN-valued properties correspond to unknown, estimable parameters.
Create a Markov-switching model template from the switching mechanism mc and the state-specific submodels mdl.
Mdl is a partially specified msVAR object representing a univariate two-state Markov-switching dynamic regression model.
msVAR converts the arima object submodels to 1-D varm object equivalents.
Mdl is prepared for estimation. You can pass Mdl, along with data and a fully specified model containing initial values for optimization, to estimate.
Create a three-state Markov-switching dynamic regression model for a 2-D response process. Specify all parameter values (this example uses arbitrary values).
Create a three-state discrete-time Markov chain model that describes the regime switching mechanism.
P = [10 1 1; 1 10 1; 1 1 10];
mc is a dtmc object. dtmc normalizes P so that each row sums to 1.
For each regime, use varm to create a VAR model that describes the response process within the regime. Specify all parameter values.
% Constants (numSeries x 1 vectors)
C1 = [1;-1];
% Autoregression coefficients (numSeries x numSeries matrices)
AR1 = {}; % 0 lags
AR2 = {[0.5 0.1; 0.5 0.5]}; % 1 lag
AR3 = {[0.25 0; 0 0] [0 0; 0.25 0]}; % 2 lags
% Innovations covariances (numSeries x numSeries matrices)
Sigma1 = [1 -0.1; -0.1 1];
% VAR Submodels
mdl1 = varm('Constant',C1,'AR',AR1,'Covariance',Sigma1);
mdl1, mdl2, and mdl3 are fully specified varm objects.
mdl = [mdl1; mdl2; mdl3];
SeriesNames: ["1" "2"]
Constant: [1 -1]'
AR: {2×2 matrix} at lag [1]
Mdl is a fully specified msVAR object representing a multivariate three-state Markov-switching dynamic regression model.
Consider including regression components for exogenous variables in each submodel of the Markov-switching dynamic regression model in Create Fully Specified Multivariate Model.
For each regime, use varm to create a VARX model that describes the response process within the regime. Specify all parameter values.
% Regression coefficients (numSeries x numRegressors matrices)
Beta1 = [1;-1]; % 1 regressor
Beta2 = [2 2;-2 -2]; % 2 regressors
Beta3 = [3 3 3;-3 -3 -3]; % 3 regressors
% VARX Submodels
mdl1 = varm('Constant',C1,'AR',AR1,'Beta',Beta1,...
'Covariance',Sigma1);
mdl1, mdl2, and mdl3 are fully specified varm objects representing the state-specified submodels.
Description: "2-Dimensional VARX(0) Model with 1 Predictor"
Description: "AR-Stationary 2-Dimensional VARX(1) Model with 2 Predictors"
Consider fitting to data a three-state Markov-switching model for a 2-D response process.
Create a discrete-time Markov chain model for the switching mechanism. Specify a 3-by-3 matrix of NaN values for the transition matrix. This setting indicates that you want to estimate all transition probabilities.
Create 2-D VAR(0), VAR(1), and VAR(2) models by using the shorthand syntax of varm. Store the models in a vector.
mdl1 = varm(2,0);
mdl = [mdl1 mdl2 mdl3];
mdl contains three state-specific varm model templates for estimation. NaN values in the properties indicate estimable parameters.
AR: {2×2 matrix of NaNs} at lag [1]
AR: {2×2 matrices of NaNs} at lags [1 2]
Mdl is a partially specified msVAR model for estimation.
Consider including regression components for exogenous variables in the submodels of the Markov-switching dynamic regression model in Create Partially Specified Multivariate Model for Estimation. Assume that the VAR(0) model includes the regressor
{\mathit{x}}_{1\mathit{t}}
, the VAR(1) model includes the regressors
{\mathit{x}}_{1\mathit{t}}
{\mathit{x}}_{2\mathit{t}}
, and the VAR(2) model includes the regressors
{\mathit{x}}_{1\mathit{t}}
{\mathit{x}}_{2\mathit{t}}
{\mathit{x}}_{3\mathit{t}}
Create the discrete-time Markov chain.
Create 2-D VARX(0), VARX(1), and VARX(2) models by using the shorthand syntax of varm. For each model, set the Beta property to a numSeries-by-numRegressors matrix of NaN values by using dot notation. Store all models in a vector.
mdl1 = varm(numSeries,0);
mdl1.Beta = NaN(numSeries,1);
Create a Markov-switching dynamic regression model from the switching mechanism mc and the state-specific submodels mdl.
Beta: [2×2 matrix of NaNs]
Consider the model in Create Partially Specified Multivariate Model for Estimation. Suppose theory dictates that states do not persist.
Create a discrete-time Markov chain model for the switching mechanism. Specify a 3-by-3 matrix of NaN values for the transition matrix. Indicate that states do not persist by setting the diagonal elements of the matrix to 0.
P(logical(eye(3))) = 0;
mc is a partially specified dtmc object.
Create the submodels and store them in a vector.
submdl = [mdl1; mdl2; mdl3];
Mdl = msVAR(mc,submdl);
Mdl.Switch.P
NaN 0 NaN
estimate treats the known diagonal elements of the transition matrix as equality constraints during estimation. For more details, see estimate.
A Markov-switching dynamic regression model of a univariate or multivariate response series yt describes the dynamic behavior of the series in the presence of structural breaks or regime changes. A collection of state-specific dynamic regression submodels describes the dynamic behavior of yt within the regimes.
{y}_{t}~\left\{\begin{array}{cc}{f}_{1}\left({y}_{t};{x}_{t},{\mathrm{θ}}_{1}\right)& ,{s}_{t}=1\\ {f}_{2}\left({y}_{t};{x}_{t},{\mathrm{θ}}_{2}\right)& ,{s}_{t}=2\\ ⋮& ⋮\\ {f}_{n}\left({y}_{t};{x}_{t},{\mathrm{θ}}_{n}\right)& ,{s}_{t}=n\end{array},
st is a discrete-time Markov chain representing the switching mechanism among regimes (Switch).
n is the number of regimes (NumStates).
{f}_{i}\left({y}_{t};{x}_{t},{\mathrm{θ}}_{i}\right)
is the regime i dynamic regression model of yt (Submodels(i)). Submodels are either univariate (ARX) or multivariate (VARX).
xt is a vector of observed exogenous variables at time t.
θi is the regime i collection of parameters of the dynamic regression model, such as AR coefficients and the innovation variances.
Hamilton [2] proposes a general model, known as Markov-switching autoregression (MSAR), allowing for lagged values of the switching state s. Hamilton [3] shows how to convert an MSAR model into a dynamic regression model with a higher-dimensional state space, supported by msVAR.
[1] Chauvet, M., and J. D. Hamilton. "Dating Business Cycle Turning Points." In Nonlinear Analysis of Business Cycles (Contributions to Economic Analysis, Volume 276). (C. Milas, P. Rothman, and D. van Dijk, eds.). Amsterdam: Emerald Group Publishing Limited, 2006.
[2] Hamilton, J. D. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle." Econometrica. Vol. 57, 1989, pp. 357–384.
[3] Hamilton, J. D. "Analysis of Time Series Subject to Changes in Regime." Journal of Econometrics. Vol. 45, 1990, pp. 39–70.
[5] Krolzig, H.-M. Markov-Switching Vector Autoregressions. Berlin: Springer, 1997.
dtmc | arima | varm | tsVAR
|
Aero-hydrodynamic loads investigations for different constructions in turbulent flows with special verification approach | JVE Journals
Saveliy Kaplunov1 , Natalya Valles2 , Lidya Shitova3 , Valeriy Foursov4
1, 2, 3, 4Federal Budget-funded Institute for Machine Science named after A. Blagonravov of the Russian Academy of Sciences, Moscow, Russia
Received 3 June 2019; accepted 18 June 2019; published 26 September 2019
Copyright © 2019 Saveliy Kaplunov, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper is devoted to the analysis of numerical modeling data exact as it possible to determine aero-hydrodynamic loads from high-speed trains on different constructions and objects of transport infrastructure. These investigations include pressure and flow velocities distributions definition on the surfaces for specified objects and also with special investigation of amplitude-frequency characteristics for the individual construction elements. According to proposed special verification methodical approach it confirms efficiency and correctness of the results obtained by authors. This includes too the application of the main original combined numerical approach as for calculation process so as for effective verification of these results with support of special analysis in case of different proposed verification schemes.
Keywords: aero-hydrodynamic loads, infrastructure facilities, Karman vortices, amplitude- frequency characteristics, turbulence models, finite volumes method, coefficients of lift and drag forces, method of discrete vortices.
This work is step forward to creation and application of mathematical models for the most dangerous oscillations excitation mechanisms for different constructions in liquid or air flow, as well as to creation of efficient computational methods for description of these models and processes. The paper contains some results of analysis for numerical modeling data, exact as it possible to determine aerodynamic loads in all these cases with constructions and objects of transport (Fig. 1). It includes definition of pressure and flow velocities distributions on the surfaces of specified objects and also with special investigation of amplitude-frequency characteristics for the individual construction elements. According to proposed special verification methodical approach [1-3] authors confirmed efficiency and correctness of the obtained results. This includes too the application of the main original combined numerical approach as for calculation process so as for effective verification of obtained results with support of special analysis.
Fig. 1. Application of the proposed approach in railway transport systems (the pedestrian bridge)
Numerical investigation method of hydrodynamic forces arising from separated flow, and construction oscillations excited by these forces, were worked out by authors. For this task decision authors proposed numerical methodic (Fig. 2) of aero-hydrodynamic loads modeling from high-speed trains (ANSYS-CFX) –2D-3D task decisions in combination with some specially chosen vortices methods, such as elaborated in Institute for Machine Science of the Russian Academy of Sciences Modernized Method of Discrete Vortices (MMDV) – for decision of 2D – tasks especially in this direction [4, 5].
Fig. 2. The Scheme of proposed combined approach functioning for complicated systems decision (elaborated by authors [2, 5])
2. Numerical methodic of aero-hydrodynamic loads modeling
2.1. (ANSYS – CFX) – for decision of 3D tasks
Basic equations. In this case aerodynamic loads on high-speed trains and on infrastructure objects are defined by use of decision of Navies-Stokes 3D non-stationary non-linear Eq. (1) of hydro-gas-dynamics with account of medium viscosity [6].
The main assumptions are the next:
1. Mass forces are out of consideration.
2. Medium is not compressible at first stage.
3. Process is isothermal:
\rho \frac{\partial u}{\partial t}+\rho u\frac{\partial u}{\partial x}+\rho v\frac{\partial u}{\partial y}+\rho w\frac{\partial u}{\partial z}=-\frac{\partial \rho }{\partial x}+\mu \left(\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}\right),
\rho \frac{\partial v}{\partial t}+\rho u\frac{\partial v}{\partial x}+\rho v\frac{\partial v}{\partial y}+\rho w\frac{\partial v}{\partial z}=-\frac{\partial \rho }{\partial y}+\mu \left(\frac{{\partial }^{2}v}{\partial {x}^{2}}+\frac{{\partial }^{2}v}{\partial {y}^{2}}+\frac{{\partial }^{2}v}{\partial {z}^{2}}\right),
\rho \frac{\partial w}{\partial t}+\rho u\frac{\partial w}{\partial x}+\rho v\frac{\partial w}{\partial y}+\rho w\frac{\partial w}{\partial z}=-\frac{\partial \rho }{\partial z}+\mu \left(\frac{{\partial }^{2}w}{\partial {x}^{2}}+\frac{{\partial }^{2}w}{\partial {y}^{2}}+\frac{{\partial }^{2}w}{\partial {z}^{2}}\right).
Besides, it is necessary to satisfy equations of indissolubility and of state:
\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u\right)}{\partial x}+\frac{\partial \left(\rho v\right)}{\partial y}+\frac{\partial \left(\rho w\right)}{\partial z}=0.
p=\rho RT,
u
v
w
– unknown velocity vector components (according to axes
x
y
z
p
t
\mu
– dynamic viscosity coefficient for air,
\rho
R
– universal gas constant,
T
– temperature.
It is also necessary to define boundary and initial conditions.
For wind flows modeling simplification it is rationally to suppose them to be incompressible (
\rho =
idem) and isothermal. Mass forces are not taken into account. As exact analytical solutions of Navies-Stokes 3D non-stationary non-linear equations are absent, for real tasks solution Finite Volume Method (FVM) [3, 5]. is successfully applied in practice.
2.2. Modernized method of discrete vortices (MMDV) – for decision of 2D – problems [3]
Method does not require grids construction and does not contain empirical parameters.
The numerical scheme is stable (there are no stops due to unlimited growth of variables number).
The developed method significantly extends possibilities of vortex formation mechanism numerical study at arbitrary motion and streamlined bodies shape changing
Existing software systems of computational fluid dynamics with grid methods are sometimes ineffective, because calculation of variable geometry structures is extremely long.
On a single mathematical and computational basis it is possible to create software hierarchy system which covers a wide range of applications. According to this basis with account of physical experiment applicability limits these schemes and models are realized successfully. Method is characterized by unique ability of vortex traces and flow formation at the same time. The dimension of problem is significantly reduced, since it is necessary to monitor not the entire area, but only vortices on the body surface and in the wake.
Apparently, it can be argued that the numerical experiment based on discrete vortices method has elements of what is commonly called “artificial intelligence” as it reproduces those features of process which were not explicitly incorporated into algorithms and models.
Calculation example with ANSYS – CFX (3D) and MMDV (2D) with application for pressure distribution definition on the surface of the pedestrian bridge and by according measuring and verification (Figs. 3-5).
Fig. 3. The pedestrian bridge
Direct calculations of flow velocity from high speed train aerodynamic effects were worked out in 3D by means of software module ANSYS (CFX), which allows on the base of Navies-Stocks equations to receive decision in form of dependence of flow velocity on the bridge from the distance between train and pedestrian bridge (all the surfaces of bridge). After that it is necessary to value magnitudes of flow velocity components near the bridge as
{V}_{1}
– velocity in horizontal plane perpendicularly to bridge,
{V}_{2}
– velocity in horizontal plane along the bridge and V3 velocity in vertical plane perpendicularly to horizontal plane.
In this case it is necessary to choose the main air flow with velocity V1 in horizontal plane perpendicularly to bridge in the direction to the bridge with corresponding values of velocity for 2D realization of task decision with MMDV application. Main results in this case of plain 2D task are in Fig. 5. Shown in Fig. 5 coefficients
{C}_{x}
{C}_{y}
give the opportunity to define dependencies of aerodynamic forces from time.
Fig. 4. Calculation of pressure on pedestrian bridge surface
Fig. 5. The estimated Karman vortices paths and aerodynamic forces for the pedestrian bridge
It is necessary also to remark, that MMDV Method allows to research aerodynamic loads on bodies making arbitrary movements and to solve problems of body motion under aerodynamic forces.
The calculation was carried out using both statements (verification of relative differences are not more than 10-15 %) for the pedestrian bridge [3, 5, 7, 8], shown in Fig. 3-5.
Initial values of the approach flow velocity are obtained from the solution of problem in ANSYS CFX in 3D setting (its initial stage) (Fig. 2) in accordance with the scheme of methods interaction in combined approach framework. In a 3D setting, furthermore, pressure distributions on the bridge structure during the passage of a high-speed train under the pedestrian bridge are obtained (Fig. 3, 4) with pressure magnitudes from 200 till 1120 Pa (
{p}_{min}=
160 Pa and
{p}_{max}=
1120 Pa) as well as according dependences of lifting and drag forces from time and frequency characteristics of the flow loading process (Fig. 5).
The paper is devoted to the analysis and verification of numerical modeling data in determining of aero-hydrodynamic loads or vibrations and impacts from dynamic problems typical for high-speed trains influence on standard infrastructure facilities of JSC “Russian Railways”, such as pedestrian bridges, wall screens and tunnels.
The purpose of this work is the analysis of data obtained by calculation of the frequency spectra components of the aero-hydrodynamic loads, pressure and speed or velocity distributions on the specified objects and amplitude-frequency characteristics for the individual construction elements.
It confirms efficiency and correctness of the results obtained by authors with the application attached to proposed original combined calculation approach. This approach is a linked system of carrying out complex research by two advanced calculation methods in their permanent interaction with an increased efficiency and with significant reduction in the cost (on 20-30 %) of expenditures realization for numerical investigations.
The obtained data were verified according to proposed scheme of interaction with the use of the bank of experimental data collected by authors, including test data at both the international standard models and the domestic ones (tests with liquid and air flows in wind tunnels and on aero-hydrodynamic stands). Relative error of such approach for the whole raw of decided tasks is about 10-17 %.
Kaplounov S. M., Valles N. G., Solonin V. I. Analysis of errors agreement for Complex numerical-experimental investigations. Engineering Aero-Hydroelasticity: 2nd International Conference EAHE, Prague, 1999, p. 223-230. [Search CrossRef]
Frolov K. V., Makhutov N. A., Kaplunov S. M., et al. Dynamic of Aero-Hydroelastic System Constructions. Science, Moscow, 2002, p. 570. [Search CrossRef]
Kaplunov S. M., Valles N. G., Makhutov N. A., Dubinsky S. I., Samsonov V. A. Aerodynamic effect of high-speed trains on the infrastructure of JSC “RZD”. The Bulletin of The Joint Academic Council of JSC “RZD”, Vols. 1-2, 2016, p. 47-57. [Search CrossRef]
Stepnov M. Statistical Methods of Processing with Results of Mechanical Tests. Reference Book. Mashinebuilding, Moscow, 1985, p. 231. [Search CrossRef]
Kaplunov S. M., Chetverushkin B. N., Doubinskiy S. I., Valles N. G., Dronova E. A. Modeling of aerodynamic loads on infrastructure elements for high speed trains. Proceedings of the 1nd International Conference on Railway Technology: Research, Development and Maintenance, 2014. [Search CrossRef]
Abramovich G. N. Applied Gasdynamics. Science, Moscow, 1969, p. 824. [Search CrossRef]
Venikov V. A., Venikov G. V. Similarity Theory and Modeling (Applied to Tasks of the Electric Power Industry): Textbook. KD Librocom, 2014, p. 439. [Search CrossRef]
Shin Y. S., Wambsganss H. W. Flow Induced Vibration in LMFBR steam generators. A state-of-the-art review. Nuclear Engineering and Design, Vol. 40, Issue 2, 1977, p. 221-285. [Publisher]
Determining the Crossflow Parameters of Two Tandem Pipelines by Numerical Methods
S. M. Kaplunov, N. G. Val’es, V. Yu. Fursov, N. A. Chentsova, V. V. Perevezentsev
|
Concentration phenomena of two-vortex solutions in a Chern-Simons model
Chen, Chiun-Chuan ; Lin, Chang-Shou ; Wang, Guofang
By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.
author = {Chen, Chiun-Chuan and Lin, Chang-Shou and Wang, Guofang},
title = {Concentration phenomena of two-vortex solutions in a {Chern-Simons} model},
AU - Chen, Chiun-Chuan
AU - Wang, Guofang
TI - Concentration phenomena of two-vortex solutions in a Chern-Simons model
Chen, Chiun-Chuan; Lin, Chang-Shou; Wang, Guofang. Concentration phenomena of two-vortex solutions in a Chern-Simons model. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 2, pp. 367-397. http://www.numdam.org/item/ASNSP_2004_5_3_2_367_0/
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\Delta u=8\pi -8\pi h{e}^{u}
on a compact Riemann surface, Asian J. Math., 1 (1997), 230-248. | MR 1491984 | Zbl 0955.58010
[16] W. Ding - J. Jost - J. Li - X. Peng - G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functional with 6th order potenliatls, Comm. Math. Phys. 217 (2001), 383-407. | MR 1821229 | Zbl 0994.58009
[17] G. Dunne, “Self-dual Chern-Simons Theories”, Lecture Notes in Physics m36, Springer-Verlag, Berlin, 1995. | Zbl 0834.58001
[18] J. Hong - Y. Kim - P. Y. Pac, Multivortex solutions of the Abelian Chern Simons theory, Phys. Rev. Letter 64 (1990), 2230-2233. | MR 1050529 | Zbl 1014.58500
[19] R. Jackiw - E. J. Weinberg, Selfdual Chern Simons vortices, Phys. Rev. Lett. 64 (1990), 2234-2237. | MR 1050530 | Zbl 1050.81595
[20] Y. Y. Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (1999), 421-444. | MR 1673972 | Zbl 0928.35057
[21] C. S. Lin, Topological degree for mean field equations on
{S}^{2}
, Duke Math. J. 104 (2000), 501-536. | MR 1781481 | Zbl 0964.35038
[22] C. S. Lin, Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Ration. Mech. Anal. 153 (2000), 153-176. | MR 1770683 | Zbl 0968.35045
[23] M. Nolasco, Non-topological
N
-vortex condensates for the self-dual chern-Simons theory, Comm. Pure Appl. Math. 56 (2003), 1752-1780. | MR 2001445 | Zbl 1032.58005
[24] M. Nolasco - G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations 9 (1999), 31-94. | MR 1710938 | Zbl 0951.58030
[25] M. Nolasco - G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal. 145 (1998), 161-195. | MR 1664542 | Zbl 0980.46022
[26] J. Prajapat - G. Tarantello, On a class of elliptic problems in
{ℝ}^{2}
: symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 967-985. | MR 1855007 | Zbl 1009.35018
[27] Spruck - Y. Yang, Topological solutions in the self-dual Chern-Simons theory: existence and approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 75-97. | Numdam | MR 1320569 | Zbl 0836.35007
[28] Taubes, Arbitrary
N
-vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys. 72 (1980), 277-292. | MR 573986 | Zbl 0451.35101
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The cube root of 0 064 _____ (steps also) - Maths - Cubes and Cube Roots - 13762253 | Meritnation.com
The cube root of 0.064 _____ (steps also)
Sushant Ranjan answered this
0.064 = \frac{64}{1000}\phantom{\rule{0ex}{0ex}}Now,\phantom{\rule{0ex}{0ex}}\sqrt[3]{0.064} = \sqrt[3]{\frac{64}{1000}} = \frac{\sqrt[3]{64}}{\sqrt[3]{1000}} = \frac{\sqrt[3]{{4}^{3}}}{\sqrt[3]{{10}^{3}}} = \frac{4}{10} = 0.4
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$$ Wikipedia
{\displaystyle {\mathsf {S}}\!\!\!{{\big |}\!{\big |}}}
One theory holds that the sign grew out of the Spanish and Spanish American scribal abbreviation "ps" for pesos. A study of late 18th- and early 19th-century manuscripts shows that the s gradually came to be written over the p, developing into a close equivalent to the "$" mark.[8][9][10][11][12] However, there are documents showing the common use of the two-stroke version in Portugal already by 1775.[13]
{\displaystyle \mathrm {CRS} \!\!\!\Vert }
{\displaystyle \mathrm {CrS} \!\!\!\Vert }
{\displaystyle \mathrm {McrS} \!\!\!\Vert }
{\displaystyle \mathrm {MS} \!\!\!\Vert }
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Linear or Quadratic Objective with Quadratic Constraints - MATLAB & Simulink
Quadratic Constrained Problem
Efficiency When Providing a Hessian
Extension to Quadratic Equality Constraints
This example shows how to solve an optimization problem that has a linear or quadratic objective and quadratic inequality constraints. The example generates and uses the gradient and Hessian of the objective and constraint functions.
