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• Chinese Optics Letters
• Vol. 21, Issue 2, 023902 (2023)
Zhiyu Chen1, Xin Zhong1, Lin Jiang2, Jiaxin Xu1, Jingxian Liu1, Yan Pan3, and Tao Zhou1、*
Author Affiliations
• 1, Chengdu 610029, China
• 2Center for Information Photonics & Communications, School of Information Science & Technology, , Chengdu 610031, China
• 3Science and Technology on Communication Security Laboratory, , Chengdu 610041, China
• show less
Zhiyu Chen, Xin Zhong, Lin Jiang, Jiaxin Xu, Jingxian Liu, Yan Pan, Tao Zhou. Linearized microwave downconversion link based on fast and intelligent impairment equalization for noncooperative systems[J]. Chinese Optics Letters, 2023, 21(2): 023902 Copy Citation Text show less

Abstract

We experimentally demonstrated the use of intelligent impairment equalization (IIE) for microwave downconversion link linearization in noncooperative systems. Such an equalizer is realized based on an artificial neural network (ANN). Once the training process is completed, the inverse link transfer function can be determined. With the inverse transformation for the detected signal after transmission, the third-order intermodulation distortion components are suppressed significantly without requiring any prior information from an input RF signal. Furthermore, fast training speed is achieved, since the configuration of ANN-based equalizer is simple. Experimental results show that the spurious-free dynamic range of the proposed link is improved to 106.5 dB · Hz2/3, which is 11.3 dB higher than that of a link without IIE. Meanwhile, the training epochs reduce to only five, which has the potential to meet the practical engineering requirement.

1. Introduction

Artificial intelligence, on the other hand, has been a topic of extensive research in signal processing[12]. Based on artificial neural networks (ANNs) or deep learning, nonlinearity monitoring[13], integrated devices design[14], modulation formats identification[15,16], and link linearization[17] in photonic systems have been demonstrated with the advantages of high speed, simple hardware, and high flexibility. Consequently, an intelligent receiver that could linearize the microwave downconversion link would be one major characteristic in next-generation microwave photonic systems. However, the training speed and the information processing speed of ANNs become a practical problem for the application of microwave downconversion in noncooperative systems.

In this study, a simple linearized photonic downconversion link for microwave signals is proposed and experimentally demonstrated by using fast and intelligent impairment equalization (IIE). Such an equalizer only consists of a three-layer ANN. With supervised learning, the inverse link transfer function can be trained successfully. Transforming the detected signals by using such an inverse function, the downconversion link linearity can be enhanced. Different from the previous approach in Ref. [17], there is no need to do the complex Fourier transforms in the proposed method, since the inputs of the equalizer are an intermediate frequency (IF) signal in the time domain, instead of a radio-frequency (RF) signal in the frequency domain. Furthermore, fast training and information processing are achieved, since the configuration of the intelligent equalizer is simple. In addition, the proposed technique does not require any prior information from the input RF signal when the training process is completed, and thus, could be applied to noncooperative systems. Experimental results show that $∼15 dB$ IMD3 suppression is achieved with the RF input power of 7 dBm per tone. Furthermore, the proposed link has an SFDR of $106.5 dB·Hz2/3$, which is 11.3 dB more than that of a link without impairment equalization. Meanwhile, the training epochs reduce to only five.

2. Principle

The principle of the proposed linearized downconversion link is shown in Fig. 1. Assuming the modulation coefficients of input RF signal and local oscillator (LO) signal are $m1$ and $m2$, respectively, the downconverted IF signal $i|Ω$ after dual-electrode MZM (DE-MZM) and direct detection can be expressed as[10]$i|Ω=4ℜPcJ1(m1)J1(m2)cos ϕ cos Ωt,$where $ℜ$, $Pc$, and $Ω$ are the responsivity of the photodetector (PD), the input optical power, and the frequency of the downconverted signal, respectively. $ϕ$ denotes the phase difference between two arms of DE-MZM induced by DC bias. As can be seen from Eq. (1), the bandwidth of the proposed link only depends on that of the PD.

Figure 1.Simplified diagram of linearized downconversion link and corresponding signal spectra and transfer characteristics. DE-MZM/DD, dual-electrode Mach–Zehnder modulated direct detection; IMD3, third-order intermodulation distortion; IIE, intelligent impairment equalization; RF, radio frequency; IF, intermediate frequency.

By Taylor-expanding the Bessel function containing the term of $m1$ in Eq. (1), the component