Suppose that your problem has the form
\underset{x}{\mathrm{min}}\left(\frac{1}{2}{x}^{T}Qx+{f}^{T}x+c\right)
\frac{1}{2}{x}^{T}{H}_{i}x+{k}_{i}^{T}x+{d}_{i}\le 0,
where 1 ≤ i ≤ m. Assume that at least one Hi is nonzero; otherwise, you can use quadprog or linprog to solve this problem. With nonzero Hi, the constraints are nonlinear, which means fmincon is the appropriate solver according to the Optimization Decision Table.
The example assumes that the quadratic matrices are symmetric without loss of generality. You can replace a nonsymmetric H (or Q) matrix with an equivalent symmetrized version (H + HT)/2.
If x has N components, then Q and Hi are N-by-N matrices, f and ki are N-by-1 vectors, and c and di are scalars.
Formulate the problem using fmincon syntax. Assume that x and f are column vectors. (x is a column vector when the initial vector x0 is a column vector.)
For consistency and easy indexing, place every quadratic constraint matrix in one cell array. Similarly, place the linear and constant terms in cell arrays.
Suppose that you have the following problem.
Create a Hessian function. The Hessian of the Lagrangian is given by the equation
{\nabla }_{xx}^{2}L\left(x,\lambda \right)={\nabla }^{2}f\left(x\right)+\sum {\lambda }_{i}{\nabla }^{2}{c}_{i}\left(x\right)+\sum {\lambda }_{i}{\nabla }^{2}ce{q}_{i}\left(x\right).
fmincon calculates an approximate set of Lagrange multipliers λi, and packages them in a structure. To include the Hessian, use the following function.
Use the fmincon interior-point algorithm to solve the problem most efficiently. This algorithm accepts a Hessian function that you supply. Set these options.
Call fmincon to solve the problem.
Examine the Lagrange multipliers.
Both nonlinear inequality multipliers are nonzero, so both quadratic constraints are active at the solution.
The interior-point algorithm with gradients and a Hessian is efficient. View the number of function evaluations.
fmincon takes only 10 function evaluations to solve the problem.
Compare this result to the solution without the Hessian.
In this case, fmincon takes about twice as many iterations and function evaluations. The solutions are the same to within tolerances.
If you also have quadratic equality constraints, you can use essentially the same technique. The problem is the same, with the additional constraints
\frac{1}{2}{x}^{T}{J}_{i}x+{p}_{i}^{T}x+{q}_{i}=0.
Reformulate your constraints to use the Ji, pi, and qi variables. The lambda.eqnonlin(i) structure has the Lagrange multipliers for equality constraints.
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This problem is a checkpoint for simplifying expressions. It will be referred to as Checkpoint 8 .
4x^2+3x-7+(-2x^2)-2x+(-3)
-3x^2-2x+5+4x^2-7x+6
Answers and extra practice for the Checkpoint problems are located in the back of your printed textbook or in the Reference Tab of your eBook. If you have an eBook for Core Connections, Course 1, login and then click the following link: Checkpoint 8: Simplifying Expressions
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Revision as of 21:19, 10 May 2022 by Wrrasba (talk | contribs) (→Pokémon GO)
{\displaystyle HP={\Biggl \lfloor }{{\Biggl (}(Base+DV)\times 2+{\biggl \lfloor }{\tfrac {{\bigl \lceil }{\sqrt {STATEXP}}{\bigr \rceil }}{4}}{\biggr \rfloor }{\Biggr )}\times Level \over 100}{\Biggr \rfloor }+Level+10}
{\displaystyle OtherStat={\Biggl \lfloor }{{\Biggl (}(Base+DV)\times 2+{\biggl \lfloor }{\tfrac {{\bigl \lceil }{\sqrt {STATEXP}}{\bigr \rceil }}{4}}{\biggr \rfloor }{\Biggr )}\times Level \over 100}{\Biggr \rfloor }+5}
{\displaystyle HP={\Bigl \lfloor }{(2\times Base+IV+\lfloor {\tfrac {EV}{4}}\rfloor )\times Level \over 100}{\Bigr \rfloor }+Level+10}
{\displaystyle OtherStat={\Biggl \lfloor }{\biggl (}{\Bigl \lfloor }{(2\times Base+IV+\lfloor {\tfrac {EV}{4}}\rfloor )\times Level \over 100}{\Bigr \rfloor }+5{\biggr )}\times Nature{\Biggr \rfloor }}
{\displaystyle Stat=(base+IV)\times cpMult}
{\displaystyle 2\times IV_{HP}+1}
{\displaystyle 2\times IV_{Attack}+1}
{\displaystyle 2\times IV_{Defense}+1}
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SortBy - Maple Help
Home : Support : Online Help : Programming : CodeTools : Profiling : SortBy
analyze and display profiling data
SortBy(proc1, proc2, ..., tab1, tab2, ..., opts)
(optional) equation(s) of the form option=value where option is one of 'number', 'order', or 'statements'; specify options for the SortBy command
The SortBy() command analyzes the profiling data for all the procedures for which data is available and prints the results.
The SortBy(proc1, proc2, ...) command analyzes the profiling data for the specified procedures and prints the results.
The SortBy(proc1, proc2, ..., tab1, tab2, ...) command analyzes the profiling data for the specified procedures and tables and prints the results.
The opts parameter can contain any of the following equations that specify options for the SortBy command.
'number' = nonnegative integer
Specifies the number of items displayed. The number option can be used to control the maximum number of items displayed. If
\mathrm{number}=n
, the top n elements are selected after the items have been sorted.
'order' = value
Specifies how to sort the table. If order is not specified, then the default is rload.
The optional argument order can be any one of the following:
alpha - sort table alphabetically by function name
ralpha - sort table reverse alphabetically by function name
time - sort table by increasing cpu time usage
rtime - sort table by decreasing cpu time usage
words - sort table by increasing memory usage
rwords - sort table by decreasing memory usage
calls - sort table by increasing number of calls to each function
rcalls - sort table by decreasing number of calls to each function
load - sort table by increasing memory^2*time
rload - sort table by decreasing memory^2*time
'statements' = true or false
Specifies whether to analyze procedures or statements. If statements=true is specified, then the SortBy function analyzes the statements instead of procedures. All the statements of the procedures are considered.
When displaying statements, the procedure name, statement number, and an abbreviation of the statement are provided with the profiling information.
It is possible that the total number of words used or the total time taken is zero. In this case, the percent times and percent words are always zero, and the total percent time and the total percent words is 100%. If either of these events occur, SortBy prints a warning message.
int( sin(exp(x^i)), x );
int( tan(exp(x^i)), x );
c := proc( )
\mathrm{with}\left(\mathrm{CodeTools}[\mathrm{Profiling}]\right):
t≔\mathrm{Build}\left(\mathrm{procs}=[a,b,c],\mathrm{commands}='c\left(\right)'\right):
\mathrm{SortBy}\left(t\right)
function calls time time% words words%
c 1 0.000 0.00 12 0.00
a 1 1.235 39.06 7276688 45.11
b 1 1.927 60.94 8854209 54.89
total: 3 3.162 100.00 16130909 100.00
\mathrm{SortBy}\left(t,\mathrm{order}=\mathrm{time}\right)
\mathrm{SortBy}\left(t,\mathrm{order}=\mathrm{words}\right)
\mathrm{SortBy}\left(t,\mathrm{order}=\mathrm{load},\mathrm{statement}=\mathrm{true}\right)
b 30 1.927 60.94 8854209 54.89
stat 2) 2 int(tan(exp(x^i)),x)
a 25 1.235 39.06 7276688 45.11
stat 2) 2 int(sin(exp(x^i)),x)
b 1 0.000 0.00 0 0.00
stat 1) 1 for i to 30 do
c 1 0.000 0.00 6 0.00
stat 1) 1 a();
a 1 0.000 0.00 0 0.00
stat 2) 2 b()
total: 59 3.162 100.00 16130909 100.00
\mathrm{SortBy}\left(t,\mathrm{order}=\mathrm{load},\mathrm{number}=3,\mathrm{statement}=\mathrm{true}\right)
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Estimate a bivariate CDF in SAS » SAS博客列表
Estimate a bivariate CDF in SAS
Statistical Graphics, Statistical Programming Add comments
This article shows how to estimate and visualize a two-dimensional cumulative distribution function (CDF) in SAS. SAS has built-in support for this computation. Although the bivariate CDF is not used as much as the univariate CDF, the bivariate version is still a useful tool in understanding the probable values of a random variate for a two-dimensional distribution.
Recall that the univariate CDF at the value x is the probability that a random observation from a distribution is less than x. In symbols, if X is a random variable, then the distribution function is \(F_X(x) = P(X\leq x)\)
F_X(x) = P(X\leq x)
. The two-dimensional CDF is similar, but it gives the probability that two random variables are both less than specified values. If X and Y are random variables, the bivariate CDF for the joint distribution is given by \(F_{XY}(x,y) = P(X \leq x ~\mbox{and}~ Y \leq y)\)
F_{XY}(x,y) = P(X \leq x ~\mbox{and}~ Y \leq y)
Univariate estimates of the CDF
Let's start with a one-dimensional example before exploring the topic in higher dimensions. For univariate probability distributions, statisticians use both the density function and the cumulative distribution function. They each contain the same information, and if you know one you can find the other. The cumulative distribution (CDF) of a continuous distribution function is the integral of the probability density function (PDF); the density function is the derivative of the cumulative distribution.
In terms of data analysis, a histogram is an estimate of a density function. So is a kernel density estimate. Consequently, there are three popular methods for estimating a CDF from data:
The empirical distribution function, which can be computed by using PROC UNIVARIATE. You can get a table of values by using the FREQ option or visualize the curve by using the CDFPLOT statement.
Add up the values in the bins of a histogram to create an ogive, which is a cumulative histogram. This estimate tends to be coarse unless there are many histogram bins.
Sum the values of a kernel density estimate to obtain an estimate of the cumulative distribution. You can use the CDF option in PROC KDE to get the cumulative estimates into a SAS data set.
Let's compare the first and last methods to estimate the CDF. For data, use the MPG_City variable from the Sashelp.Cars data.
/* Use PROC UNIVARIATE to compute the empirical CDF from data */
proc univariate data=sashelp.Cars; /* use the FREQ option to get a table */
cdfplot MPG_City;
ods select CDFPlot;
ods output CDFPlot=UNIOut;
/* Use PROC KDE to estimate the CDF from a kernel density estimate */
proc kde data=sashelp.Cars;
univar MPG_City / CDF out=KDEOut;
/* combine the estimates and plot on the same graph */
set UNIOut(rename=(ECDFY=ECDF ECDFX=X))
KDEOut(rename=(distribution=kerCDF Value=X));
ECDF = ECDF / 100; /* convert from percent to proportion */
keep X ECDF kerCDF;
title "Estimates of CDF";
label X = "MPG_City";
series x=X y=ECDF / legendlabel="Empirical CDF";
series x=X y=kerCDF / legendlabel="Kernel CDF";
yaxis label="Cumulative Distribution" grid;
xaxis values=(5 to 60 by 5) grid valueshint;
The two estimates are very close. The empirical CDF is more "jagged," whereas the estimate from the kernel density is smoother. You could use either curve to estimate quantiles or probabilities.
Bivariate estimates of the CDF
Now onto the bivariate case. For bivariate data, you cannot use PROC UNIVARIATE (obviously, from the name!), but you can still use PROC KDE. Instead of using the UNIVAR statement, you can use the BIVAR statement to compute a bivariate KDE. Let's look at the joint distribution of the MPG_City and Weight variables in the Sashelp.Cars data:
scatter x=MPG_City y=Weight;
The scatter plot shows that the density of these points is concentrated near the lower-left corner of the plot. You can use PROC KDE to visualize the density. The PLOTS= option can create a contour plot and a two-dimensional histogram, as follows:
title "2-D Histogram";
bivar MPG_City Weight / plots=(Contour Histogram)
CDF out=KDEOut(rename=(Distribution=kerCDF Value1=X Value2=Y));
The OUT= option writes a SAS data set that contains the estimates for the density; the CDF option adds the cumulative distribution estimates. The column names are Value1 (for the first variable on the BIVAR statement), Value2 (for the second variable), and Distribution (for the CDF). I have renamed the variables to X, Y, and CDF, but that is not required. You can plot use a heat map (or a contour plot) to visualize the bivariate CDF, as follows:
title "Estimate of 2-D CDF";
label X="MPG_City" Y="Weight" kerCDF="Kernel CDF";
heatmapparm x=X y=Y colorresponse=kerCDF;
The CDF of bivariate normal data
The CDF is not used very often for bivariate data because the density function shows more details. In some sense, all CDFs look similar. In the previous example, the two variables are negatively correlated and are not linearly related. For comparison, the following program uses the RANDNORMAL function in SAS/IML to generate bivariate normal data that has a positive correlation of 0.6. (I have previously shown the empirical CDF for this data, and I have also drawn the graph of the bivariate CDF for the population.) The calls to PROC KDE and PROC SGPLOT are the same as for the previous example:
call randseed(123456);
N = 1e5; /* sample size */
Sigma = ( 1 || rho)// /* correlation matrix */
(rho|| 1 );
mean = {0 0};
Z = randnormal(N, mean, Sigma); /* sample from MVN(0, Sigma) */
create BivNorm from Z[c={X Y}]; append from Z; close;
proc kde data=BivNorm;
bivar X Y / plots=NONE
title "Estimate of 2-D Normal CDF";
proc sgplot data=KDEOut aspect=1;
label kerCDF="Kernel CDF";
If you compare the bivariate CDF for the Cars data to the CDF for the bivariate normal data, you can see differences in the range of the red region, but the overall impression is the same. The CDF is near 0 in the lower-left corner, near 1 in the upper-right corner, and is approximately 0.5 along an L-shaped curve near the middle of the data. This is the basic shape of a generic 2-D CDF, just as an S-shape or sigmoid-shaped curve is the basic shape of a generic 1-D CDF.
You can use the CDF option on the BIVAR statement in PROC KDE to obtain an estimate for a bivariate CDF. The estimate is based on a kernel density estimate of the data. You can use the HEATMAPPARM statement in PROC SGPLOT to visualize the CDF surface. If you prefer a contour plot, you can use the graph template language (GTL) to create a contour plot.
The post Estimate a bivariate CDF in SAS appeared first on The DO Loop.
The probability integral transform Compute 2-D cumulative sums and ogives
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Reward Farming - PIP
Send money to others on the PIP extension. Earn PIP tokens. PIP has an interactive token distribution model. Users who send tokens to other users with PIP extension, earn regular rewards of $PIP token. The goal is to accelerate the transition to Web3 by attracting more users. Rewards may be subject to user-specific limits.
Reward farming period: Rewards are calculated daily, based on the past 24 hours’ transaction activity. ( 00:00 UTC )
Reward distribution: Daily farming rewards will be distributed to users in 6 hours after the end of each day.
All the rewards will be calculated based on successful transactions.
Refunds, Pending or Cancelled transactions will not be included to reward farming calculations.
Transaction time is based on the receiver's receiving time.
A notification will be sent in advance if any changes occur to reward farming (rate, etc.).
A user can earn a maximum of 200 PIPs with daily rewards.
Daily PIP reward limits may change due to user participation and the balance of the farming system.
PIP.ME transactions have no fees and are not included in Farming Rewards.
PIP token transactions are not included in Reward Farming.
Multi-use of wallet addresses and social media profiles are not allowed. The users who try to exploit or abuse the system will be banned permanently from the PIP ecosystem.
Rewardable Actions
Send money with PIP extension on supported platforms
Spend money with PIP extension on supported platforms
User Reward = (Individual Points / All Points ) * Daily Reward
Individual Points = Network Rate * Volume Rate
A. Network Rate
Applicable only for Send/Spend transactions.
Network Rate = Platform/Tag * Referral * New Member
1. Platform / Tag Rate
Point rates are based on which channel is used for the transaction.
.sol Domains
The referral rate is applicable when the receiver (based on the platform) receives "for the first time" through "PIP escrow".
Accounts that have been connected to PIP at least once are not eligible.
Tag and .sol payments are not included in the referral rate.
3. New Member Rate
The new member rate is applicable when the user succeeds in sending a transaction to the new counterpart (based on a social platform account)
Each time you send tokens to the same user, the new member rate will be decremented by 1.
Tag and .sol payments are not included in the new member rate.
B. Volume Rate
Users who have a higher volume will get higher points.
Volume Rate = Asset Rate * Fee Rate
1. Asset Rate
SOL, USDC, RAY, SRM, FIDA, KIN, WOOF, SDOGE, ZBC, MEAN
PIP transactions are not included in Reward Farming.
2. Fee Rate
Transaction Amount ( USD ) * 0.01
Fees are applicable only for sending money.
Processing fee is %1
The fee is collected by the asset to be transmitted.
PIP foundation will burn the PIP tokens which acquired through PIP transactions
You can claim your rewards after 02:00 am ( UTC ) any time from the PIP extension. You need at least 1 PIP as collectible reward to claim. Claimed rewards will be sent directly to your PIP wallet address. Any rewards you don't claim will still be available the next day.
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Mr. Crow, the head groundskeeper at High Tech Middle School, mows the lawn along the side of the gym. The lawn is rectangular, and the length is
5
feet more than twice the width. The perimeter of the lawn is
250
Define a variable and write an equation for this problem.
Recall that the equation of a perimeter is
2l+2w
. Use the information to make an equation for the perimeter with only one variable and set it equal to
250
, which is the given perimeter.
w
represent the width of the lawn:
2w+2(2w+5)=250
Solve the equation that you wrote in part (a) and find the dimensions of the lawn.
Start by distributing the two to the terms in parentheses and solve from there for
w
w
, substitute it into an equation for
l
2(2w+5)=\text{length}
Use the dimensions you calculated in part (a) to find the area of the lawn.
\text{length}\cdot\text{width}
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Effect of Longitudinal Vortex Generator Location on Thermoelectric-Hydraulic Performance of a Single-Stage Integrated Thermoelectric Power Generator | J. Thermal Sci. Eng. Appl. | ASME Digital Collection
e-mail: sdesh@vt.edu
e-mail: bharvish@vt.edu
e-mail: jpandit@vt.edu
e-mail: mating715@mail.xjtu.edu.cn
e-mail: huxtable@vt.edu
e-mail: sekkad@vt.edu
Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received May 15, 2017; final manuscript received March 15, 2018; published online June 14, 2018. Assoc. Editor: Samuel Sami.
Deshpande, S., Ravi, B. V., Pandit, J., Ma, T., Huxtable, S., and Ekkad, S. (June 14, 2018). "Effect of Longitudinal Vortex Generator Location on Thermoelectric-Hydraulic Performance of a Single-Stage Integrated Thermoelectric Power Generator." ASME. J. Thermal Sci. Eng. Appl. October 2018; 10(5): 051016. https://doi.org/10.1115/1.4040033
Vortex generators have been widely used to enhance heat transfer in various heat exchangers. Out of the two types of vortex generators, transverse vortex generators and longitudinal vortex generators (LVGs), LVGs have been found to show better heat transfer performance. Past studies have shown that the implementation of these LVGs can be used to improve heat transfer in thermoelectric generator systems. Here, a built in module in COMSOL Multiphysics® was used to study the influence of the location of LVGs in the channel on the comprehensive performance of an integrated thermoelectric device (TED). The physical model under consideration consists of a copper interconnector sandwiched between p-type and n-type semiconductors and a flow channel for hot fluid in the center of the interconnector. Four pairs of LVGs are mounted symmetrically on the top and bottom surfaces of the flow channel. Thus, using numerical methods, the thermo-electric-hydraulic performance of the integrated TED with a single module is examined. By fixing the material size D, the fluid inlet temperature
Tin
, and attack angle β, the effects of the location of LVGs and Reynolds number were investigated on the heat transfer performance, power output, pressure drop, and thermal conversion efficiency. The location of LVGs did not have significant effect on the performance of TEGs in the given model. However, the performance parameters show a considerable change with Reynold's number and best performance is obtained at Reynold number of Re = 500.
Energy efficiency, Energy systems, Forced convection, Heat exchangers, Heat recovery, Low temperature heat transfer, Thermal systems
Flow (Dynamics), Generators, Heat, Heat transfer, Pressure drop, Reynolds number, Vortices, Temperature, Fluids, Heat exchangers, Copper
), Clemson, SC, June 19–23, pp.
Power and Efficiency Calculation in Commercial TEG and Application in Wasted Heat Recovery in Automobile
Fifth European Conference on Thermoelectrics
), Odessa, Ukraine, Sept. 10–12, pp. 1–4.https://pdfs.semanticscholar.org/21ec/ac8683e6d69886ed4363fb01aefb56145978.pdf
A Mathematic Model of Thermoelectric Module With Applications on Waste Heat Recovery From Automobile Engine
A Three-Dimensional Numerical Modeling of Thermoelectric Device With Consideration of Coupling of Temperature Field and Electric Potential Field
B. V. K.
Convective Heat Transfer and Contact Resistances Effects on Performance of Conventional and Composite Thermoelectric Devices
Thermoelectric-Hydraulic Performance of a Multistage Integrated Thermoelectric Power Generator
Thermoelectric Performance of Novel Composite and Integrated Devices Applied to Waste Heat Recovery
Effect of Pin Fin to Channel Height Ratio and Pin Fin Geometry on Heat Transfer Performance for Flow in Rectangular Channels
Experimental Study of Heat Transfer Enhancement in Narrow Rectangular Channel With Longitudinal Vortex Generators
Numerical Study on Laminar Convection Heat Transfer in a Rectangular Channel With Longitudinal Vortex Generator—Part A: Verification of Field Synergy Principle
, Teng, J. T., Cheng, C. H., Jin, S., Huang, S., Liu, C., Lee, M. T., Pan, H. H., and Greif, R.,
A Study on Fluid Flow and Heat Transfer in Rectangular Microchannels With Various Longitudinal Vortex Generators
Numerical Study on Laminar Convection Heat Transfer in a Channel With Longitudinal Vortex Generator. Part B: Parametric Study of Major Influence Factors
Three-Dimensional Multiphysics Coupled Field Analysis of an Integrated Thermoelectric Device
Numer. Heat Transf. Part A
Effect of Inclined Vortex Generators on Heat Transfer Enhancement in a Three-Dimensional Channel
Lalande-Bertrand
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The optional filter parameter, passed as the index to the Map or Map2 command, restricts the application of
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
A≔\mathrm{Matrix}\left([[1,2,3],[0,1,4]],\mathrm{shape}=\mathrm{triangular}[\mathrm{upper},\mathrm{unit}]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\end{array}]
M≔\mathrm{Map}\left(x↦x+1,A\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\end{array}]
\mathrm{evalb}\left(\mathrm{addressof}\left(A\right)=\mathrm{addressof}\left(M\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
B≔〈〈1,2,3〉|〈4,5,6〉〉
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map2}[\left(i,j\right)↦\mathrm{evalb}\left(i=1\right)]\left(\left(x,a\right)↦a\cdot x,3,B\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{12}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map}\left(x↦x+1,g\left(3,A\right)\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\end{array}]\right)
C≔\mathrm{Matrix}\left([[1,2],[3]],\mathrm{scan}=\mathrm{triangular}[\mathrm{upper}],\mathrm{shape}=\mathrm{symmetric}\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{Map}\left(x↦x+1,C\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{9}\\ \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{4}\end{array}]
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A health state assessment method for ship propulsion system based on fuzzy theory and variable weight theory | JVE Journals
Pian Hu1 , Taotao Zhou2
1, 2China Ship Development and Design Center, Wuhan, China
It is hard to determine the equipment weight in the ship propulsion system health status evaluation. A device weight determination method based on fuzzy theory and variable weight theory is presented. In this method, expert knowledge is used to determine the initial weight of each device; variable weights theory is used to appropriately adjust devices weights combining with the actual health status of each device. Simulation analysis results show that the proposed propulsion system integrated status assessment method could reasonably reflect actual status, which proves it to be scientific and valid in engineering application.
Keywords: fuzzy theory, variable weight theory, ship propulsion system, health status evaluation.
The purpose of propulsion system health state assessment is to directly reflect the actual operating conditions of the system and provide supports for ship maintenance. In order to evaluate the health state of complex system which composed of a lot of equipment, it is necessary to divide the system according to the structure of the system, and then use an appropriate method to synthesize the its health state.
Ship propulsion is a typical complex system with complex hierarchical structure and diverse working conditions. At present, the grading evaluation [1, 2], single-level fuzzy processing [3, 4] and multi-layer fuzzy technology [5, 6] have been studied for complex system health status evaluation [7]. Geng et al. [8] studied a ship system and equipment technology status evaluation method based on fuzzy model. From the perspective of engineering application, the ship is divided into equipment, subsystem, system and other levels, the fuzzy theory and weighted synthesis method are used to assess the propulsion device technical status. Abou et al. [9] solved fuzzy characteristics problem of the ship propulsion system condition monitoring parameters by using the fuzzy logic method and realized the ship propulsion system health state assessment. Li et al. [10] proposed an improved fuzzy health evaluation method using the degradation degree function, dynamic threshold and variable weight, and evaluated the degradation status of the wind turbine with the actual monitoring data. The results show that the improved method is accurate and reliable.
Aiming at the health status evaluation of propulsion system, the key problem is to determine each equipment weight coefficient scientifically and rationally. Current, the weight determining method is less, single subjective weighting method is usually used which with a strong subjectivity [11].
2. Expert fuzzy scoring method and variable weight health assessment method
2.1. Expert fuzzy scoring method
Considering the diversity and fuzziness of the propulsion system, the importance and the failure probability of equipment, each device weight coefficient is determined based on the expert fuzzy scoring method. In one condition, the propulsion system needs to start seven devices, the expert fuzzy score table is designed as shown in Table 1.
Table 1. Expert fuzzy grading sheet
Assuming the influence of the health status of the equipment on the propulsion system status is divided into five levels showed by 1 to 5. As shown in the Table 1, the expert I considers that the health state of the equipment 2 has a particularly large effect on the integrated state of the propulsion system, and its weight value is 5, and the health state of the device 1 has little effect on the propulsion system state, and its weight value is 1. Here 10 experts give the device weight coefficient of the system.
Assuming that the influence of the
i
th expert on the
j
th device to the propulsion system health state is
{\alpha }_{i, j}
{\alpha }_{i, j}
is an integer of 1 to 5. Then, the evaluation value of the
j
th device in the advancing system is:
{\lambda }_{j}=\sum _{i=1}^{10}{\alpha }_{i,j}
i=
1, 2,…, 10 which is the number of experts. Then, the weighting coefficients
{b}_{j}={\lambda }_{j}/\sum _{j=1}^{7}{\lambda }_{j}
are obtained by normalizing the score results, and the fuzzy weight value of the
j
th device is obtained. The weighting coefficients of the device are obtained by
B=\left\{{b}_{1},{b}_{2},\dots ,{b}_{j}\right\}
{b}_{j}
is the weight coefficient of the
j
th device.
2.2. Variable weight health assessment method
The initial weight of each device is determined by the subjective weighting method. The variable weight synthesis principle is used to consider the imbalance of assessment result, dynamically reflect the equipment health status, assess the propulsion system health state objectively and accurately.
According to the health status of each device, the variable value
C
of the propulsion system health state is determined by:
C=\sum _{j=1}^{m}100{w}_{j}\left({x}_{1}, {x}_{2},\dots , {x}_{m}, {\omega }_{1}^{\left(0\right)}, {\omega }_{2}^{\left(0\right)},\dots , {\omega }_{m}^{\left(0\right)}\right){x}_{j},
{\omega }_{m}^{\left(0\right)}
is the initial weight of the device, and satisfies
\sum _{j=1}^{m}{\omega }_{j}^{\left(0\right)}=1
{w}_{j}
is the variable weight of the device;
{x}_{j}
is the state value of the
j
{w}_{j}\left({x}_{1}, {x}_{2}, \dots , {x}_{n}, {\omega }_{1}^{\left(0\right)}, {\omega }_{2}^{\left(0\right)}, \dots , {\omega }_{n}^{\left(0\right)}\right)=\frac{{\omega }_{j}^{\left(0\right)}\frac{\partial B\left({x}_{1}, {x}_{2},\dots , {x}_{n}\right)}{\partial {x}_{j}}}{\sum _{j=1}^{n}{\omega }_{k}^{\left(0\right)}\frac{\partial B\left({x}_{1}, {x}_{2},\dots , {x}_{n}\right)}{\partial {x}_{k}}},
B\left({x}_{1}, {x}_{2}, \dots , {x}_{n}\right)
represents the equilibrium function, that is, the gradient vector of the state variable weight vector
S\left(X\right)
The state variable weight vector is a very important factor in the variable weight evaluation. If the state variable weight vector is selected, the variable weight law is determined. According to the gradient vector of the literature, the empirical formula and the equilibrium function
B\left(X\right)=\sum _{j=1}^{m}{x}_{j}^{\alpha }
\left(\alpha \in \left[0, 1\right]\right)
are the most widely used variable weight vector. However, when the status of some devices is particularly small or even 0, this time is not suitable for the use of empirical formula. Therefore, in order to consider the equilibrium relations among the factors, we choose the gradient vector of the equilibrium function
B\left(X\right)=\sum _{j=1}^{m}{x}_{j}^{\alpha }
\left(\alpha \in \left[0, 1\right]\right)
to construct the state variable weight vector.
From the Eq. (2) and
B\left(X\right)=\sum _{j=1}^{m}{x}_{j}^{\alpha }
\left(\alpha \in \left[0, 1\right]\right)
{w}_{j}\left({x}_{1}, {x}_{2}, \dots , {x}_{n}, {\omega }_{1}^{\left(0\right)}, {\omega }_{2}^{\left(0\right)}, \dots , {\omega }_{n}^{\left(0\right)}\right)=\frac{{w}_{j}^{\left(0\right)}{x}_{j}^{\alpha -1}}{\sum _{j=1}^{n}{w}_{k}^{\left(0\right)}{x}_{k}^{\alpha -1}}.
Therefore, the health assessment of variable weight is:
C=\sum _{j=1}^{m}\frac{{w}_{j}^{\left(0\right)}{x}_{j}^{\alpha }}{\sum _{j=1}^{n}{w}_{k}^{\left(0\right)}{x}_{k}^{\alpha -1}}.
In the health evaluation method, the variable weight principle is used to solve the problem that the health state of some equipment will affect the overall state of the whole system, and the evaluation result is more reliable and reasonable. The variable weight value reflects the equilibrium requirement, which affects the health evaluation result to a large extent. The practical application of a large number of engineering shows that
\alpha =
0.2 can be applied to general engineering situation.
2.3. The method of weight determination combined expert fuzzy score with variable weight theory
In order to ensure the safety of the system, assessing the system health state scientifically and reasonably, a weight determining method combined expert fuzzy score with variable weight theory is proposed. According to the equipment important degree in the propulsion system and the historical experience of the equipment failures, the initial weight coefficient of each device is determined, and the weight coefficient is adjusted according to the equipment health variable theory.
Compared with the conventional method, the variable weight synthesis method can highlight the serious defects of the equipment and avoid the serious damage of one or several of the equipment cannot be reflected in the health state assessment, affecting the accuracy of the assessment results. It will show the serious defects of the equipment in the propulsion system health evaluation and provide a more accurate and effective guidance for the using and maintenance of the propulsion system.
In order to illustrate the effectiveness of the proposed method, assume that the propulsion system consisted by six devices: gearbox, shaft, sliding bearing, propeller, oil pump, cooling water pump, fuel pump. According to the expert fuzzy scoring method, the score results of five industry experts are showed in Table 2.
Consider the propulsion system is in the following five typical cases: 1) all equipment are in the optimal state; 2) all equipment are in a bad state; 3) all equipment are in a serious state of damage; 4) other equipment are in the normal state only sliding bearing is serious damaged; 5) Other equipment are in the normal state only oil pump is serious damaged. The validity of the health evaluation method is verified in Table 3, Table 4, Table 5, Table 6 and Table 7.
Table 3. Propulsion system health status evaluation sheet in condition 1
Propulsion system health status
Note: All devices are in the optimal state
Note: All devices are in bad condition
Note: All equipment is severely damaged
Note: The sliding bearing has been severely damaged
It can be concluded from Table 3, Table 4 and Table 5 that the results of the variable weight method are the same as those obtained by the health assessment method when the health status of all the equipment is consistent. It indicates that the two methods can effectively reflect the actual state of the propulsion system.
Observe Table 6, the other equipment is in the normal state only sliding bearing serious damage, you can find, often the weight of the health assessment of the results of 74.80, said the propulsion system is in a normal state; the proposed health assessment method to assess the results of 39.28, show the propulsion system is in a bad state. And then observe the Table 7, the other equipment is in the normal state only a serious damage to the case of oil pump, you can see, often the weight of the health assessment of the results of 80.40, said the propulsion system in a normal state; proposed health assessment method to assess the results of 63.18, Indicating that the propulsion system is in a bad state. In both cases, there is a serious damage to the equipment; we need to carry out maintenance. The results of the conventional method cannot show the propulsion system actual state obviously, and the proposed evaluation method can effectively assess the actual state of the propulsion system and can provide the health evaluation value of the ship allegation system according to the actual situation.
Note: The fuel pump has been severely damaged
In the case of sliding bearings or fuel pumps of the propulsion system is severely damaged, compare the results of the two health state assessment methods, we can find that the health status results of the propulsion system by conventional method to are normal and required on maintenance. The results by the proposed method are in bad state and need maintenance. It can be found that the conventional method ignores the failure in the system and cannot reflect correctly the propulsion system state. The proposed health assessment method can effectively characterize the actual state of the propulsion system.
In order to determining the equipment weight of the propulsion system, a weight determination method combining fuzzy theory and variable weight theory is proposed. This method uses the expert knowledge to determine the initial weight of equipment according to the importance and the historical operation experience of the equipment. On this basis, combined with the actual health status of equipment, the weights are adjusted appropriately by the variable weight theory, which could reflect the system’s actual state more reasonable. The simulation results showed that the proposed method could reflect actual status of propulsion system reasonably and effectively.
Barger T. S., Brown D. E., Alwan M. Health-status monitoring through analysis of behavioral patterns. IEEE Transactions on Systems, Man, and Cybernetics – Part A: Systems and Humans, Vol. 35, Issue 1, 2005, p. 22-27. [Search CrossRef]
Miao Q., Wang D., Pecht M. A probabilistic description scheme for rotating machinery health evaluation. Journal of Mechanical Science and Technology, Vol. 24, Issue 12, 2010, p. 2421-2430. [Search CrossRef]
Sohn H., Farrar C. R., Hemez F. M., et al. A Review of Structural Health Monitoring Literature: 1996-2001. Los Alamos National Laboratory, 2003. [Search CrossRef]
Shen C., Wang D., Kong F., et al. Fault diagnosis of rotating machinery based on the statistical parameters of wavelet packet paving and a generic support vector regressive classifier. Measurement, Vol. 46, Issue 4, 2013, p. 1551-1564. [Search CrossRef]
Yan R., Gao R. X. An efficient approach to machine health diagnosis based on harmonic wavelet packet transform. Robotics and Computer-Integrated Manufacturing, Vol. 21, Issue 4, 2005, p. 291-301. [Search CrossRef]
Zeng S., Pecht M. G., Wu J. Status and perspectives of prognostics and health management technologies. ACTA Aeronautica et Astronautica Sinica – Series A and B, Vol. 26, Issue 5, 2005, p. 626-632. [Search CrossRef]
Widodo A., Yang B. S. Support vector machine in machine condition monitoring and fault diagnosis. Mechanical Systems and Signal Processing, Vol. 21, 2007, p. 2560-2574. [Search CrossRef]
Geng J. B. Research on Technical Condition Integrated Evaluation for Power Plant of Ships Based on Information Fusion. Huazhong University of Science and Technology, 2007. [Search CrossRef]
Abou S. C., Stachowicz M. Ship propulsion system health monitoring based on safety control functions. ELMAR, 50th International Symposium, Vol. 2, 2008, p. 393-396. [Search CrossRef]
Li H., Hu Y. G., Yang C., et al. An improved fuzzy synthetic condition assessment of a wind turbine generator system. International Journal of Electrical Power and Energy Systems, Vol. 45, Issue 1, 2013, p. 468-476. [Search CrossRef]
Liang S. T., Hao C. X., Wang M. L. The application of fuzzy neural network in the marine electric propulsion system condition assessment. Chinese Journal of Ship Research, Vol. 9, Issue 5, 2014, p. 99-104. [Search CrossRef]
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monodromy - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Polynomials : Algebraic Curves : monodromy
Compute the monodromy of an algebraic curve
monodromy(f, x, y, opt)
This procedure computes the monodromy of a Riemann surface represented as a plane algebraic curve; that is, as a polynomial f in two variables x and y. The Riemann surface is the covering surface for y as an N-valued function of x, where
N=\mathrm{degree}\left(f,y\right)
is the degree of covering. Curves with singularities are allowed as input.
The output is a list containing the following:
\mathrm{x0}
for x for which y takes N different values, so that
\mathrm{x0}
is not a branchpoint nor a singularity.
L=[\mathrm{fsolve}\left(\mathrm{subs}\left(x=\mathrm{x0},f\right),y,\mathrm{complex}\right)]
of pre-images of
\mathrm{x0}
. This list of y-values at
x=\mathrm{x0}
effectively labels the sheets of the Riemann surface at
x=\mathrm{x0}
. Sheet 1 is
{L}_{1}
, sheet 2 is
{L}_{2}
[[{b}_{1},{m}_{1}],[{b}_{2},{m}_{2}],\mathrm{...}]
of branchpoints
{b}_{i}
with their monodromy
{m}_{i}
. The monodromy
{m}_{i}
of branchpoint
{b}_{i}
is the permutation of
L
obtained by applying analytic continuation on
L
following a path from
\mathrm{x0}
{b}_{i}
, going around
{b}_{i}
counter-clockwise, and returning to
\mathrm{x0}
The permutations
{m}_{i}
will be given in disjoint cycle notation. The branchpoints
{b}_{i}
\mathrm{discrim}\left(f,y\right)
The order of the branchpoints is chosen in such a way that the complex numbers
{b}_{1}-\mathrm{x0},...
have increasing arguments. The point x0 is chosen on the left of the branchpoints, so all arguments are between
-\frac{\mathrm{\pi }}{2}
\frac{\mathrm{\pi }}{2}
. If the arguments coincide, branchpoints that are closer to x0 are considered first. The point infinity will be given last, if it is a branchpoint.
It can take some time for this procedure to finish. To have monodromy print information about the status of the computation while it is working, give the variable infolevel[algcurves] an integer value > 1.
If the optional argument showpaths is given, then a plot is generated displaying the paths used for the analytic continuation. If the optional argument group is given, then the output is the monodromy group G, the permutation group generated by the
{m}_{i}
. This group G is the Galois group of f as a polynomial over
C\left(x\right)
. G is a subgroup of galois(f,y), which is the Galois group of f over Q(x).
\mathrm{with}\left(\mathrm{algcurves}\right):
\mathrm{monodromy}\left({y}^{3}-x,x,y\right)
[\textcolor[rgb]{0,0,1}{-1.}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{-1.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.500000000000000}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.866025403784439}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.500000000000000}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{0.866025403784439}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{\infty }}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]]]]
f≔{\left({y}^{4}-x\right)}^{2}+1
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\left({\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
G≔\mathrm{monodromy}\left(f,x,y,\mathrm{group}\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{permgroup}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}{[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]}\right)
Note: G is not transitive, which means that f is reducible.
\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)
\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\right)
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How to fool the plot inspection heuristic? - PhotoLens
When teaching TCS, there is always the student who asks: “What do I need formal proof for if I can just do X which always works?” It is up to his teacher(s) to point out and illustrate the fallacy. There is a brilliant set of examples of apparent patterns that eventually fail over at math.SE, but those are fairly mathematical scenarios.
Are there examples of (relative) asymptotic growth where the truth is not obvious from the function definiton and plot inspection for reasonably large n
gives you a completely wrong idea? Mathematical functions and real data sets (e.g. runtime of a specific algorithm) are both welcome; please refrain from piecewise defined functions, though.
Speaking from experience, when trying to figure out the growth rate for some observed function (say, Markov chain mixing time or algorithm running time), it is very difficult to tell factors of (\log n)^a
(\log n)^a
from n^b
. For example, O(\sqrt{n} \log n)
O(\sqrt{n} \log n)
looks a lot like O(n^{0.6})
O(n^{0.6})
For example, in “Some unexpected expected behavior results for bin packing” by Bentley et al., the growth rate of empty space for the Best Fit and First Fit bin packing algorithms when packing items uniform on [0,1]
was estimated empirically as n^{0.6}
n^{0.6}
and n^{0.7}
n^{0.7}
, respectively. The correct expressions are n^{1/2}\log^{3/4}n
n^{1/2}\log^{3/4}n
and n^{2/3}
n^{2/3}
Source : Link , Question Author : Raphael , Answer Author : dfeuer
Categories asymptotics, didactics Tags asymptotics, didactics Post navigation
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Revision as of 19:25, 13 April 2015 by MathAdmin (talk | contribs) (Created page with "<span class="exam">Find the slope of the tangent line to the graph of <math style="vertical-align: -14%">f(x)=x^{3}-3x^{2}-5x+7</math> at the point <math style="vertical-align...")
Find the slope of the tangent line to the graph of
{\displaystyle f(x)=x^{3}-3x^{2}-5x+7}
{\displaystyle (3,-8)}
Recall that for a given value,
{\displaystyle f'(x)}
is precisely the point of the tangent line through the point
{\displaystyle \left(x,f(x)\right)}
. Once we have the slope, we can then use the point-slope form for a line:
{\displaystyle y-y_{0}=m(x-x_{0}),}
{\displaystyle m}
is the known slope and
{\displaystyle \left(x_{0},y_{0}\right)}
Finding the slope:
{\displaystyle f'(x)\,\,=\,\,3x^{2}-6x-5,}
so the tangent line through
{\displaystyle (3,-8)}
{\displaystyle m\,\,=\,\,f'(3)\,\,=\,\,3(3)^{2}-6(3)-5\,\,=\,\,4.}
Using the point-slope form listed in foundations, along with the point
{\displaystyle (3,-8)}
{\displaystyle m=4}
{\displaystyle y-(-8)\,\,=\,\,4(x-3),}
{\displaystyle y\,\,=\,\,4x-20.}
{\displaystyle y\,\,=\,\,4x-20.}
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\textcolor[rgb]{0.407843137254902,0.250980392156863,0.36078431372549}{\mathrm{ω}}
\mathrm{ω}
\mathrm{π}:E→M
be a fiber bundle, with base dimension
m
{\mathrm{π}}^{\mathrm{∞}}:{J}^{\mathrm{∞}}\left(E\right) → M
E
({x}^{i}, {u}^{\mathrm{α}}, {u}_{{i}_{}}^{\mathrm{α}}, {u}_{{i}_{}j}^{\mathrm{α}}
{u}_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{α}}, ....)
{\mathrm{Θ}}^{\mathrm{α}} = {\mathrm{du}}^{\mathrm{α}}-{u}_{\mathrm{ℓ}}^{\mathrm{α}}{\mathrm{dx}}^{\mathrm{ℓ}}
{\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)
n
s.
\mathrm{ω} ∈{\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)
{E}_{\mathrm{α}}\left(\mathrm{ω}\right) ∈ {\mathrm{Ω}}^{\left(n-1,s\right)}\left({J}^{\infty }\left(E\right)\right)
\mathrm{ω}
I: {\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)→{\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)
I\left(\mathrm{ω}\right) = \frac{1}{s}{\mathrm{Θ}}^{\mathrm{α} }∧{E}_{\mathrm{α}}\left(\mathrm{ω}\right).
I
\mathrm{η}
\left(n-1, s\right),
I\left({d}_{H}\mathrm{η}\right) = 0
{d}_{H }\mathrm{η}
\mathrm{η}
\mathrm{ω}
\left(n,s\right)
I\left(\mathrm{ω}\right) =0,
\left(n-1, s\right)
\mathrm{ω} = {d}_{H }\mathrm{η}
I
I∘I = I
\textcolor[rgb]{0.407843137254902,0.250980392156863,0.36078431372549}{\mathrm{\omega }}
\left(n, s\right)
I\left(\mathrm{ω}\right)
{J}^{3}\left(E\right)
E
\left(x,u\right)→ x.
{\mathrm{ω}}_{1}
\textcolor[rgb]{0,0,1}{\mathrm{ω1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{d}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]]]\right)
{\mathrm{ω}}_{2}
\textcolor[rgb]{0,0,1}{\mathrm{ω2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ω3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}]]]\right)
{\mathrm{ω}}_{3}
{\mathrm{ω}}_{3}
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{b}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}]]]\right)
{J}^{3}\left(E\right)
E
\left(x,y, u, v\right)→ \left(x,y\right)
{\mathrm{ω}}_{4}.
\textcolor[rgb]{0,0,1}{\mathrm{ω4}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{e}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{d}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{f}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{e}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}]]]\right)
{\mathrm{ω}}_{5}.
\textcolor[rgb]{0,0,1}{\mathrm{ω5}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]]]\right)
{\mathrm{ω}}_{6}
\mathrm{η}.
\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ω6}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{16}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{13}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]]]\right)
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Löwenheim–Skolem theorem - formulasearchengine
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.
2 Precise statement
4.1 Downward part
4.2 Upward part
A signature consists of a set of function symbols Sfunc, a set of relation symbols Srel, and a function
{\displaystyle \operatorname {ar} :S_{\operatorname {func} }\cup S_{\operatorname {rel} }\rightarrow \mathbb {N} _{0}}
representing the arity of function and relation symbols. (A nullary function symbol is called a constant symbol.) In the context of first-order logic, a signature is sometimes called a language. It is called countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all the symbols it contains.
A first-order theory consists of a fixed signature and a fixed set of sentences (formulas with no free variables) in that signature. Theories are often specified by giving a list of axioms that generate the theory, or by giving a structure and taking the theory to consist of the sentences satisfied by the structure.
The modern statement of the theorem is both more general and stronger than the version for countable signatures stated in the introduction.
In its general form, the Löwenheim–Skolem Theorem states that for every signature σ, every infinite σ-structure M and every infinite cardinal number κ ≥ |σ|, there is a σ-structure N such that |N| = κ and
if κ < |M| then N is an elementary substructure of M;
if κ > |M| then N is an elementary extension of M.
The theorem is often divided into two parts corresponding to the two bullets above. The part of the theorem asserting that a structure has elementary substructures of all smaller infinite cardinalities is known as the downward Löwenheim–Skolem Theorem. The part of the theorem asserting that a structure has elementary extensions of all larger cardinalities is known as the upward Löwenheim–Skolem Theorem.
The statement given in the introduction follows immediately by taking M to be an infinite model of the theory. The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem. For historical variants of the theorem, see the notes below.
Let N denote the natural numbers and R the reals. It follows from the theorem that the theory of (N, +, ×, 0, 1) (the theory of true first-order arithmetic) has uncountable models, and that the theory of (R, +, ×, 0, 1) (the theory of real closed fields) has a countable model. There are, of course, axiomatizations characterizing (N, +, ×, 0, 1) and (R, +, ×, 0, 1) up to isomorphism. The Löwenheim–Skolem theorem shows that these axiomatizations cannot be first-order. For example, the completeness of a linear order, which is used to characterize the real numbers as a complete ordered field, is a non-first-order property.
A theory is called categorical if it has only one model, up to isomorphism. This term was introduced by Template:Harvtxt, and for some time thereafter mathematicians hoped they could put mathematics on a solid foundation by describing a categorical first-order theory of some version of set theory. The Löwenheim–Skolem theorem dealt a first blow to this hope, as it implies that a first-order theory which has an infinite model cannot be categorical. Later, in 1931, the hope was shattered completely by Gödel's incompleteness theorem.
Many consequences of the Löwenheim–Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the existence of uncountable models of true arithmetic, which satisfy every first-order induction axiom but have non-inductive subsets. Another consequence that was considered particularly troubling is the existence of a countable model of set theory, which nevertheless must satisfy the sentence saying the real numbers are uncountable. This counterintuitive situation came to be known as Skolem's paradox; it shows that the notion of countability is not absolute.
Downward part
For each first-order
{\displaystyle \sigma \,}
{\displaystyle \varphi (y,x_{1},\ldots ,x_{n})\,,}
the axiom of choice implies the existence of a function
{\displaystyle f_{\varphi }:M^{n}\to M}
{\displaystyle a_{1},\ldots ,a_{n}\in M}
{\displaystyle M\models \varphi (f_{\varphi }(a_{1},\dots ,a_{n}),a_{1},\dots ,a_{n})}
{\displaystyle M\models \neg \exists y\varphi (y,a_{1},\dots ,a_{n})\,.}
Applying the axiom of choice again we get a function from the first order formulas
{\displaystyle \varphi }
to such functions
{\displaystyle f_{\varphi }\,.}
{\displaystyle f_{\varphi }}
gives rise to a preclosure operator
{\displaystyle F\,}
on the power set of
{\displaystyle M\,}
{\displaystyle F(A)=\{b\in M\mid b=f_{\varphi }(a_{1},\dots ,a_{n});\,\varphi \in \sigma ;\,a_{1},\dots ,a_{n}\in A\}}
{\displaystyle A\subseteq M\,.}
Iterating
{\displaystyle F\,}
countably many times results in a closure operator
{\displaystyle F^{\omega }\,.}
Taking an arbitrary subset
{\displaystyle A\subseteq M}
{\displaystyle \left\vert A\right\vert =\kappa }
, and having defined
{\displaystyle N=F^{\omega }(A)\,,}
one can see that also
{\displaystyle \left\vert N\right\vert =\kappa \,.}
{\displaystyle N\,}
is an elementary substructure of
{\displaystyle M\,}
by the Tarski–Vaught test.
The trick used in this proof is essentially due to Skolem, who introduced function symbols for the Skolem functions
{\displaystyle f_{\varphi }}
into the language. One could also define the
{\displaystyle f_{\varphi }}
as partial functions such that
{\displaystyle f_{\varphi }}
is defined if and only if
{\displaystyle M\models \exists y\varphi (y,a_{1},\dots ,a_{n})\,.}
The only important point is that
{\displaystyle F\,}
is a preclosure operator such that
{\displaystyle F(A)\,}
contains a solution for every formula with parameters in
{\displaystyle A\,}
which has a solution in
{\displaystyle M\,}
{\displaystyle \left\vert F(A)\right\vert \leq \left\vert A\right\vert +\left\vert \sigma \right\vert +\aleph _{0}\,.}
Upward part
First, one extends the signature by adding a new constant symbol for every element of M. The complete theory of M for the extended signature σ' is called the elementary diagram of M. In the next step one adds κ many new constant symbols to the signature and adds to the elementary diagram of M the sentences c ≠ c' for any two distinct new constant symbols c and c'. Using the compactness theorem, the resulting theory is easily seen to be consistent. Since its models must have cardinality at least κ, the downward part of this theorem guarantees the existence of a model N which has cardinality exactly κ. It contains an isomorphic copy of M as an elementary substructure.
This account is based mainly on Template:Harvtxt. To understand the early history of model theory one must distinguish between syntactical consistency (no contradiction can be derived using the deduction rules for first-order logic) and satisfiability (there is a model). Somewhat surprisingly, even before the completeness theorem made the distinction unnecessary, the term consistent was used sometimes in one sense and sometimes in the other.
For every countable signature σ, every σ-sentence which is satisfiable is satisfiable in a countable model.
Löwenheim's paper was actually concerned with the more general Peirce–Schröder calculus of relatives (relation algebra with quantifiers). He also used the now antiquated notations of Ernst Schröder. For a summary of the paper in English and using modern notations see Template:Harvtxt.
According to the received historical view, Löwenheim's proof was faulty because it implicitly used König's lemma without proving it, although the lemma was not yet a published result at the time. In a revisionist account, Template:Harvtxt considers that Löwenheim's proof was complete.
Template:Harvtxt gave a (correct) proof using formulas in what would later be called Skolem normal form and relying on the axiom of choice:
Every countable theory which is satisfiable in a model M, is satisfiable in a countable substructure of M.
Template:Harvtxt also proved the following weaker version without the axiom of choice:
Template:Harvtxt simplified Template:Harvtxt. Finally, Anatoly Ivanovich Maltsev (Анато́лий Ива́нович Ма́льцев, 1936) proved the Löwenheim–Skolem theorem in its full generality Template:Harv. He cited a note by Skolem, according to which the theorem had been proved by Alfred Tarski in a seminar in 1928. Therefore the general theorem is sometimes known as the Löwenheim–Skolem–Tarski theorem. But Tarski did not remember his proof, and it remains a mystery how he could do it without the compactness theorem.
"I follow custom in calling Corollary 6.1.4 the upward Löwenheim-Skolem theorem. But in fact Skolem didn't even believe it, because he didn't believe in the existence of uncountable sets." – Template:Harvtxt.
"Skolem [...] rejected the result as meaningless; Tarski [...] very reasonably responded that Skolem's formalist viewpoint ought to reckon the downward Löwenheim-Skolem theorem meaningless just like the upward." – Template:Harvtxt.
"Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association of his name to a result of this type, which he considered an absurdity, nondenumerable sets being, for him, fictions without real existence." – Template:Harvtxt.
|CitationClass=citation }} (Template:Google books)
|CitationClass=citation }}; A more concise account appears in chapter 9 of {{#invoke:citation/CS1|citation |CitationClass=citation }}
Sakharov, Alex and Weisstein, Eric W., "Löwenheim-Skolem Theorem", MathWorld.
Burris, Stanley N., Contributions of the Logicians, Part II, From Richard Dedekind to Gerhard Gentzen
Burris, Stanley N., Downward Löwenheim–Skolem theorem
Simpson, Stephen G. (1998), Model Theory
Template:Metalogic
Retrieved from "https://en.formulasearchengine.com/index.php?title=Löwenheim–Skolem_theorem&oldid=228751"
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Vascular resistance - Wikipedia
(Redirected from Total peripheral resistance)
1 Units for measuring
2.1 Systemic calculations
2.2 Pulmonary calculations
4 Systemic
6 Coronary
Units for measuringEdit
Units for measuring vascular resistance are dyn·s·cm−5, pascal seconds per cubic metre (Pa·s/m3) or, for ease of deriving it by pressure (measured in mmHg) and cardiac output (measured in L/min), it can be given in mmHg·min/L. This is numerically equivalent to hybrid resistance units (HRU), also known as Wood units (in honor of Paul Wood, an early pioneer in the field), frequently used by pediatric cardiologists. The conversion between these units is:[1]
{\displaystyle 1\,{\frac {{\text{mmHG}}\cdot {\text{min}}}{\text{ L }}}({\text{HRUs}})=8\,{\frac {{\text{MPa}}\cdot {\text{s}}}{{\text{m}}^{3}}}=80\,{\frac {{\text{dyn}}\cdot {\text{sec}}}{{\text{cm}}^{5}}}}
The basic tenet of calculating resistance is that flow is equal to driving pressure divided by flow rate.[citation needed]
{\displaystyle R=\Delta P/Q}
This is the hydraulic version of Ohm's law, V=IR (which can be restated as R=V/I), in which the pressure differential is analogous to the electrical voltage drop, flow is analogous to electric current, and vascular resistance is analogous to electrical resistance.
Systemic calculationsEdit
{\displaystyle {\frac {80\cdot (mean\ arterial\ pressure-mean\ right\ atrial\ pressure)}{cardiac\ output}}}
Pulmonary calculationsEdit
The pulmonary vascular resistance can be calculated in units of dyn·s·cm−5 as[citation needed]
{\displaystyle {\frac {80\cdot (mean\ pulmonary\ arterial\ pressure-mean\ pulmonary\ artery\ wedge\ pressure)}{cardiac\ output}}}
where the pressures are measured in units of millimetres of mercury (mmHg) and the cardiac output is measured in units of litres per minute (L/min). The pulmonary artery wedge pressure (also called pulmonary artery occlusion pressure or PAOP) is a measurement in which one of the pulmonary arteries is occluded, and the pressure downstream from the occlusion is measured in order to approximately sample the left atrial pressure.[4] Therefore, the numerator of the above equation is the pressure difference between the input to the pulmonary blood circuit (where the heart's right ventricle connects to the pulmonary trunk) and the output of the circuit (which is the input to the left atrium of the heart). The above equation contains a numerical constant to compensate for the units used, but is conceptually equivalent to the following:[citation needed]
{\displaystyle R={\frac {\Delta P}{Q}}}
As an example: If Systolic pressure: 120 mmHg, Diastolic pressure: 80 mmHg, Right atrial mean pressure: 3 mmHg, Cardiac output: 5 L/min, Then mean arterial pressure would be: (2 diastolic pressure + systolic pressure)/3 = 93.3 mmHg, and systemic vascular resistance: (93 - 3) / 5 = 18 Wood units, or equivalently 1440 dyn·s/cm5.
There are many factors that alter the vascular resistance. Vascular compliance is determined by the muscle tone in the smooth muscle tissue of the tunica media and the elasticity of the elastic fibers there, but the muscle tone is subject to continual homeostatic changes by hormones and cell signaling molecules that induce vasodilation and vasoconstriction to keep blood pressure and blood flow within reference ranges.[citation needed]
In a first approach, based on fluids dynamics (where the flowing material is continuous and made of continuous atomic or molecular bonds, the internal friction happen between continuous parallel layers of different velocities) factors that influence vascular resistance are represented in an adapted form of the Hagen–Poiseuille equation:[citation needed]
{\displaystyle R={\frac {8L\eta }{\pi r^{4}}}}
In Hagen–Poiseuille equation, the flow layers start from the wall and, by viscosity, reach each other in the central line of the vessel following a parabolic velocity profile.[citation needed]
In a second approach, more realistic and coming from experimental observations on blood flows, according to Thurston,[5] there is a plasma release-cell layering at the walls surrounding a plugged flow. It is a fluid layer in which at a distance δ, viscosity η is a function of δ written as η(δ), and these surrounding layers do not meet at the vessel centre in real blood flow. Instead, there is the plugged flow which is hyperviscous because holding high concentration of RBCs. Thurston assembled this layer to the flow resistance to describe blood flow by means of a viscosity η(δ) and thickness δ from the wall layer.[citation needed]
{\displaystyle R={\frac {cL\eta (\delta )}{\pi \delta r^{3}}}}
Blood viscosity increases as blood is more hemoconcentrated, and decreases as blood is more dilute. The greater the viscosity of blood, the larger the resistance will be. In the body, blood viscosity increases as red blood cell concentration increases, thus more hemodilute blood will flow more readily, while more hemoconcentrated blood will flow more slowly.[citation needed]
Counteracting this effect, decreased viscosity in a liquid results in the potential for increased turbulence. Turbulance can be viewed from outside of the closed vascular system as increased resistance, thereby countering the ease of flow of more hemodilute blood. Turbulence, particularly in large vessels, may account for some pressure change across the vascular bed.
The blood flow resistance in a vessel is mainly regulated by the vessel radius and viscosity when blood viscosity too varies with the vessel radius. According to very recent results showing the sheath flow surrounding the plug flow in a vessel,[7] the sheath flow size is not neglectible in the real blood flow velocity profile in a vessel. The velocity profile is directly linked to flow resistance in a vessel. The viscosity variations, according to Thurston,[5] are also balanced by the sheath flow size around the plug flow. The secondary regulators of vascular resistance, after vessel radius, is the sheath flow size and its viscosity.
Vascular resistance depends on blood flow which is divided into 2 adjacent parts : a plug flow, highly concentrated in RBCs, and a sheath flow, more fluid plasma release-cell layering. Both coexist and have different viscosities, sizes and velocity profiles in the vascular system.[citation needed]
{\displaystyle F={\frac {QcL\eta (\delta )}{\pi \delta r}}}
Many of the platelet-derived substances, including serotonin, are vasodilatory when the endothelium is intact and are vasoconstrictive when the endothelium is damaged.[citation needed]
Cholinergic stimulation causes release of endothelium-derived relaxing factor (EDRF) (later it was discovered that EDRF was nitric oxide) from intact endothelium, causing vasodilation. If the endothelium is damaged, cholinergic stimulation causes vasoconstriction.[8]
Adenosine most likely does not play a role in maintaining the vascular resistance in the resting state. However, it causes vasodilation and decreased vascular resistance during hypoxia. Adenosine is formed in the myocardial cells during hypoxia, ischemia, or vigorous work, due to the breakdown of high-energy phosphate compounds (e.g., adenosine monophosphate, AMP). Most of the adenosine that is produced leaves the cell and acts as a direct vasodilator on the vascular wall. Because adenosine acts as a direct vasodilator, it is not dependent on an intact endothelium to cause vasodilation.[citation needed]
Adenosine causes vasodilation in the small and medium-sized resistance arterioles (less than 100 μm in diameter). When adenosine is administered it can cause a coronary steal phenomenon,[9] where the vessels in healthy tissue dilate as much as the ischemic tissue and more blood is shunted away from the ischemic tissue that needs it most. This is the principle behind adenosine stress testing. Adenosine is quickly broken down by adenosine deaminase, which is present in red cells and the vessel wall.[10]
A decrease in SVR (e.g., during exercising) will result in an increased flow to tissues and an increased venous flow back to the heart. An increased SVR will decrease flow to tissues and decrease venous flow back to the heart.[citation needed]
The major determinant of vascular resistance is small arteriolar (known as resistance arterioles) tone. These vessels are from 450 μm down to 100 μm in diameter. (As a comparison, the diameter of a capillary is about 5 to 10 μm.)[citation needed]
Another determinant of vascular resistance is the pre-capillary arterioles. These arterioles are less than 100 μm in diameter. They are sometimes known as autoregulatory vessels since they can dynamically change in diameter to increase or reduce blood flow.[citation needed]
Any change in the viscosity of blood (such as due to a change in hematocrit) would also affect the measured vascular resistance.[citation needed]
Pulmonary vascular resistance (PVR) also depends on the lung volume, and PVR is lowest at the functional residual capacity (FRC). The highly compliant nature of the pulmonary circulation means that the degree of lung distention has a large effect on PVR. This results primarily due to effects on the alveolar and extra-alveolar vessels. During inspiration, increased lung volumes cause alveolar expansion and lengthwise stretching of the interstitial alveolar vessels. This increases their length and reduces their diameter, thus increasing alveolar vessel resistance. On the other hand, decreased lung volumes during expiration cause the extra-alveolar arteries and veins to become narrower due to decreased radial traction from adjacent tissues. This leads to an increase in extra-alveolar vessel resistance. PVR is calculated as a sum of the alveolar and extra-alveolar resistances as these vessels lie in series with each other. Because the alveolar and extra-alveolar resistances are increased at high and low lung volumes respectively, the total PVR takes the shape of a U curve. The point at which PVR is the lowest is near the FRC.[citation needed]
CoronaryEdit
The regulation of tone in the coronary arteries is a complex subject. There are a number of mechanisms for regulating coronary vascular tone, including metabolic demands (i.e. hypoxia), neurologic control, and endothelial factors (i.e. EDRF, endothelin).[citation needed]
Local metabolic control (based on metabolic demand) is the most important mechanism of control of coronary flow. Decreased tissue oxygen content and increased tissue CO2 content act as vasodilators. Acidosis acts as a direct coronary vasodilator and also potentiates the actions of adenosine on the coronary vasculature.[citation needed]
^ Fuster, V.; Alexander, R.W.; O'Rourke, R.A. (2004) Hurst's the heart, book 1. 11th Edition, McGraw-Hill Professional, Medical Pub. Division. Page 513. ISBN 978-0-07-143224-5.
^ a b Table 30-1 in: Trudie A Goers; Washington University School of Medicine Department of Surgery; Klingensmith, Mary E; Li Ern Chen; Sean C Glasgow (2008). The Washington manual of surgery. Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins. ISBN 978-0-7817-7447-5. {{cite book}}: CS1 maint: multiple names: authors list (link)
^ a b c d Derived from values in dyn·s/cm5
^ University of Virginia Health System."The Physiology: Pulmonary Artery Catheters"
^ a b c d e GB Thurston, Viscosity and viscoelasticity of blood in small diameter tubes, Microvasular Research 11, 133 146, 1976
^ "Cardiac Output and Blood Pressure". biosbcc. Retrieved 7 April 2011.
^ Measurement of real pulsatile blood flow using X-ray PIV technique with CO2 microbubbles, Hanwook Park, Eunseop Yeom, Seung-Jun Seo, Jae-Hong Lim & Sang-Joon Lee, NATURE, Scientific Reports 5, Article number: 8840 (2015), doi:10.1038/srep08840.
^ Satoskar, RS; Bhandarkar, SD (2020). Pharmacology and Pharmacotherapeutics. Elsevier Health Sciences. p. 268. ISBN 978-8131257067.
^ Masugata H, Peters B, Lafitte S, et al. (2003). "Assessment of adenosine-induced coronary steal in the setting of coronary occlusion based on the extent of opacification defects by myocardial contrast echocardiography". Angiology. 54 (4): 443–8. doi:10.1177/000331970305400408. PMID 12934764. S2CID 42646704.
^ Opie, Lionel H. (2004). Heart Physiology: From Cell to Circulation. Lippincott Williams & Wilkins. p. 286. ISBN 0781742781.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Vascular_resistance&oldid=1076124334"
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(Redirected from ↪)
Collection of maps which give the same result
The commutative diagram used in the proof of the five lemma.
1.1 Arrow symbols
1.2 Verifying commutativity
4 Diagram chasing
5 In higher category theory
6 Diagrams as functors
A commutative diagram often consists of three parts:
objects (also known as vertices)
morphisms (also known as arrows or edges)
paths or composites
Arrow symbols[edit]
A monomorphism (injective homomorphism) may be labeled with a
{\displaystyle \hookrightarrow }
.[2] or a
{\displaystyle \rightarrowtail }
An epimorphism (surjective homomorphism) may be labeled with a
{\displaystyle \twoheadrightarrow }
An isomorphism (bijective homomorphism) may be labeled with a
{\displaystyle {\overset {\sim }{\rightarrow }}}
The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as
{\displaystyle \exists }
If the morphism is in addition unique, then the dashed arrow may be labeled
{\displaystyle !}
{\displaystyle \exists !}
Verifying commutativity[edit]
In the left diagram, which expresses the first isomorphism theorem, commutativity of the triangle means that
{\displaystyle f={\tilde {f}}\circ \pi }
. In the right diagram, commutativity of the square means
{\displaystyle h\circ f=k\circ g}
{\displaystyle r\circ h\circ g=H\circ G\circ l}
{\displaystyle m\circ g=G\circ l}
{\displaystyle r\circ h=H\circ m}
Diagram chasing[edit]
In higher category theory[edit]
In higher category theory, one considers not only objects and arrows, but arrows between the arrows, arrows between arrows between arrows, and so on ad infinitum. For example, the category of small categories Cat is naturally a 2-category, with functors as its arrows and natural transformations as the arrows between functors. In this setting, commutative diagrams may include these higher arrows as well, which are often depicted in the following style:
{\displaystyle \Rightarrow }
. For example, the following (somewhat trivial) diagram depicts two categories C and D, together with two functors F, G : C → D and a natural transformation α : F ⇒ G:
Diagrams as functors[edit]
Main article: Diagram (category theory)
a node for every object in the index category,
an arrow for a generating set of morphisms (omitting identity maps and morphisms that can be expressed as compositions),
the commutativity of the diagram (the equality of different compositions of maps between two objects), corresponding to the uniqueness of a map between two objects in a poset category.
the objects are the nodes,
there is a morphism between any two objects if and only if there is a (directed) path between the nodes,
with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom).
However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram). As a simple example, the diagram of a single object with an endomorphism (
{\displaystyle f\colon X\to X}
), or with two parallel arrows (
{\displaystyle \bullet \rightrightarrows \bullet }
{\displaystyle f,g\colon X\to Y}
, sometimes called the free quiver), as used in the definition of equalizer need not commute. Further, diagrams may be messy or impossible to draw, when the number of objects or morphisms is large (or even infinite).
^ Weisstein, Eric W. "Commutative Diagram". mathworld.wolfram.com. Retrieved 2019-11-25.
^ a b "Maths - Category Theory - Arrow - Martin Baker". www.euclideanspace.com. Retrieved 2019-11-25.
^ Riehl, Emily (2016-11-17). "1". Category Theory in Context (PDF). Dover Publications. p. 11.
^ Weisstein, Eric W. "Diagram Chasing". mathworld.wolfram.com. Retrieved 2019-11-25.
Adámek, Jiří; Horst Herrlich; George E. Strecker (1990), Abstract and Concrete Categories (PDF), John Wiley & Sons, ISBN 0-471-60922-6 Now available as free on-line edition (4.2MB PDF).
Barr, Michael; Wells, Charles (2002), Toposes, Triples and Theories (PDF), ISBN 0-387-96115-1 Revised and corrected free online version of Grundlehren der mathematischen Wissenschaften (278) Springer-Verlag, 1983).
WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms, categories, functors, natural transformations.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Commutative_diagram&oldid=1084911961#Arrow_symbols"
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Dolby Stereo - Wikipedia
Find sources: "Dolby Stereo" – news · newspapers · books · scholar · JSTOR (June 2016) (Learn how and when to remove this template message)
Dolby Stereo is a sound format made by Dolby Laboratories. It is a unified brand for two completely different basic systems: the Dolby SVA (stereo variable-area) 1976 system used with optical sound tracks on 35mm film,[1] and Dolby Stereo 70mm noise reduction on 6-channel magnetic soundtracks on 70mm prints.[2]
Dolby SVA significantly improves the development of sound effects in films and theorization of sound design by Walter Murch.[1] In 1982, it was adapted for home use as Dolby Surround when hi-fi capable consumer VCRs were introduced, and further improved in 1987 with the Dolby Pro Logic home decoding system.
1 Dolby SVA
1.2 The Dolby Stereo Matrix
1.3 Dolby Surround/Dolby Pro Logic (home decoders)
2 Dolby Stereo 70 mm Six Track
3 Ultra Stereo
Dolby SVA[edit]
Of the two, Dolby SVA is by far the more significant, bringing high-quality stereo sound within the reach of virtually every cinema. Though 6-track magnetic stereo had been used in Cinerama films since 1952, and Fox had introduced 4-track stereo magnetic sound as part of the CinemaScope system in 1953, the technology had proved to be expensive and unreliable. [3] Except in large cities, most movie theaters did not have facilities for playing back magnetic soundtracks, and a majority of films continued to be produced with mono optical soundtracks. Dolby SVA provided a method for putting high-quality stereo soundtracks on optical sound prints.
The optical soundtrack on a Dolby Stereo encoded 35 mm film carries not only left and right tracks for stereophonic sound, but also—through a matrix decoding system (Dolby Motion Picture matrix or Dolby MP[4]) similar to that developed for "quadraphonic" or "quad" sound in the 1970s—a third center channel, and a fourth surround channel for speakers on the sides and rear of the theater for ambient sound and special effects. This yielded a total of four sound channels, as in the 4-track magnetic system, in the track space formerly allocated for one mono optical channel. Dolby also incorporated its A-Type noise reduction into the Dolby Stereo system.
Dolby Labs became involved in movie sound when film studios used Dolby A type noise reduction on studio magnetic film recordings. The first film to use Dolby noise reduction throughout the production process is A Clockwork Orange (1971), though much of the benefit was lost when it was released with a standard "Academy" optical soundtrack. This led to a proposal from Dolby that A type noise reduction be applied to the optical soundtrack on release prints.[5]
Through the early 1970s, there was renewed interest in improving the quality of optical soundtracks, which had changed little since the 1930s. In particular, the infamous "Academy curve" (the standard frequency response for cinema playback of optical tracks as specified by the Academy of Motion Picture Arts and Sciences in 1938) was still in use. It involved a drastic roll-off in the high-frequency response of the theater system with the intention of reducing the audibility of noise and distortion. Dolby proposed replacing the Academy curve with Dolby A type noise reduction on the track. Starting with the 1974 film Callan, ten films were released with a Dolby encoded mono soundtrack. Theaters were equipped with a Dolby A type noise reduction module and a third-octave equalizer to equalize the electro-acoustic frequency response of the speakers/auditorium. It created a new international standard for cinema sound.[6][7]
Though the system worked well, theater owners were reluctant to invest in the technology until stereo was added to the mix. The idea of putting a two-channel optical stereo soundtrack in the normal soundtrack area of a film print was not new; the stereo pioneer Alan Blumlein had made experimental stereophonic films using such a system as early as 1933, and J. G. Frayne of Westrex had proposed a similar system to the SMPTE in 1955.[8]
By 1970, however, it was apparent that magnetic recording methods were not going to displace optical soundtracks on most release prints, and in the early 1970s Eastman Kodak revived the idea, as described by R. E. Uhlig in a paper presented to the SMPTE in 1972.[9] Initially using a two-channel, 16m film recorder built for them by RCA, Kodak recorded two-channel stereo soundtracks much as Blumlein and Frayne had done before, but added Dolby noise reduction to improve the limited dynamic range available from these half-width tracks.[7]
The remaining problem was the lack of a center channel, regarded as essential to lock dialogue to the middle of the screen. Uhlig discussed this issue in a follow-up paper.[10] He considered the possibility of splitting the soundtrack area three ways to provide a third center channel, but dismissed it because of the negative impact it would have on dynamic range and the problems involved in converting film projectors. Instead he suggested feeding a center-channel speaker with a simple mix of the left and right channels; however, this is not entirely satisfactory as it degrades the stereo separation. After he brought his idea and worked with Dolby, Dolby SVA, a 35mm stereo variable-area optical encoding and decoding system, came up and later was adopted as an industry standard--ISO 2969. [1]
At this time, Dolby joined forces with Kodak in developing this system. Dolby's solution to the center-channel problem was to use a "directionally enhanced" matrix decoder, based on those developed for domestic "Quadraphonic" systems, to recover a center channel from left and right channels recorded on the film. The matrix decoder originally employed for this used the Sansui QS matrix under license. This system was used for the 1975 Ken Russell film Lisztomania.
The matrix was then extended to provide a fourth channel for surround loudspeakers, allowing for a 4-channel system with the same speaker layout as the CinemaScope 4-track magnetic stereo system of the 1950s, but at a far lower cost.
Dolby Stereo, as this 4-channel system was now branded, was first used in 1976's A Star is Born. From spring 1979 onward, a new custom matrix replaced the Sansui QS matrix. It was first used in that year's Hair and Hurricane.[11][7]
At first, Dolby Stereo equipment was installed mainly in larger theaters already equipped with amplifiers and speakers for CinemaScope 4-track stereo. But the success of 1977's Star Wars, which used the 4-channel system to great effect, encouraged owners of smaller theaters to install stereo equipment for the first time.
A key feature of this system was its backward compatibility: the same print could play anywhere, from an old drive-in theater with mono sound to a Dolby Stereo-equipped cinema, eliminating the need for a costly double inventory of prints for distribution. The success of Dolby Stereo resulted in the final demise of magnetic stereo on 35mm release prints. From then on, only 70mm prints used magnetic sound.
In the early 1990s, Dolby SR noise reduction began to replace Dolby A type NR in 35mm motion picture exhibition. All release prints encoded with Dolby Digital include a Dolby SR analog soundtrack, both as a backup in case the digital track malfunctions and for theaters not equipped for Dolby Digital playback.
Today, as of 2021; Dolby Stereo still exists, although its market share in the cinema space has quite declined rapidly, with the advent of digital cinema. It serves as a backup for movie theaters, still exhibiting films on 35mm film.
The Dolby Stereo Matrix[edit]
The Dolby Stereo Matrix is straightforward. The four original channels of Left (L), Center (C), Right (R), and Surround (S), are combined into two, known as Left-total (LT) and Right-total (RT) by this formula:
{\displaystyle 1}
{\displaystyle 0}
{\displaystyle {\frac {1}{\sqrt {2}}}}
{\displaystyle +j{\frac {1}{\sqrt {2}}}}
{\displaystyle 0}
{\displaystyle 1}
{\displaystyle {\frac {1}{\sqrt {2}}}}
{\displaystyle -j{\frac {1}{\sqrt {2}}}}
where j = 90° phase-shift
This center channel information is carried by both LT and RT in phase, and surround channel information by both LT and RT but out of phase. This gives good compatibility with both mono playback, which reproduces L, C and R from the mono speaker with C at a level 3dB higher than L or R, but surround information cancels out. It also gives good compatibility with two-channel stereo playback where C is reproduced from both left and right speakers to form a phantom center and surround is reproduced from both speakers but in a diffuse manner.
A simple 4-channel decoder could simply send the sum signal (L+R) to the center speaker, and the difference signal (L-R) to the surrounds. But such a decoder would provide poor separation between adjacent speaker channels, thus anything intended for the center speaker would also reproduce from left and right speakers only 3dB below the level in the center speaker. Similarly anything intended for the left speaker would be reproduced from both the center and surround speakers, again only 3dB below the level in the left speaker. There is, however, complete separation between left and right, and between center and surround channels.
To overcome this problem the cinema decoder uses so-called "logic" circuitry to improve the separation. The logic circuitry decides which speaker channel has the highest signal level and gives it priority, attenuating the signals fed to the adjacent channels. Because there already is complete separation between opposite channels there is no need to attenuate those, in effect the decoder switches between L and R priority and C and S priority. This places some limitations on mixing for Dolby Stereo and to ensure that sound mixers mixed soundtracks appropriately they would monitor the sound mix via a Dolby Stereo encoder and decoder in tandem.[11] In addition to the logic circuitry the surround channel is also fed via a delay, adjustable up to 100 ms to suit auditoria of differing sizes, to ensure that any leakage of program material intended for left or right speakers into the surround channel is always heard first from the intended speaker. This exploits the "Precedence effect" to localize the sound to the intended direction.
Dolby Surround/Dolby Pro Logic (home decoders)[edit]
Main article: Dolby Pro Logic
Dolby Surround is the earliest consumer version of Dolby's multichannel analog film sound format Dolby Stereo.
Due to the compatibility of the Dolby Stereo matrix with mono and stereo playback, when films originally made in Dolby Stereo were released on stereo domestic video formats - such as VHS-HiFi, laserdisc or broadcast on stereo TV - the original two-channel Dolby Stereo soundtrack could be used. Some domestic listeners were keen to hear these soundtracks in a manner more akin to how they would have sounded in the theater and for that market some manufacturers produced simplified surround decoders. To keep the cost down these decoders dispensed with a center speaker output and the logic circuitry found on the professional decoder, but did include the surround delay. To distinguish these decoders from the professional units found in cinemas they were given the name "Dolby Surround" decoders. The term "Dolby Surround" was also licensed by Dolby for use on TV programs or straight-to-video movies recorded through the Dolby Stereo matrix.
By the late 1980s integrated-circuit manufacturers were working on designing Dolby matrix decoders. A typical early example is the SSM-2125 from PMI.[12] The SSM-2125 is a complete Dolby Stereo matrix decoder (except for the surround delay) on a single chip, it allowed domestic decoders which used the same logic system found in professional decoders to be marketed to the consumer. These decoders were thus given the name Dolby Pro Logic.
Dolby Stereo 70 mm Six Track[edit]
Dolby Stereo 70 mm Six Track is the use of Dolby noise reduction on the six magnetic soundtracks of a 70 mm print. This was first used on some prints of the MGM film Logan's Run released in 1976.
The Todd-AO format was introduced in 1955 and included multi-channel magnetic sound from the start, it does not have an optical soundtrack (although in recent years some 70mm prints have used a DTS digital track in place of the analogue magnetic one).
The original layout was for 5 front channels and one surround. But by the 1970s the use of the intermediate (left-center and right-center) tracks had been largely abandoned, these channels either being left blank, or filled with a simple mix of the adjacent channels. Dolby did not approve of this later practice, which results in loss of separation, but instead used these channels for LFE (low-frequency enhancement) utilizing the bass units of the otherwise redundant intermediate front speakers. Later the unused HF capacity of these channels was used to provide for stereo surround in place of the mono surround of the Todd-AO layout[13] giving the modern 5.1 channel allocation retained today by Dolby Digital.
Ultra Stereo[edit]
By 1984, Dolby Stereo had a competitor. Ultra Stereo Labs had introduced a comparable stereo optical sound system, Ultra Stereo. Its cinema processor introduced improvements in matrix decoding and convolution matching with greater channel separation. An included balancing circuit compensated for film weave and some imbalances between the left and right tracks that previously caused voice leakage into the surround channel. The Ultra Stereo sound system won a 1984 Technical Achievement Award from the Academy of Motion Picture Arts and Sciences.[14]
^ a b c Beck, Jay (2016). Designing sound: audiovisual aesthetics in 1970s American cinema. New Brunswick, New Jersey. ISBN 978-0-8135-6415-9. OCLC 945447753.
^ Dienstfrey (2016). "The Myth of the Speakers: A Critical Reexamination of Dolby History". Film History. 28 (1): 167. doi:10.2979/filmhistory.28.1.06.
^ Bragg, Herbert E.; Belton, John (1988). "The Development of CinemaScope". Film History. 2 (4): 359–371. ISSN 0892-2160.
^ "Dolby Surround Sound". Margo.student.utwente.nl. Retrieved 2015-07-23.
^ Rothman, Lily. "How Did Dolby Sound Change the Movies?". Retrieved 1 January 2017 – via entertainment.time.com.
^ "The production of Wide-Range, Low-Distortion Optical Soundtracks Utilising the Dolby Noise Reduction System" by Ioan Allen – Journal of the SMPTE Vol84 September 1975.
^ a b c "The History of Surround Sound - Surround Sound in the Movies - Que". Retrieved 1 January 2017.
^ "A Compatible Photographic Stereophonic Sound System" by J. G. Frayne, Journal of the SMPTE, Vol 64 June 1955
^ "Stereophonic Photographic Soundtracks" by R. E. Uhlig, Journal of the SMPTE Vol 82 April 1973
^ "Two and Three Channel Stereophonic Soundtracks for Theaters and Television" by R. E. Uhlig, Journal of the SMPTE Vol 83 September 1974
^ a b "Mixing Dolby Stereo Film Sound" by Larry Blake, Recording Engineer/Producer Vol 12, No.1 - February 1981
^ PMI Audio Handbook Vol 1, 1990
^ "The CP200 - A Comprehensive Cinema Theater Audio Processor" by David Robinson. Journal of the SMPTE September 1981
^ "www.uslinc.com". www.uslinc.com. Archived from the original on 2012-07-08. Retrieved 2015-07-23.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Dolby_Stereo&oldid=1084136411"
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This is markdown unity test! | StoryHub
Files named README.md are automatically displayed below the file’s directory listing. For the top level directory this mirrors the standard GitHub presentation.
*** promo README.md files are meant to provide orientation for developers browsing the repository, especially first-time users.
headerFontFamily: ['Avenir Next', 'Helvetica Neue', 'Arial', 'sans-serif'],
// JS web apps.
<!-- Embedded content here ... -->
var rows = prompt('How many rows for your multiplication table?')
var cols = prompt('How many columns for your multiplication table?')
if (rows == '' || rows == null) rows = 10
if (cols == '' || cols == null) cols = 10
createTable(rows, cols)
var output = "<table border='1' width='500' cellspacing='0'cellpadding='5'>"
output = output + '<tr>'
while (j <= cols) {
output = output + '<td>' + i * j + '</td>'
output = output + '</tr>'
output = output + '</table>'
You can render LaTeX mathematical expressions.
\Gamma(n) = (n-1)!\quad\forall n\in\mathbb N
\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.
You can render UML diagrams, this will produce a sequence diagram:
And this will produce a flow chart:
Libraries often attempt to remain neutral in the resulting debates, but that neutrality is predicated on the idea that the debates are taking place on a post-enlightenment playing field and that, eventually, the best ideas will succeed. Or at least that, over time, reasonable people will develop a shared set of facts and tools for assessing and discussing that reality. This allows the library to accomplish good for its users by providing access
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find the quotient and remainder:6x^6 - 3x^2 + 7x - 152 ÷ x^2 - 3 - Maths - Algebraic Expressions and Identities - 8811585 | Meritnation.com
Answer : Find quotient and remainder for :
\frac{6{x}^{6} - 3{x}^{2} + 7x - 152}{{x}^{2} - 3}
We using long division method , As :
Quotient = 6 x4 + 18 x2 + 51
Remainder = 7 x + 1
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High Index SMES Device for Gravitomagnetic Field Generation
Seculine Consulting, Cupertino, CA, USA.
Abstract: A method is described for creating a measurable unbalanced gravitational acceleration using a gravitomagnetic field surrounding a superconducting toroid as described by Forward (1962). An experimental Superconducting Magnetic Energy Storage (SMES) toroid configuration of wound superconducting nanowire is proposed to create a measurable acceleration field along the axis of symmetry, providing experimental confirmation of the additive nature of a Lense-Thirring derived gravitomagnetic field. In the present paper, gravitational coupling enhancement of this effect is explored using a high index or high permittivity material, as predicted by Sarfatti (2020) using his modification to Einstein’s General Relativity Field Equations for gravitational coupling in matter.
Keywords: Gravitational, Gravitomagnetic, Lense-Thirring, Superconducting Magnetic Energy Storage, SMES, Nanorods, Nanowires, Super Dielectric Materials, SDM, Super Capacitors, High Permittivity, Gravitomagnetic Permeability, Gravitational Coupling
When Forward [1] first proposed a gravitomagnetic toroid for unbalanced gravitational force production in 1962 that any experimental realization was quite impractical. However, recent advances in high temperature superconducting (HTSC) nanorod wire (nanowire) technology, described recently by Stephenson et al. [2] and Rieken et al. [3], have enabled a new class of superconducting magnetic energy storage (SMES) devices operating at current densities sufficient to develop measurable gravitomagnetic fields.
In the present paper, an experimental SMES toroid configuration is proposed that uses a super dielectric material (e.g. high permittivity super capacitor) to substantially improve gravitational coupling of mass flow to the curvature of spacetime as predicted by Sarfatti [4]. Depending on the nature of gravitational coupling, it is predicted that a set of standard accelerometers could measure acceleration fields along the axis of symmetry of the toroidal coil, thus providing experimental confirmation of the additive nature of the gravitomagnetic fields, as well as the production of a linear component of the overall acceleration field, see Figure 1 for details.
In the instantiation of Forward’s gravitational generation coil described in this paper, superconducting electron flow provides the change in mass current in the toroid. A high permittivity material such as a super dielectric material (SDM) [5] is added to the space in the center of the toroid to improve gravitational coupling. Alternatively or additionally high permittivity insulators may be used as an exterior jacket to the conducting portion of the nanowire so as to further boost gravitational coupling, as gravitomagnetic effects are largest adjacent to the mass flow.
As first developed in Forward 1962 Ref. [1] the linear force Gf developed by gravitomagnetic force in the mass flow toroid of Figure 1 is given by Equation (1):
{G}_{f}=\left(\frac{\eta }{4\pi }\right)\left(\frac{N\stackrel{˙}{T}{r}^{2}}{{R}^{2}}\right)
Figure 1. Gravitational force generation coil from Forward 1962 [1] with an inspiralling mass current, with a vector potential P, creating gravitomagnetic field G, with high index material in the center to improve coupling.
\eta
\eta ={\eta }_{o}{\eta }_{r}
as described in reference [2], N is number of turns in the toroid coil winding,
\stackrel{˙}{T}
represents the mass flow, r is the cross sectional radius of the toroid, and R is the overall radius of the toroid.
Mass flow can be generated by means of generating an electrical current by virtue of putting the mass of electrons in motion.
Electron Mass Flow
{T}_{e}={p}_{e}=\left(\Omega \times r\right){m}_{e}
where Ω = angular rate, angular velocity is
v=\Omega \times r
in the classical case [6].
{\stackrel{\dot{}}{T}}_{e}={\stackrel{\dot{}}{p}}_{e}=a\cdot {m}_{e}=\left(\Omega \times v\right){m}_{e}
{\stackrel{˙}{T}}_{e}={F}_{e}=\frac{{m}_{e}{v}^{2}}{r}={m}_{e}{a}_{e}={m}_{e}\left({\omega }^{2}r\right)
\omega =\frac{2\pi }{{t}_{p}}=\frac{\text{d}\theta }{\text{d}t}
We now attempt to estimate the possible inspiralling current flux enabled by the emerging SMES technology as it relates to the core geometry constraints described in Figure 3, a toroid with torus geometry. We start with the assumptions needed to calculate the number of turns N.
From Equation (1) the torus assumptions made in Figure 3 can be factored into Equation (6) as follows:
{G}_{f}=\left(\frac{{\eta }_{o}{\eta }_{r}}{4\pi }\right)S\left(\frac{N\stackrel{˙}{T}{r}^{2}}{{R}^{2}}\right)
Figure 3. Toroid with a torus shaped core geometry, including high index scaling matter in center.
ηo = absolute gravitomagnetic permeability;
ηr = relative gravitomagnetic permeability;
S = Sarfatti scaling factor;
\stackrel{˙}{T}
= change in mass, or mass flow;
Note that Equation (6) differs from Forward’s original Equation (1) formulation in that it includes the scaling factor S which accounts for the scaling of gravitational coupling due to high index matter or metamaterials, which was first expressed by Sarfatti as an additional zero rank tensor in Einstein’s Field Equations [4] [7]:
{R}_{uv}-\frac{1}{2}R{g}_{uv}=\left(\frac{8\pi G}{{c}^{4}}\right)S{T}_{uv}
Ruv = Ricci curvature tensor;
Rguv = Ricci scalar curvature;
c = speed of light;
S = Sarfatti scaling tensor;
Tuv = Stress Energy Tensor.
For the purposes of describing an idealized case with a realistic geometry we develop a description of a device bounded by a 10 cm toroid centerline diameter, shown in Figure 3 and Figure 4, with a cross-sectional diameter of 1 cm. Furthermore 16 sectors are defined as shown in Figure 4.
We further add additional assumptions regarding to what extent conductors are wrapped around the toroid shaped device to determine constraints on the number of conductive loops that can be accommodated using the technology described in [2]. As shown in Figure 5 via cross section we assume here a conductor winding depth of 0.1 cm.
The segmented share of the inside of cross section Cs is 1/16th of the overall inner circumference as given in Equation (8) as follows:
{C}_{s}=\frac{2\pi {r}_{i}}{16}=1.77\text{\hspace{0.17em}}\text{cm}
with a depth D the minimum inner loop cross sectional area can be described in Equation (9) as follows:
{A}_{sec}=D\cdot {C}_{s}=0.1\text{\hspace{0.17em}}\text{cm}\times 1.77\text{\hspace{0.17em}}\text{cm}=0.177\text{\hspace{0.17em}}{\text{cm}}^{\text{2}}
{A}_{c}=\pi {r}_{c}^{2}=\text{7}.\text{854}\times {10}^{-\text{9}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\text{m}}^{\text{2}}
For packing nanowire conductors in a cross-sectional area described in Figure 6, assume as a worst case rectangular area described by the shortest edges such that a number of conductors in depth, Nd, may be packed in one dimension, with the number of conductors, Ncs, packed in the other dimension. These packing counts may be calculated in Equation (11a) and Equation (11b) as follows:
{N}_{d}={D}_{c}/{d}_{c}=\frac{0.1\text{\hspace{0.17em}}\text{cm}}{100\text{\hspace{0.17em}}\mu \text{m}}=10
{N}_{cs}={C}_{s}/{d}_{c}=\frac{1.77\text{\hspace{0.17em}}\text{cm}}{100\text{\hspace{0.17em}}\mu \text{m}}=177
{N}_{sec}={N}_{d}\cdot {N}_{cs}=10\times 177=1770
N=16{N}_{cs}=28320
\stackrel{˙}{T}
{c}_{r}=2\pi r=3.14\text{\hspace{0.17em}}\text{cm}
Assume further a supply voltage of 16 KV, resulting in 16 KeV of kinetic energy for each electron, which corresponds with the upper limit of a non-relativistic case, where v = 0.25c, so thatγ = 1.06 - 1.0.
{\stackrel{˙}{T}}_{e}=\frac{{m}_{e}{v}^{2}}{r}
r = 0.5 cm for the assumed geometry.
{a}_{e}=\frac{{v}^{2}}{r}=1.125\times {10}^{18}\text{m}/{\text{s}}^{\text{2}}
{\stackrel{\dot{}}{T}}_{e}={m}_{e}\cdot {a}_{e}=10.25\times {10}^{-13}\text{\hspace{0.05em}}\text{N}
\stackrel{˙}{T}={\stackrel{˙}{T}}_{e}\cdot {N}_{e}
What is the number of electrons Ne in one loop in motion (part of the mass flow) at a given time for an assumed velocity of v = 0.25c? Ne in one loop can be described by the current I times the period of a single loop circulation ∆t:
{N}_{e}=I\cdot \Delta t
\Delta t=\frac{{c}_{r}}{v}=\frac{2\pi r}{v}=\text{}0.4189\text{\hspace{0.17em}}\text{ns}
what is the possible current inside the idealized device for the case where the entire winding is in series? We make the assumption about max current to stay below critical current density of 250 MA/m2as described in [2] [3].
Current is limited by the maximum permissible current density and the cross section of the conductort:
I=J\cdot {A}_{c}
where J is material dependent. For the nanowire assumed in Ref. [2], J = 250 MA/m2. Cross sectional area Ac = 7.854 × 10−9 m2 as given in Equation (10). Therefore, maximum current for this conductor diameter isI = 1.96 Amps.
{N}_{e}=\left(\frac{\text{electrons}}{\text{Coulomb}}\right)I\left(\frac{\text{Coulombs}}{\text{sec}}\right)\cdot \Delta t=5.12\times {10}^{9}\text{\hspace{0.05em}}\text{electrons}
\stackrel{˙}{T}={\stackrel{˙}{T}}_{e}\left(\frac{\text{Newtons}}{\text{electron}}\right)\cdot {N}_{e}\left(\text{#electrons}\right)=5.25\text{\hspace{0.17em}}\text{mN}
Thus, each loop experiences about 5.25 mN of integrated centripetal force (
\stackrel{˙}{T}
Revisiting Equation (6) which describes the overall linear force developed at the center of the toroidal coil, total gravitomagnetically developed force will be:
{G}_{f}=\left({\eta }_{o}{\eta }_{r}\right)\left(S\right)\left(\frac{N\stackrel{˙}{T}{r}^{2}}{4\pi {R}^{2}}\right)=\left({\eta }_{o}{\eta }_{r}\right)\left(S\right)\left(0.118\text{\hspace{0.17em}}\text{N}\right)
where known variables have been grouped on the right and unknown variables have beencollected on the left. This raises the question what are the correct values for ηo and ηr?
Predicted force using gravitational potential scaling
If ηo goes as G/2c as does gravitomagnetic potential (Ref. [8], Equation (1.5)), then:
{\eta }_{o}=-\frac{G}{2c}=1.11\times {10}^{-19}
{G}_{f}{}_{}=\left({\eta }_{r}\right)1.31\times {10}^{-20}
. Values of ηr are experimentally unknown at this time. However if values of ηr track values of
{\epsilon }_{r}^{2}
then values as high as ηr = 1016 may be possible, yielding Gf = 0.131 mN. For a test mass of 1 Kg this is equivalent to an easily measurable 13.4 micro-g’s. This is for a scaling factor of G/2c. If the scaling factor is G/c2 then the effect drops to an unmeasurably low level of 4.47 × 10−14 g’s, even for ηr = 1016.
Predicted force using gravitational field scaling
If the scaling factor is G/c3 or G/c4 then the effect is negligible for ηr = 1016, and a much higher index material will need to be found to improve coupling, or other means must be found for improving either the S tensor and/or gravitomagnetic permeability ηr.
Consider the case where ηo goes as 8πG/c4 as does the coupling constant in Einstein’s Field Equations, then:
{\eta }_{o}=\frac{8\pi G}{{c}^{4}}=2.07\times {10}^{-43}
which gives the following expression for total gravitational force developed:
{G}_{f}=\left({\eta }_{o}{\eta }_{r}\right)\left(S\right)\left(0.118\text{\hspace{0.17em}}\text{N}\right)=\left({\eta }_{r}\right)\left(S\right)2.44\times {10}^{-44}\text{\hspace{0.05em}}\text{N}
A value of ηr=1016 gives
{G}_{f}=\left(S\right)4.43\times {10}^{-23}
Newtons, which requires a scalar value of S of at least an additional 1016 to obtain a measurable value for Gf. Per [4] for an isotropic material S can be written as:
S=\frac{1}{2}\left({ϵ}^{2}+\frac{1}{{\mu }^{2}}\right)=\frac{1}{2}\left({n}^{4}+1\right)/{\mu }^{2}
Therefore either a permittivity of ε = 108 or an index of n = 104 with μ = 1 would be sufficient to provide the necessary 16 orders of magnitude in coupling improvement. However care should be taken that effects captured by the S factor are not duplicative of those accounted for in the value of ηr.
Per [5] super dielectrics can provide up to ε = 109 which, if we simultaneously had ηr = 1016 via some unrelated phenomena, would improve the overall generated gravitational force to a marginally measurable level:
{G}_{f}=\left({\eta }_{r}\right)\left(S\right)2.44\times {10}^{-44}=0.244\text{\hspace{0.17em}}\text{nN}
Alternatively metamaterials could also be investigated to establish whether non-isotropic materials may create a more advantageous gravitational coupling improvement.
Lenz’s Law Implications
Is there an equivalent of Lenz’s Law for gravitomagnetics? Consider a hypothetical case where a Forward toroid that is capable of supporting sufficient current and coupling to develop 1 G of acceleration. As a thought experiment imagine disconnecting the power supply and shorting the toroidal coil to itself, such that it is one continuous closed loop of nanowire, and equip a flying craft with such a coil.
At the surface of the Earth such a properly oriented toroidal coil would be immersed in a 1g gravitational field. Should not such a field generate a counter-current in a toroidal coil capable of supporting a current of that level, essentially providing a self-powered counter field? If so, a craft so equipped would be able to hover using a counter-field equal but opposite to the Earth’s gravitational field without need of additional power sources.
Similarly, additional closed loop toroidal coils, properly oriented, could also be used to “current charge,” collecting energy from a static gravitational field, potentially providing power that could be directed to other propulsive or non-propulsive functions of such a hypothetical craft.
An argument is made for using high permittivity materials in the core or donut hole and/or nanowire insulators of an SMES toroid to improve gravitomagnetic coupling for the creation of an acceleration field, possibly of measurable amplitude. Improved coupling will be beneficial for shrinking device scale and complexity and overcoming the very weak coupling between mass flow and gravitomagnetic spacetime curvature.
The authors wish to acknowledge Jack Sarfatti for helpful correspondence on his insights into altering Einstein’s Field Equations in the presence of matter and/or metamaterials. The financial support of Seculine Consulting is gratefully acknowledged.
HTSC: High temperature superconductor
SC: Superconductor
SDM: Super dielectric material
Cite this paper: Stephenson, G. (2021) High Index SMES Device for Gravitomagnetic Field Generation. Journal of High Energy Physics, Gravitation and Cosmology, 7, 367-376. doi: 10.4236/jhepgc.2021.72020.
[1] Forward, R. (1962) Guidelines to Antigravity. Proceedings of the Gravity Research Foundation, New Boston.
[2] Stephenson, G., Rieken, W. and Bhargava, A. (2019) Extended Cases of Laboratory Generated Gravitomagnetic Field Measurement Devices. Journal of High Energy Physics, Gravitation and Cosmology, 5, 375-394.
[3] Rieken, W., Bhargava, A., et al. (2018) YBa2Cu3Ox Superconducting NanoRods. Japanese Journal of Applied Physics, 57, 023101.
[4] Stephenson, G. (2020) 2000-2020 Summary of Gravitational Work. APEC Proceedings, Blaine, 12 December 2020.
[5] Fromille, S. and Phillips, J. (2014) Super Dielectric Materials. Materials, 7, 8197-8212.
[7] Sarfatti, J. (2020) Explaining US Navy Close Encounters with Tic Tac UAV Metric Engineering. Proceedings of the Estes Park Advanced Propulsion Workshop, Space Studies Institute, North Hollywood, October 2020.
[8] Mashhoon, B. (2008) Gravitoelectromagnetism: A Brief Review.
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Let’s return to the logistic regression equation and demonstrate how this works by fitting a model in sklearn. The equation is:
ln(\frac{p}{1-p}) = b_{0} + b_{1}x_{1} + b_{2}x_{2} +\cdots + b_{n}x_{n}
Suppose that we want to fit a model that predicts whether a visitor to a website will make a purchase. We’ll use the number of minutes they spent on the site as a predictor. The following code fits the model:
model.fit(purchase, min_on_site)
Next, just like linear regression, we can use the right-hand side of our regression equation to make predictions for each of our original datapoints as follows:
log_odds = model.intercept_ + model.coef_ * min_on_site
print(log_odds)
Notice that these predictions range from negative to positive infinity: these are log odds. In other words, for the first datapoint, we have:
ln(\frac{p}{1-p}) = -3.28394203
We can turn log odds into a probability as follows:
\begin{aligned}
ln(\frac{p}{1-p}) = -3.28 \\
\frac{p}{1-p} = e^{-3.28} \\
p = e^{-3.28} (1-p) \\
p = e^{-3.28} - e^{-3.28}*p \\
p + e^{-3.28}*p = e^{-3.28} \\
p * (1 + e^{-3.28}) = e^{-3.28} \\
p = \frac{e^{-3.28}}{1 + e^{-3.28}} \\
p = 0.96
\end{aligned}
In Python, we can do this simultaneously for all of the datapoints using NumPy (loaded as np):
np.exp(log_odds)/(1+ np.exp(log_odds))
The calculation that we just did required us to use something called the sigmoid function, which is the inverse of the logit function. The sigmoid function produces the S-shaped curve we saw previously:
In the workspace, we’ve fit a logistic regression on the Codecademy University data and saved the intercept and coefficient on hours_studied as intercept and coef, respectively.
For each student in the dataset, use the intercept and coefficient to calculate the log odds of passing the exam. Save the result as log_odds.
Now, convert the predicted log odds for each student into a predicted probability of passing the exam. Save the predicted probabilities as pred_probability_passing.
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Symmetry Considerations When Using Proper Orthogonal Decomposition for Predicting Wind Turbine Yaw Loads | J. Sol. Energy Eng. | ASME Digital Collection
Saranyasoontorn, K., and Manuel, L. (July 18, 2006). "Symmetry Considerations When Using Proper Orthogonal Decomposition for Predicting Wind Turbine Yaw Loads." ASME. J. Sol. Energy Eng. November 2006; 128(4): 574–579. https://doi.org/10.1115/1.2349541
In an earlier study, the authors discussed the efficiency of low-dimensional representations of inflow turbulence random fields in predicting statistics of wind turbine loads that included blade and tower bending moments. Both root-mean-square and
10-min
extreme statistics for these loads were approximated very well when time-domain simulations were carried out on a
600kW
two-bladed turbine and only a limited number of inflow “modes” were employed using proper orthogonal decomposition (POD). Here, turbine yaw loads are considered and the conventional ordering of POD modes is seen to be not as efficient in predicting full-field load statistics for the same turbine. Based on symmetry arguments, reasons for a different treatment of yaw loads are presented and reasons for observed deviation from the expected monotonic convergence to full-field load statistics with increasing POD mode number are illustrated.
wind turbines, load forecasting, statistics, turbulence, blades
Blades, Inflow, Principal component analysis, Simulation, Statistics as topic, Stress, Turbulence, Wind turbines, Yaw, Turbines
Cambridge Monogr. Mech., Cambridge University Press
The Dynamic Response of a Westinghouse 600-kW Wind Turbine
,” Report SERI/STR-217–3405,
SNwind User’s Guide
, IEC/TC 88 61400–1.
, Report NREL/EL-500–29798, Golden, CO.
,” Report NREL/TP-442–4822,
Measured Structural Loads for the Micon 65∕13
Wind Energy: Proceedings of the Energy-Sources Technology Conference
, SED-Vol. 15, January 23–26,
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Revision as of 10:09, 24 March 2015 by MathAdmin (talk | contribs) (Created page with "<br> <span style="font-size:135%"> <font face=Times Roman>8. (a) Find the linear approximation <math style="vertical-align: -14%;">L(x)</math> to the function <math style="ver...")
{\displaystyle L(x)}
{\displaystyle f(x)=\sec x}
{\displaystyle x=\pi /3}
{\displaystyle L(x)}
{\displaystyle \sec \,(3\pi /7)}
Recall that the linear approximation L(x) is the equation of the tangent line to a function at a given point. If we are given the point x0, then we will have the approximation
{\displaystyle L(x)=f'(x_{0})\cdot (x-x_{0})+f(x_{0})}
. Note that such an approximation is usually only good "fairly close" to your original point x0.
Note that f '(x) = sec x tan x. Since sin (π/3) = √3/2 and cos (π/3) = 1/2, we have
{\displaystyle f'(\pi /3)=2\cdot {\frac {{\sqrt {3}}/2}{1/2}}=2{\sqrt {3}}.}
Similarly, f(π/3) = sec (π/3) = 2. Together, this means that
{\displaystyle L(x)=f'(x_{0})\cdot (x-x_{0})+f(x_{0})}
{\displaystyle =2{\sqrt {3}}(x-\pi /3)+2.}
This is simply an exercise in plugging in values. We have
{\displaystyle L\left({\frac {3\pi }{7}}\right)=2{\sqrt {3}}\left({\frac {3\pi }{7}}-{\frac {\pi }{3}}\right)+2}
{\displaystyle =2{\sqrt {3}}\left({\frac {9\pi -7\pi }{21}}\right)+2}
{\displaystyle =2{\sqrt {3}}\left({\frac {2\pi }{21}}\right)+2}
{\displaystyle ={\frac {4{\sqrt {3}}\pi }{21}}+2.}
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Effect of Drying Air Velocity on Drying Kinetics of Tomato Slices in a Forced-Convective Solar Tunnel Dryer
1Energy-Environment-Development Niger (ENDA Energy Niger), Niamey, Niger
2Laboratoire d’Energétique, d’Electronique, d’Electrotechnique, d’Automatique et d’Informatique Industrielle (LEEII), Université Abdou Moumouni, Niamey, Niger
The objective of this work is to analyse the extent to which a change in the drying air velocity may affect the drying kinetics of tomato in a forced-convective solar tunnel dryer. 2 m∙s−1 (V1) and 3 m∙s−1 (V2) air speeds were applied in similar drying air temperature and humidity conditions. Main drying constants calculated included the drying rate, the drying time and the effective water diffusivity based on the derivative form of the Fick’s second law of diffusion. Henderson and Pabis Model and Page Model were used to describe the drying kinetics of tomato. We found that solar drying of tomato occurred in both constant and falling-rate phases. The Page Model appeared to give a better description of tomato drying in a forced-convective solar tunnel dryer. At t = 800 min, the drying rate was approximately 0.0023 kg of water/kg dry matter when drying air velocity was at 2 m/s. At the same moment, the drying rate was higher than 0.0032 kg of water/kg dry matter when the drying air velocity was 3 m/s. As per the effective water diffusivity, its values changed from 2.918E−09 m2∙s−1 to 3.921E−09 m2∙s−1 when drying air velocity was at 2 and 3 m∙s−1 respectively, which is equivalent to a 25% increase. The experimentations were conducted in Niamey, on the 1st and 5th of January 2019 for V2 and V1 respectively. For both two experiments, the starting time was 9:30 local time.
X=0.622\ast \frac{\phi \ast {P}_{vsat}}{P-\phi \ast {P}_{vsat}}
\phi
{P}_{vsat}
{P}_{vsat}
{P}_{vsat}=23.1964-\frac{3816.44}{{T}_{a}+273.18}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}{T}_{a}>45˚\text{C}
MR=\frac{{M}_{t}-{M}_{e}}{{M}_{0}-{M}_{e}}
{M}_{t}
{M}_{e}
{M}_{0}
\frac{\partial M}{\partial t}={D}_{eff}\left[\frac{{\partial }^{2}M}{\partial {x}^{2}}+\frac{{a}_{1}}{x}\frac{\partial M}{\partial x}\right]
\frac{\partial T}{\partial t}=\alpha \left[\frac{{\partial }^{2}T}{\partial {x}^{2}}+\frac{{a}_{1}}{x}\frac{\partial T}{\partial x}\right]
MR=\frac{{M}_{t}-{M}_{e}}{{M}_{cr}-{M}_{e}}={A}_{1}{\sum }_{i=1}^{\infty }\frac{1}{{\left(2i-1\right)}^{2}}\text{exp}\left(-\frac{{\left(2i-1\right)}^{2}{\text{π}}^{2}{D}_{eff}t}{{A}_{2}}\right)
{M}_{cr}
{A}_{1}
{A}_{2}
MR=\frac{{M}_{t}-{M}_{e}}{{M}_{cr}-{M}_{e}}={A}_{1}\mathrm{exp}\left(-\frac{{\text{π}}^{2}{D}_{eff}}{{A}_{2}}t\right)
{D}_{eff}
MR=\frac{{M}_{t}-{M}_{e}}{{M}_{cr}-{M}_{e}}=a\cdot \mathrm{exp}\left(-k\cdot t\right)
\frac{8}{{\text{π}}^{2}}
MR=\mathrm{exp}\left(-k{t}^{y}\right)
\mathrm{ln}\left(MR\right)=\mathrm{ln}\left(\frac{8}{{\text{π}}^{2}}\right)-\left(\frac{{\text{π}}^{2}{D}_{eff}}{4{H}^{2}}t\right)
{D}_{eff}
\frac{{\text{π}}^{2}{D}_{eff}}{4{H}^{2}}
\mathrm{ln}\left(MR\right)=f\left(t\right)
{D}_{eff}
{M}_{0}
{M}_{f}
{D}_{eff}
{D}_{eff}
\mathrm{ln}\left(MR\right)=f\left(t\right)
{D}_{eff}
{D}_{eff}
{D}_{eff}
Moussa Na Abou, M., Madougou, S. and Boukar, M. (2019) Paper Title. Journal of Sustainable Bioenergy Systems, 9, 64-78. https://doi.org/10.4236/jsbs.2019.92005
1. Dorouzi, M., Mortezapour, H., Akhavan, H.-R. and Moghaddam, A.G. (2018) Tomato Slices Drying in a Liquid Desiccant-Assisted Solar Dryer Coupled with a Photovoltaic-Thermal Regeneration System. Solar Energy, 162, 364-371. https://doi.org/10.1016/j.solener.2018.01.025
2. FAO (2019) Niger—Production Alimentaire: Tomates. Perspectives Monde, Université de Sherbrooke, Lettreset Sciences Humaines, Ecole de Politique Appliquée. http://perspective.usherbrooke.ca/bilan/servlet/BMTendanceStatPays?langue=fr&codePays=NER&codeStat=RSA.FAO.Tomatoes&codeStat2=x
3. FAO (2015) SAVE FOOD: Global Initiative on Food Loss and Waste Reduction. Food and Agriculture Organization of the United Nations (FAO), Vialedelle Terme di Caracalla 00153, Rome, Italy.
4. Troger, K., Henselb, O. and Burkert, A. (2007) Conservation of Onion and Tomato in Niger—Assessment of Post-Harvest Losses and Drying Methods. Conference on International Agricultural Research for Development, University of Kassel-Witzenhausen and University of Gottingen, 9-11 October 2007.
5. Kemp, I.C., et al. (2001) Methods for Processing Experimental Drying Kinetics Data. Drying Technology, 19, 15-34. https://doi.org/10.1081/DRT-100001350
6. Steinfeld, A. and Segal, I. (1986). A Simulation Model for Solar Thin Layer Drying Process. Drying Technology, 4, 535-554. https://doi.org/10.1080/07373938608916349
7. Nadeau, J.P. (1995) Séchage: des processus physiques aux processus industriels. Tec & Doc Lavoisier, Cachan.
8. Bonazzi, C. and Bimbenet, J.-J. (2003) Séchage des produitsalimentaires: Principes. Institut national agronomique de Paris-Grignon, école nationale supérieure des industries agricoles et alimentaires (ENSIA).
9. Ekechukwu, O.V. (1995) Drying Principles and Theory: An Overview. University of Nigeria and the International Centre for Theoretical Physics, Trieste, Italy.
10. Luikov, A.V. (1966) Heat and Mass Transfer in Capillary-Porous Bodies. Pergamon Press, London. https://doi.org/10.1016/B978-1-4832-0065-1.50010-6
11. Dissa, A.O. (2007) Séchage Convectif et solaire de la mangue (Mangifera Indica L.): Caractérisation expérimentale, modélisation et simulation du procédé telecharger. Editions Universitaires Européennes.
12. Henderson, S.M. and Pabis, S. (1961) Grain Drying Theory II: Temperature Effects on Drying Coefficients. Journal of Agricultural Engineering Research, 6, 169-174.
13. Page, C. (1949) Factors Influencing the Maximum Rates of Air-Drying of Shelled Corn in Thin Layer. Unpublished M.S. Thesis, Purdue University, Lafayette, IN.
14. Crank, J. (1975) Mathematics of Diffusions. 2nd Edition, Oxford University Press, London.
15. Villa-Corrales, L., Flores-Prieto, J.J., Xamán-Villasenor, J.P. and García-Hernández, E. (2010) Numerical and Experimental Analysis of Heat and Moisture Transfer during Drying of Ataulfo Mango. Journal of Food Engineering, 98, 198-206.
16. RECA (2016) La tomate au Niger. Présentation préparée par le RECA.
17. Ahouannou, C., Jannot, Y., Lips, B. and Lallemand, A. (2000) CaractérisationetModélisation du Séchage de trios Produits tropicaux: Manioc, gingembre et gombo. Sciences de Aliments, 20, 413-422. https://doi.org/10.3166/sda.20.413-432
18. Krischer, O. (1963) Die Wissenschaftlichen Grundlagen der Trocknungstechnik. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-662-26011-1
19. Samimi-Akhijahani, H. and Arabhosseini, A. (2018) Accelerating Drying Process of Tomato Slices in a PV-Assisted Solar Dryer Using a Sun Tracking System. Renewable Energy, 123, 428-438. https://doi.org/10.1016/j.renene.2018.02.056
20. Sacilik, K., Keskin, R. and Elicin, A.K. (2005) Mathematical Modelling of Solar Tunnel Drying of Thin Layer Organic Tomato. Journal of Food Engineering, 73, 231-238.
21. Doymaz, I. (2006) Air-Drying Characteristics of Tomatoes. Journal of Food Engineering, 78, 1291-1297. https://www.elsevier.com/locate/jfoodeng
22. Reyes, A., Mahn, A., Huenulaf, P. and González, T. (2014) Tomato Dehydration in a Hybrid-Solar Dryer. Journal of Chemical Engineering & Process Technology, 5, 4. https://doi.org/10.4172/2157-7048.1000196
23. Gaware, T.J., Sutar, N. and Thorat, B.N. (2010) Drying of Tomato Using Different Methods: Comparison of Dehydration and Rehydration Kinetics. Drying Technology, 28, 651-658. https://doi.org/10.1080/07373931003788759
24. Charreau, A. and Cavaillé, R. (1995) Séchage, Théorie et calculs. Techniques de l’Ingénieur, traité Génie des procédés.
25. Yagcioglu, A., Demir, V. and Gunhan, T. (2007) Effective Moisture Diffusivity Estimation from Drying Data. Tarim Makinalari Bilimi Dergisi, 3, 249-256.
26. Zogzas, N.P. and Maroulis, Z.B. (1996) Effective Moisture Diffusivity Estimation from Drying Data: A Comparison between Various Methods of Analysis. Drying Technology, 14, 1543-1573. https://doi.org/10.1080/07373939608917163
27. Lewis, W.K. (1921) The Rate of Drying of Solid Materials. Journal of Industrial and Engineering Chemistry, 13, 427-432. https://doi.org/10.1021/ie50137a021
28. Ben Mariem, S. and Ben Mabrouk, S. (2014) Drying Characteristics of Tomato Slices and Mathematical Modeling. International Journal of Energy Engineering, 4, 17-24.
30. Erbay, Z. and Icier, F. (2010) A Review of Thin Layer Drying of Foods: Theory, Modeling, and Experimental Results. Critical Reviews in Food Science and Nutrition, 50, 441-464.
31. Barati, E. and Esfahani, J.A. (2011) A New Solution Approach for Simultaneous Heat and Mass Transfer during Convective Drying of Mango. Journal of Food Engineering, 102, 302-309.
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LU matrix factorization - MATLAB lu - MathWorks Deutschland
Compute the LU factorization of a matrix and examine the resulting factors. LU factorization is a way of decomposing a matrix
\mathit{A}
into an upper triangular matrix
\mathit{U}
, a lower triangular matrix
\mathit{L}
, and a permutation matrix
\mathit{P}
\mathrm{PA}=\mathrm{LU}
. These matrices describe the steps needed to perform Gaussian elimination on the matrix until it is in reduced row echelon form. The
\mathit{L}
matrix contains all of the multipliers, and the permutation matrix
\mathit{P}
accounts for row interchanges.
Create a 5-by-5 magic square matrix and solve the linear system
\mathrm{Ax}=\mathit{b}
with all of the elements of b equal to 65, the magic sum. Since 65 is the magic sum for this matrix (all of the rows and columns add to 65), the expected solution for x is a vector of 1s.
Using a permutation vector also saves on execution time in subsequent operations. For instance, you can use the previous LU factorizations to solve a linear system
\mathrm{Ax}=\mathit{b}
. Although the solutions obtained from the permutation vector and permutation matrix are equivalent (up to roundoff), the solution using the permutation vector typically requires a little less time.
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Long Short-Term Memory It is an abstraction of how computer memory works. It is “bundled” with whatever processing unit is implemented in the Recurrent Network, although outside of its flow, and is responsible for keeping, reading, and outputting information for the model. The way it works is simple: you have a linear unit, which is the information cell itself, surrounded by three logistic gates responsible for maintaining the data. One gate is for inputting data into the information cell, one is for outputting data from the input cell, and the last one is to keep or forget data depending on the needs of the network.
Thanks to that, it not only solves the problem of keeping states, because the network can choose to forget data whenever information is not needed, it also solves the gradient problems, since the Logistic Gates have a very nice derivative. Long Short-Term Memory Architecture As seen before, the Long Short-Term Memory is composed of a linear unit surrounded by three logistic gates. The name for these gates vary from place to place, but the most usual names for them are:
the “Input” or “Write” Gate, which handles the writing of data into the information cell,
the “Output” or “Read” Gate, which handles the sending of data back onto the Recurrent Network, and
the “Keep” or “Forget” Gate, which handles the maintaining and modification of the data stored in the information cell.
Building a LSTM with TensorFlow Although RNN is mostly used to model sequences and predict sequential data, we can still classify images using a LSTM network. If we consider every image row as a sequence of pixels, we can feed a LSTM network for classification. Lets use the famous MNIST dataset here. Because MNIST image shape is 28*28px, we will then handle 28 sequences of 28 steps for every sample.
mnist = input_data.read_data_sets(".", one_hot=True)
# Defining variables for train & test data
trainings = mnist.train.images
trainlabels = mnist.train.labels
testings = mnist.test.images
testlabels = mnist.test.labels
ntrain = trainings.shape[0]
ntest = testings.shape[0]
dim = trainings.shape[1]
nclasses = trainlabels.shape[1]
print ("Train Images: ", trainings.shape)
print ("Train Labels ", trainlabels.shape)
print ("Test Images: " , testings.shape)
print ("Test Labels: ", testlabels.shape)
Train Images: (55000, 784)
Train Labels (55000, 10)
Test Images: (10000, 784)
Test Labels: (10000, 10)
# Defining Network Parameters
# The input should be a Tensor of shape: [batch_size, time_steps, input_dimension], but in our case it would be (?, 28, 28)
x = tf.placeholder(dtype="float", shape=[None, n_steps, n_input], name="x") # Current data input shape: (batch_size, n_steps, n_input) [100x28x28]
y = tf.placeholder(dtype="float", shape=[None, n_classes], name="y")
# Randoming initializing weights & biases
weights = { 'out': tf.Variable(tf.random_normal([n_hidden, n_classes])) }
biases = {'out': tf.Variable(tf.random_normal([n_classes])) }
{'out': <tf.Variable 'Variable_8:0' shape=(128, 10) dtype=float32_ref>}
Let’s Understand the parameters, inputs and outputs
{\Delta{S}}= S({\mu \Delta{t}}+{\sigma \epsilon \sqrt{\Delta{t}}})
We will treat the MNIST image
\in \mathcal{R}^{28 \times 28}
28
sequences of a vector
\mathbf{x} \in \mathcal{R}^{28}
Our simple RNN consists of
One input layer which converts a $28*28$ dimensional input to an $128$ dimensional hidden layer,
One intermediate recurrent neural network (LSTM)
One output layer which converts an $128$ dimensional output of the LSTM to $10$ dimensional output indicating a class label.
#Lets design our LSTM Model
#Lets define a lstm cell with tensorflow
#__dynamic_rnn__ creates a recurrent neural network specified from __lstm_cell__:
outputs, states = tf.nn.dynamic_rnn(lstm_cell, inputs=x, dtype=tf.float32)
Tensor("rnn_3/transpose:0", shape=(?, 28, 128), dtype=float32)
The output of the rnn would be a [100x28x128] matrix. we use the linear activation to map it to a [?x10 matrix]
output = tf.reshape(tf.split(outputs, 28, axis=1, num=None, name='split')[-1],[-1,128])
pred = tf.matmul(output, weights['out']) + biases['out']
Tensor("Reshape_1:0", shape=(?, 128), dtype=float32)
#Now, we define the cost function and optimizer:
cost = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(labels=y, logits=pred ))
#Here we define the accuracy and evaluation methods to be used in the learning process:
#Running the tensorflow graph
#Keep training until reach max iterations
#We will read a batch of 100 images [100 x 784] as batch_x
#batch_y is a matrix of [100x10]
#We consider each row of the image as one sequence
#Reshape data to get 28 seq of 28 elements, so that, batxh_x is [100x28x28]
#Run optimization op (backprop)
#Calculate batch accuracy
#Calculate batch loss
print("Iter " + str(step*batch_size) + ", Minibatch Loss= " + \
#Calculate accuracy for 128 mnist test images
This is the end of the Recurrent Neural Networks with TensorFlow learning notebook. There are multiple application of RNN in Language Modeling and Generating Text, Machine Translation, Speech Recognition, Generating Image Descriptions, etc. Hopefully you now have a better understanding of Recurrent Neural Networks and how to implement one utilizing TensorFlow. Thank you for reading this notebook and in the next blog I shall share image classification using CNN and we will use tensorboard to visualize how to tune the hyperparameters of the network.
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Not to be confused with Harshad number (derived from Sanskrit harśa meaning "great joy").
In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because
{\displaystyle 1^{2}+3^{2}=10}
{\displaystyle 1^{2}+0^{2}=1}
. On the other hand, 4 is not a happy number because the sequence starting with
{\displaystyle 4^{2}=16}
{\displaystyle 1^{2}+6^{2}=37}
eventually reaches
{\displaystyle 2^{2}+0^{2}=4}
, the number that started the sequence, and so the process continues in an infinite cycle without ever reaching 1. A number which is not happy is called sad or unhappy.
More generally, a
{\displaystyle b}
-happy number is a natural number in a given number base
{\displaystyle b}
that eventually reaches 1 when iterated over the perfect digital invariant function for
{\displaystyle p=2}
The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and senior lecturer in pure mathematics at Leeds University) by his daughter, who had learned of them at school. However, they "may have originated in Russia" (Guy 2004:§E34).
1 Happy numbers and perfect digital invariants
1.1 Natural density of b-happy numbers
1.2 Happy bases
2 Specific b-happy numbers
2.1 4-happy numbers
2.3 10-happy numbers
3 Happy primes
3.1 6-happy primes
3.2 10-happy primes
Happy numbers and perfect digital invariantsEdit
See also: Perfect digital invariant
{\displaystyle n}
be a natural number. Given the perfect digital invariant function
{\displaystyle F_{p,b}(n)=\sum _{i=0}^{\lfloor \log _{b}{n}\rfloor }{\left({\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}\right)}^{p}}
for base
{\displaystyle b>1}
, a number
{\displaystyle n}
{\displaystyle b}
-happy if there exists a
{\displaystyle j}
{\displaystyle F_{2,b}^{j}(n)=1}
{\displaystyle F_{2,b}^{j}}
{\displaystyle j}
-th iteration of
{\displaystyle F_{2,b}}
{\displaystyle b}
-unhappy otherwise. If a number is a nontrivial perfect digital invariant of
{\displaystyle F_{2,b}}
{\displaystyle b}
-unhappy.
For example, 19 is 10-happy, as
{\displaystyle F_{2,10}(19)=1^{2}+9^{2}=82}
{\displaystyle F_{2,10}^{2}(19)=F_{2,10}(82)=8^{2}+2^{2}=68}
{\displaystyle F_{2,10}^{3}(19)=F_{2,10}(68)=6^{2}+8^{2}=100}
{\displaystyle F_{2,10}^{4}(19)=F_{2,10}(100)=1^{2}+0^{2}+0^{2}=1}
For example, 347 is 6-happy, as
{\displaystyle F_{2,6}(347)=F_{2,6}(1335_{6})=1^{2}+3^{2}+3^{2}+5^{2}=44}
{\displaystyle F_{2,6}^{2}(347)=F_{2,6}(44)=F_{2,6}(112_{6})=1^{2}+1^{2}+2^{2}=6}
{\displaystyle F_{2,6}^{3}(347)=F_{2,6}(6)=F_{2,6}(10_{6})=1^{2}+0^{2}=1}
There are infinitely many
{\displaystyle b}
-happy numbers, as 1 is a
{\displaystyle b}
-happy number, and for every
{\displaystyle n}
{\displaystyle b^{n}}
{\displaystyle 10^{n}}
{\displaystyle b}
{\displaystyle b}
-happy, since its sum is 1. The happiness of a number is preserved by removing or inserting zeroes at will, since they do not contribute to the cross sum.
Natural density of b-happy numbersEdit
By inspection of the first million or so 10-happy numbers, it appears that they have a natural density of around 0.15. Perhaps surprisingly, then, the 10-happy numbers do not have an asymptotic density. The upper density of the happy numbers is greater than 0.18577, and the lower density is less than 0.1138.[2]
Happy basesEdit
Are base 2 and base 4 the only bases that are happy?
A happy base is a number base
{\displaystyle b}
where every number is
{\displaystyle b}
-happy. The only happy bases less than 5×108 are base 2 and base 4.[3]
Specific b-happy numbersEdit
4-happy numbersEdit
{\displaystyle b=4}
, the only positive perfect digital invariant for
{\displaystyle F_{2,b}}
is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are preperiodic points for
{\displaystyle F_{2,b}}
, all numbers lead to 1 and are happy. As a result, base 4 is a happy base.
{\displaystyle b=6}
{\displaystyle F_{2,b}}
is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle
5 → 41 → 25 → 45 → 105 → 42 → 32 → 21 → 5 → ...
and because all numbers are preperiodic points for
{\displaystyle F_{2,b}}
, all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 6 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits.
In base 10, the 74 6-happy numbers up to 1296 = 64 are (written in base 10):
1, 6, 36, 44, 49, 79, 100, 160, 170, 216, 224, 229, 254, 264, 275, 285, 289, 294, 335, 347, 355, 357, 388, 405, 415, 417, 439, 460, 469, 474, 533, 538, 580, 593, 600, 608, 628, 638, 647, 695, 707, 715, 717, 767, 777, 787, 835, 837, 847, 880, 890, 928, 940, 953, 960, 968, 1010, 1018, 1020, 1033, 1058, 1125, 1135, 1137, 1168, 1178, 1187, 1195, 1197, 1207, 1238, 1277, 1292, 1295
10-happy numbersEdit
{\displaystyle b=10}
{\displaystyle F_{2,b}}
4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 → ...
{\displaystyle F_{2,b}}
, all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 10 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits.
In base 10, the 143 10-happy numbers up to 1000 are:
The distinct combinations of digits that form 10-happy numbers below 1000 are (the rest are just rearrangements and/or insertions of zero digits):
The first pair of consecutive 10-happy numbers is 31 and 32.[4] The first set of three consecutive is 1880, 1881, and 1882.[5] It has been proven that there exist sequences of consecutive happy numbers of any natural number length.[6] The beginning of the first run of at least n consecutive 10-happy numbers for n = 1, 2, 3, ... is[7]
1, 31, 1880, 7839, 44488, 7899999999999959999999996, 7899999999999959999999996, ...
As Robert Styer puts it in his paper calculating this series: "Amazingly, the same value of N that begins the least sequence of six consecutive happy numbers also begins the least sequence of seven consecutive happy numbers."[8]
The number of 10-happy numbers up to 10n for 1 ≤ n ≤ 20 is[9]
3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238, 12005034444292997294.
Happy primesEdit
{\displaystyle b}
-happy prime is a number that is both
{\displaystyle b}
-happy and prime. Unlike happy numbers, rearranging the digits of a
{\displaystyle b}
-happy prime will not necessarily create another happy prime. For instance, while 19 is a 10-happy prime, 91 = 13 × 7 is not prime (but is still 10-happy).
All prime numbers are 2-happy and 4-happy primes, as base 2 and base 4 are happy bases.
6-happy primesEdit
In base 6, the 6-happy primes below 1296 = 64 are
10-happy primesEdit
In base 10, the 10-happy primes below 500 are
7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 (sequence A035497 in the OEIS).
The palindromic prime 10150006 + 7426247×1075000 + 1 is a 10-happy prime with 150007 digits because the many 0s do not contribute to the sum of squared digits, and 12 + 72 + 42 + 22 + 62 + 22 + 42 + 72 + 12 = 176, which is a 10-happy number. Paul Jobling discovered the prime in 2005.[10]
As of 2010[update], the largest known 10-happy prime is 242643801 − 1 (a Mersenne prime).[dubious – discuss] Its decimal expansion has 12837064 digits.[11]
In base 12, there are no 12-happy primes less than 10000, the first 12-happy primes are (the letters X and E represent the decimal numbers 10 and 11 respectively)
11031, 1233E, 13011, 1332E, 16377, 17367, 17637, 22E8E, 2331E, 233E1, 23955, 25935, 25X8E, 28X5E, 28XE5, 2X8E5, 2E82E, 2E8X5, 31011, 31101, 3123E, 3132E, 31677, 33E21, 35295, 35567, 35765, 35925, 36557, 37167, 37671, 39525, 4878E, 4X7X7, 53567, 55367, 55637, 56357, 57635, 58XX5, 5X82E, 5XX85, 606EE, 63575, 63771, 66E0E, 67317, 67371, 67535, 6E60E, 71367, 71637, 73167, 76137, 7XX47, 82XE5, 82EX5, 8487E, 848E7, 84E87, 8874E, 8X1X7, 8X25E, 8X2E5, 8X5X5, 8XX17, 8XX71, 8E2X5, 8E847, 92355, 93255, 93525, 95235, X1X87, X258E, X285E, X2E85, X85X5, X8X17, XX477, XX585, E228E, E606E, E822E, EX825, ...
The examples below implement the perfect digital invariant function for
{\displaystyle p=2}
and a default base
{\displaystyle b=10}
described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and repeating a number.
A simple test in Python to check if a number is happy:
def pdi_function(number, base: int = 10):
"""Perfect digital invariant function."""
total += pow(number % base, 2)
number = number // base
def is_happy(number: int) -> bool:
"""Determine if the specified number is a happy number."""
while number > 1 and number not in seen_numbers:
seen_numbers.add(number)
number = pdi_function(number)
^ "Sad Number". Wolfram Research, Inc. Retrieved 16 September 2009.
^ Gilmer, Justin (2013). "On the Density of Happy Numbers". Integers. 13 (2): 2. arXiv:1110.3836. Bibcode:2011arXiv1110.3836G.
^ Sloane, N. J. A. (ed.). "Sequence A161872 (Smallest unhappy number in base n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A035502 (Lower of pair of consecutive happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 April 2011.
^ Sloane, N. J. A. (ed.). "Sequence A072494 (First of triples of consecutive happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 April 2011.
^ Pan, Hao (2006). "Consecutive Happy Numbers". arXiv:math/0607213.
^ Sloane, N. J. A. (ed.). "Sequence A055629 (Beginning of first run of at least n consecutive happy numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Styer, Robert (2010). "Smallest Examples of Strings of Consecutive Happy Numbers". Journal of Integer Sequences. 13: 5. 10.6.3 – via University of Waterloo. Cited in Sloane "A055629" harvtxt error: no target: CITEREFSloane_"A055629" (help).
^ Sloane, N. J. A. (ed.). "Sequence A068571 (Number of happy numbers <= 10^n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Chris K. Caldwell. "The Prime Database: 10150006 + 7426247 · 1075000 + 1". utm.edu.
^ Chris K. Caldwell. "The Prime Database: 242643801 − 1". utm.edu.
Guy, Richard (2004). Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. ISBN 0-387-20860-7.
Schneider, Walter: Mathews: Happy Numbers.
Weisstein, Eric W. "Happy Number". MathWorld.
calculate if a number is happy
Happy Numbers at The Math Forum.
145 and the Melancoil at Numberphile.
Symonds, Ria. "7 and Happy Numbers". Numberphile. Brady Haran. Archived from the original on 15 January 2018. Retrieved 2 April 2013.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Happy_number&oldid=1068797542"
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Improvements in Command Completion
A number of changes make authoring mathematical documents in Maple easier.
Maple 2016 includes better user controls of math in documents. You can now toggle a 2-D expression, which by default is executable math, to nonexecutable math.
Click in the expression, and then hover over the expression. A small circular popup appears.
Tip: If the popup does not appear, move your mouse away, and then move back over the expression.
Click the circular popup to toggle between executable and nonexecutable math.
Alternatively, use the shortcut key Shift + F5 or, under the Edit menu, clear the check box beside Executable Math.
When editing a 2-D math expression, executable math is displayed with a blue background, such as:
Nonexecutable math is displayed with a gray background:
Use nonexecutable math for expressions that are only for display purposes, that you do not want to execute. Often these are mathematical formulas which are not valid Maple syntax. If an expression, such as
\sum \mathrm{x__i}
, is executed accidentally, it can lead to error messages or unwanted output. If this happens, toggling the expression to nonexecutable math removes the error message or output and changes the math to nonexecutable.
Maple 2016 enhances 2-D math equation editing by displaying a visible multiplication symbol (
\cdot
) between side-by-side closing and opening parentheses. For example, if you write
\left(f+g\right) \left(x,y\right)
, Maple inserts the multiplication symbol:
This makes it clear that the expression with adjacent parentheses is interpreted as multiplication. This change is designed to help you avoid common errors in writing expressions and troubleshoot your work more quickly.
If you did not mean multiplication (for example you meant to apply the functions
f+g
x,y )
, delete the dot.
\left(f+g\right)\cdot \left(x,y\right)
\left(\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{g}\right)\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
\left(f+g\right)\left(x,y\right)
\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{g}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
You can control whether an expression with adjacent parentheses in 2-D math is interpreted to mean multiplication or function application. For details, see the Smart Operators section of the Options dialog.
For more details about these and other features of 2-D math, see 2-D Math Details.
As you are typing a Maple name or command, Maple checks both your current session and Maple's archive of known names and commands to try to anticipate what you are planning to enter. This process goes by the name command completion. For Maple 2016, several adjustments have been made to this process to make entering names and commands into your Maple session easier:
If the command completion mechanism determines that all known completions of what you have typed thus far share a prefix which is longer than what you have typed, a tooltip showing that prefix, followed by "...", appears. For example, if you type
\mathrm{Lin}
, then a tooltip showing "Linear..." will appear, as all of the stored Maple commands which start "Lin" also share the full prefix "Linear". To accept this prefix completion, just press Enter.
If the command completion mechanism anticipates that you are entering a command which has been marked as deprecated, a tooltip appears showing the completion of what you are entering but it also indicates that this is deprecated and suggests a more modern replacement.
If you use the command completion shortcut key Esc to display a popup menu of all possible completions of what you have entered thus far and Maple determines that you are entering a deprecated command, the menu will include the modern replacement(s) above the direct completion of your command.
The command completion system will not suggest as a possible completion any undocumented commands.
For more information on command completion, see Complete Command.
The redesigned interface icons offer improved scaling on high resolution monitors for Maple 2016. Icons across the Maple user interface have been updated, including all toolbars.
New Look for the Worksheet Toolbar
New Look for the Plotting Toolbar
For more information on the icons, see Worksheet Toolbar.
The icons for zooming on the main toolbar have been changed to Zoom In, Zoom Out, and Default Zoom ( ), respectively. Previously, there were three icons that set the zoom level to 100%, 150%, or 200%.
The current zoom level is now shown in the status bar at the bottom of the Maple window.
The hotkey Ctrl + 0 has been changed from setting the zoom level to 50% to setting the current zoom level to the default zoom level. Other previous behavior, such as the ability to use Ctrl + 1, Ctrl + 2, ... to set a defined zoom level, is unchanged. Similarly, the ability to zoom in or out using the combination Ctrl + Mouse Scroll wheel is also unchanged.
Another change for zoom is that the zoom level is no longer stored in the Maple worksheet. Any worksheet that is opened in a Maple session is now opened to the zoom level as set in the Tools menu > Options... > Interface tab > Default Zoom (Windows and Linux) or Maple 2016 menu > Preferences > Options Dialog > Default Zoom level (Mac).
For more information on zoom levels, see Maple Preferences or for more comprehensive shortcut key lists by platform, see lists for Windows, Mac, and Linux.
Show or Hide Table Borders
The display of interior and exterior table borders can be optionally turned off in Maple worksheets using the Table Properties dialog. When borders are turned off, Maple provides a worksheet-level setting to control the display of "invisible" borders when hovering over a table with the mouse (View > Show/Hide Contents > Hidden Table Borders ). However, there can be cases where it is desirable to set the table border visibility on a per table basis. As such, table borders can be changed using the following options found in the Table Properties dialog:
Always – hidden table borders are always displayed on mouse hover.
Never – hidden table borders are never displayed.
According to Show/Hide Setting – the table obeys the global worksheet setting. (This is the default.)
Full Screen Mode on Mac
For Mac users, Maple can now be used in full screen mode. To switch to full screen mode, select View > Full Screen Mode or use the shortcut key for toggling full screen mode, Control + Command + F.
Sections are one of the main tools to control the flow of content in a Maple document. Maple 2016 offers a new optional way to display a section. By default, sections are delimited with an arrow beside the section heading and a line along the left border. The section is collapsible by clicking the arrow or line.
Default look of sections
If preferred, you can change the look so sections are not collapsible and no visible delimiter is shown. From the context-sensitive menu, select Section > Hide arrow and stay expanded.
New look of sections
Several new shortcut keys have been added to Maple 2016:
To move the preceding term to the denominator in a rational expression, type // and then enter the numerator. Examples:
Entering a//b produces
\frac{b}{a}
Entering (1+x)//1 produces
\frac{1}{\left(1+x\right)}
For more information on shortcut keys for entering rationals and other 2-D math, see 2-D Math Shortcut Keys.
To toggle between executable and nonexecutable math, use Shift + F5.
To toggle to full screen mode on Mac, use the key combination Control + Command + F.
For more comprehensive shortcut key lists by platform, see lists for Windows, Mac, and Linux.
Inserting Rows or Columns into a Matrix
It is now possible to insert rows and columns into Matrices using the context-sensitive menu for 2-D math input. Previously, a new row or column could only be added to an existing matrix using the Ctrl + Shift + R (row) or Ctrl + Shift + C (column) shortcut keys. Right-click on a matrix and look for the context menu options Insert Column and Insert Row.
DataFrame Context Menu
The Maple programming language provides many commands that are useful for exploring DataFrames. The right-click context menu provides easy access to a selection of these commands, displaying context specific commands that can be applied to DataFrames or DataSeries.
The DataFrame context menu contains many commands that can be applied to entire DataFrames as well as to a single DataSeries in a DataFrame. The second section of the DataFrame context menu includes commands for conversions, operations, queries, and visualization of DataFrames and DataSeries. The third section includes more commands relating to statistics and data analysis, including data analysis, data manipulation, properties and quantities, and summary and tabulation.
A useful feature of the context menu is the ability to quickly filter the DataFrame by value or to select columns to apply operations to. This can be beneficial when dealing with heterogeneous data that includes non-numeric DataSeries. In many cases, commands in Statistics assume an entirely numeric DataFrame. Selectively removing the non-numeric data makes it possible to use the routines natively on the given DataFrame.
More Operations in the Context Menu
Many more commands are now available in the context menu, including operations for number theory, multivariate calculus, and more.
A new Group Constructors Palette contains buttons for constructing groups.
By default, the Group Constructors palette will not be visible on start-up in the left pane of the Maple window. To enable the palette, click the View menu, and select Palettes>Show Palette>Group Constructors.
Four new entries in the File > New submenu pertain to content inside of a Workbook:
Document in Workbook: Create a new document inside of the current workbook file.
Worksheet in Workbook: Create a new worksheet inside of the current workbook file.
Folder in Workbook: Create a new folder in the Workbook Navigator tree.
Maple Code Attachment: Create a new code attachment inside of the current workbook file.
Under the Edit menu, the new entry Executable Math enables toggling between executable and nonexecutable math.
For ease of use, some menu items have been rearranged. All menu items pertaining to Document Blocks are now found under Edit > Document Blocks. One menu item has a new name: Expand Document Block is now Show Command.
This new submenu also includes the following menu items that were previously found under other menus:
View: Group and Block Management
View: Toggle Input/Output Display
View: Inline Document Output
Format: Create Document Block
Format: Remove Document Block
A new menu item for Mac users puts Maple in Full Screen Mode.
The View > Workbook Navigator submenu collects all commands relevant to the display of content in Workbook Navigator palette.
|
The impact of Cesarean section on female fertility: a narrative review
Lorenz Hinterleitner, Herbert Kiss, Johannes Ott
Problems in gynaecologic oncology in girls and young women—an outline of selected issues
Katarzyna Plagens-Rotman, Maria Połocka-Molińska, Piotr Merks, Matylda Gwoździcka-Piotrowska, ... Grażyna Jarząbek-Bielecka
Treatment of vulvar pain caused by atrophy: a systematic review of clinical studies
Sonia Sánchez, Laura Baquedano, Nicolás Mendoza
Efficacy of tamoxifen for infertile women with thin endometrium undergoing frozen embryo transfer: a meta-analysis
Zhongying Huang, Zhun Xiao, Qianhong Ma, Yu Bai, Feilang Li
Real-world implementation and adaptation to local settings of first trimester preeclampsia screening in Italy: a systematic review
Silvia Amodeo, Giulia Bonavina, Anna Seidenari, Paolo Ivo Cavoretto, Antonio Farina
Antimicrobial resistance and epidemiology of extended spectrum-
\beta
-lactamases (ESBL)-producing Escherichia coli and Enterobacter cloacae isolates from intensive care units at obstetrics & gynaecology departments: a retrospective analysis
Kun Chen, Guo-Liang Yang, Wen-Ping Li, Ming-Cheng Li, Xue-Ying Bao
Can venous cord blood neutrophil to lymphocyte ratio and platelet to lymphocyte ratio predict early-onset sepsis in preterm infants?
Shu-Jun Chen, Xie-Xia Zheng, Hong-Xing Jin, Jian-Hua Chen, ... Cui-E Chen
Age and anti-Műllerian hormone: prediction of cumulative pregnancy outcome in in vitro fertilization with diminished ovarian reserve
Yu Deng, Zhan-Hui Ou, Min-Na Yin, Pei-Ling Liang, ... Ling Sun
Mechanical bowel preparation prior to gynaecological laparoscopy enables better operative field visualization, lower pneumoperitoneum pressure and Trendelenburg angle during the surgery: a perspective that may add to patient safety
Üzeyir Kalkan, Murat Yassa, Kadir Bakay, Şafak Hatırnaz
Long-term impact of chronic pelvic pain on quality of life in women with and without endometriosis
Sayuli Bhide, Rebecca Flyckt, Meng Yao, Tommaso Falcone
Increased nuchal translucency and fetal outcomes: a population-based study in Thailand
Kuntharee Traisrisilp, Supatra Sirichotiyakul, Fuanglada Tongprasert, Kasemsri Srisupundit, ... Theera Tongsong
Reference values of fetal atrioventricular time intervals derive from antegrade late diastolic arterial blood flow (ALDAF) from 14 to 40 weeks of gestation
Thanakorn Heetchuay, Thotsapon Trakulmungkichkarn, Noel Pabalan, Nutthaphon Imsom-Somboon
Ex vivo myolysis with dual wavelengths diode laser system: macroscopic and histopathological examination
Maurizio Nicola D’Alterio, Francesco Scicchitano, Daniela Fanni, Gavino Faa, ... Stefano Angioni
Neža Sofija Pristov, Ela Rednak, Ksenija Geršak, Andreja Trojner Bregar, Miha Lučovnik
The effect of ultraviolet index measurements on levels of vitamin D and inflammatory markers in pregnant women
Obstetric outcomes in women of advanced maternal age after assisted reproduction
Anna M. Rubinshtein, Oleg V. Golyanovskiy
The role of platelet-to-lymphocyte ratio and neutrophil-to-lymphocyte ratio as a supplemental tool for differential diagnosis of uterine myoma and sarcoma
Yoon Young Jeong, Eun Ji Lee, Eun Byeol Cho, Jung Min Ryu, Youn Seok Choi
Diminished ovarian reserve and ectopic ovaries in patients with Mayer-Rokitansky-Küster-Hauser syndrome candidates for Uterus Transplantation: our experience
Basilio Pecorino, Giuseppe Scibilia, Placido Borzì, Maria Elena Vento, ... Paolo Scollo
Epidural analgesia at trial of labour after caesarean section. A retrospective cohort study over 12 years
Valeria Filippi, Luigi Raio, Sophia Amylidi-Mohr, Rudolf Tschudi, Daniele Bolla
Evaluation of satisfaction level of women with labiaplasty
Comparison of an estradiol patch and GnRH-antagonist protocol with a letrozole/antagonist protocol for patients without oocyte development, fertilization and/or embryo development in previous IVF cycles
Aybike Pekin, Ayşe Gül Kebapçılar, Ersin Çintesun, Setenay Arzu Yılmaz, Özlem Seçilmiş Kerimoğlu
Microbiological pattern of laboratory confirmed vaginal infections among Saudi women
Dalia Saad ElFeky, Rasha Assiri, Hanadi Bakhsh, Ruba Almubaraz, ... Raghad Alomairy
Anatomical and clinical outcomes of vaginally assisted laparoscopic lateral suspension in comparison with laparoscopic lateral suspension
Eren Akbaba, Burak Sezgin, Ahmet Akın Sivaslıoğlu
A retrospective series of homologous intracytoplasmic sperm injection cycle results of 99 women with mosaic Turner syndrome
Nur Dokuzeylul Gungor, Kagan Güngör, Mustecep Kavrut, Arzu Yurci
Comparison of application of Fenton, Intergrowth-21st and WHO growth charts in a population of Polish newborns
Dominik Jakubowski, Daria Salloum, Marek Maciejewski, Magdalena Bednarek-Jędrzejek, ... Sebastian Kwiatkowski
(This article belongs to the Special Issue Modern trends in reproductive surgery)
Correlative factors associated with the recurrence of ovarian endometriosis: a retrospective study
Xi-Wa Zhao, Meng-Meng Zhang, Jian Zhao, Wei Zhao, Shan Kang
The possible association of uterine fibroid formation with copper intrauterine device use: a cross-sectional study
Sevcan Arzu Arinkan, Hilal Serifoglu
Diagnosis and management of intramural ectopic pregnancy
Qian Hu, Mohammed Sharooq Paramboor, Tao Guo
Laparotomic manual replacement for uterine inversion following vaginal birth: a case report
Xiao-Ying Chen, Chang Yu, Jian An, Mian Pan
Posterior reversible encephalopathy syndrome with reversible cerebral vasoconstriction syndrome in a normal primigravida woman at the 35-week gestational stage: a case report
Shingo Tanaka, Maki Goto, Saya Watanabe, Sachino Kira, ... Hiroshi Tsujioka
Postpartum hemorrhage and prolonged hyperfibrinolysis as complications of uterine cavernous hemangioma: a case report and literature review
Xue-Li Bai, Xia Cao
The brainstem-tentorium angle revisited. Difficulties encountered and possible solutions
Laura Joigneau, Yolanda Ruiz, Coral Bravo, Julia Bujan, ... Juan De León-Luis
|
Marginal, Average and Total Revenue: Meaning & Calculation
How do you know how well a company is operating? What does it mean for a company to have had a billion pounds in total revenue in a single year? What does that mean for the company’s average revenue and marginal revenue? What do these concepts mean in economics, and how do firms use them in their day-to-day business operations?
This explanation will teach you what you need to know about total revenue, average revenue, and marginal revenue.
Marginal, average, and total revenue: a definition
To understand the meaning of marginal and average revenue, you have to start by understanding the meaning of total revenue.
Total revenue is all the money a firm makes during a period by selling the goods and services it produces.
The total revenue doesn’t take into account the cost that the firm incurs during a production process. Instead, it only takes into account the money coming from selling what the firm produces. As the name suggests, total revenue is all the money coming into the firm from selling its products. Any additional unit of output sold would increase the total revenue.
Average revenue shows how much revenue there is per unit of output. In other words, it calculates how much revenue a firm receives, on average, from each unit of product they sell. To calculate the average revenue, you have to take the total revenue and divide it by the number of output units.
Average revenue shows how much revenue there is per unit of output.
Marginal revenue refers to the increase in total revenue from increasing one output unit. To calculate the marginal revenue, you have to take the difference in total revenue and divide it by the difference in total output.
Marginal revenue is the increase in total revenue from increasing one output unit.
Let’s say that the firm has a total revenue of £100 after producing 10 units of output. The firm hires an additional worker, and the total revenue increases to £110, while the output increases to 12 units.
What’s the marginal revenue in this case?
Marginal revenue = (£110-£100)/(12-10) = £5.
That means that the new worker generated £5 of revenue for an additional unit of output produced.
Why is the average revenue the firm’s demand curve?
The average revenue curve is also the firm’s demand curve. Let's see why.
Figure 1. Average revenue and the demand curve- StudySmarter.
Figure 1 above illustrates how the demand curve for the firm’s output equals the average revenue a firm experiences. Imagine there’s a firm that sells chocolate. What do you think would happen when the firm charges £6 per chocolate?
By charging £6 per unit of chocolate the firm can sell 30 units of chocolate. That suggests that the firm makes £6 per chocolate sold. The firm then decides to decrease the price to £2 per chocolate, and the number of chocolates it sells at this price increases to 50.
Note that the amount of sales at each price is equal to the firm’s average revenue. As the demand curve also shows the average revenue the firm makes at each price level, the demand curve equals the firm’s average revenue.
You can also calculate the firm’s total revenue by simply multiplying the quantity by the price. When the price equals £6, the quantity demanded is 20 units. Therefore, the firm's total revenue is equal to £120.
Marginal, average, and total revenue formula and examples
Let’s look at some formulas and examples of the marginal, average, and total revenue.
The total revenue formula helps firms calculate the amount of the total money that entered the company during a given sales period. The total revenue formula equals the amount of output sold multiplied by the price.
Totalrevenue=Price×TotalOutputSold
A firm sells 200,000 candies in a year. The price per candy is £1.5. What’s the firm’s total revenue?
Total revenue = the amount of candies sold x the price per candy
Thus, total revenue = 200,000 x 1.5 = £300,000.
We calculate the average revenue, which is the firm’s revenue per unit of output sold by dividing the total revenue by the total amount of output.
Averagerevenue=\frac{Totalrevenue}{Totaloutput}
Assume that a firm that sells microwaves makes £600,000 in total revenues in a year. The number of microwaves sold that year is 1,200. What’s the average revenue?
Average revenue = total revenue/number of microwaves sold = 600,000/1,200 = £500. The firm makes £500 on average from selling one microwave.
Marginal revenue, which is the firm’s additional increase in total revenue from selling an extra unit of output, is equal to the difference of total revenues divided by the difference in total output sold.
Marginalrevenue=\frac{∆Totalrevenue}{∆Output}
Assume that a firm has generated £5,000 in total revenues in year 1. The total amount of output sold was 400 units. The next year the firm has increased its production to 500 units, and the total revenue was £5,500. What’s the marginal revenue?
Marginal revenue = (£5,500-£5,000)/(500-400) = 500/100 = £5.
An additional unit of output produced resulted in an increase in revenue of £5.
The relationship between marginal and total revenue
Total revenue refers to the total sales a firm experiences from selling its output. In contrast, the marginal revenue calculates how much the total revenue increases by when an additional unit of goods or services is sold.
Total revenue is extremely important for firms: they always try to increase it as it would result in an increase in profits. But an increase in total revenue doesn’t always lead to profit maximisation.
Sometimes, an increase in total revenue can be harmful to a firm. The increase in revenue could decrease productivity or increase the cost associated with producing the output to generate sales. That’s when the situation becomes complex for firms.
The relationship between total revenue and marginal revenue is important because it helps firms make better decisions when profit maximising.
Remember that marginal revenue calculates the increase in total revenue when additional output is sold. Although, initially, marginal revenue from selling an additional unit of a product continues increasing, there comes the point where the marginal revenue starts to fall due to the law of diminishing marginal returns.
This point where the diminishing marginal returns kick in is shown at point B in Figure 2 below. This is the point at which total revenue is maximised and marginal revenue is equal to zero.
After that point, although the total revenue of a firm is increasing, it increases by less and less. This is because an additional output sold is not adding as much to the total revenue after that point.
Figure 2. Relationship between marginal and total revenue - StudySmarter.
All in all, as the marginal revenue measures the increase in total revenue from selling an additional unit of output, it helps firms decide whether it’s wise to increase their total sales by producing more.
The relationship between marginal and average revenue
The relationship between marginal revenue and average revenue can be contrasted between the two opposite market structures: perfect competition and monopoly.
In perfect competition, there is a huge number of firms supplying goods and services that are homogenous. As a result, firms can’t influence the market price as even the slightest increase would lead to no demand for their product. This means that there is perfectly elastic demand for their product. Due to the perfectly elastic demand, the rate at which total revenue increases is constant.
Since the price remains constant, an additional product sold will always increase the total sales by the same amount. Marginal revenue shows how much total revenue increases as a result of an additional unit sold. As total revenues increase at a constant rate, the marginal revenue will be constant. Additionally, average revenue shows the revenue per product sold, which is also constant. This leads to marginal revenue being equal to the average revenue in a perfectly competitive market structure.
In contrast, in an imperfectly competitive market structure, such as a monopoly, you can observe a different relationship between average revenue and marginal revenue. In such a market, a firm faces a downward-sloping demand curve equal to average revenue in Figure 2. The marginal revenue will always be equal to or smaller than the average revenue in an imperfectly competitive market. That’s due to the change in output sold when prices change.
Marginal, Average and Total Revenue - Key takeaways
As the name suggests, total revenue is all the money coming into a firm from selling its products.
Average revenue shows how much revenue a single unit of output brings on average.
Marginal revenue refers to the increase in total revenue from increasing output sold by one unit.
As the demand curve also shows the average revenue the firm makes at each price level, the demand curve equals the firm’s average revenue.
The total revenue formula equals the amount of output sold multiplied by the price.
Average revenue is calculated by dividing the total revenue by the total amount of output.
Marginal revenue is equal to the difference of total revenues divided by the difference in total quantity.
Marginal revenue is equal to the average revenue in a perfectly competitive market structure.
The marginal revenue will always be equal to or smaller than the average revenue in an imperfectly competitive market.
What is the meaning of marginal, average, and total revenue?
As the name suggests, total revenue is all the money coming into the firm from selling their products.
Average revenue shows how much revenue a single unit of output brings.
Marginal revenue refers to the increase in total revenue from increasing one unit of output.
How do you calculate MR and TR?
What is the relationship between marginal and total revenue?
As the marginal revenue measures the increase in total sales revenue from selling an additional unit of output, it helps a firm decide whether it’s wise to increase their total sales by producing more.
Final Marginal, Average and Total Revenue Quiz
Total revenue is all the money a firm makes during a period by selling the goods and services they produce.
Revenue is all the money a firm makes during a period by selling the goods and services they produce.
What is the formula of total revenue?
What is the formula of marginal revenue?
What is the formula of average revenue?
What is the relationship between MR and AR in a perfectly competitive market?
What is the relationship between MR and AR in an imperfect market?
The marginal revenue will always be equal to or smaller than the average in an imperfect market.
Why does marginal revenue begin to fall at some point?
Due to the law of diminishing returns.
Why is the average revenue also the firm's demand curve?
As the demand curve also shows the average revenue the firm makes at each price level, the demand curve equals the firm's average revenue.
Why is the relationship between marginal revenue and total revenue important?
If a firm sells 50 units of output at 10 pounds what is the total revenue of the firm?
If the firm has 40000 pounds in total revenue and sells 5000 of output, what's the average revenue?
8 pounds per output sold.
of the users don't pass the Marginal, Average and Total Revenue quiz! Will you pass the quiz?
Long Run Entry and Exit Decisions Learn
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(-)-camphene synthase Wikipedia
(-)-camphene synthase
(-)-camphene synthase (EC 4.2.3.117, CS) is an enzyme with systematic name geranyl-diphosphate diphosphate-lyase (cyclizing, (-)-camphene-forming).[1][2][3][4] This enzyme catalyses the following chemical reaction
{\displaystyle \rightleftharpoons }
(-)-camphene + diphosphate
(-)-Camphene is the major product in Abies grandis (grand fir) with traces of other monoterpenoids.
^ Bohlmann J, Phillips M, Ramachandiran V, Katoh S, Croteau R (August 1999). "cDNA cloning, characterization, and functional expression of four new monoterpene synthase members of the Tpsd gene family from grand fir (Abies grandis)". Archives of Biochemistry and Biophysics. 368 (2): 232–43. doi:10.1006/abbi.1999.1332. PMID 10441373.
^ Huber DP, Philippe RN, Godard KA, Sturrock RN, Bohlmann J (June 2005). "Characterization of four terpene synthase cDNAs from methyl jasmonate-induced Douglas-fir, Pseudotsuga menziesii". Phytochemistry. 66 (12): 1427–39. doi:10.1016/j.phytochem.2005.04.030. PMID 15921711.
^ Falara V, Akhtar TA, Nguyen TT, Spyropoulou EA, Bleeker PM, Schauvinhold I, Matsuba Y, Bonini ME, Schilmiller AL, Last RL, Schuurink RC, Pichersky E (October 2011). "The tomato terpene synthase gene family". Plant Physiology. 157 (2): 770–89. doi:10.1104/pp.111.179648. PMC 3192577. PMID 21813655.
(-)-camphene+synthase at the US National Library of Medicine Medical Subject Headings (MeSH)
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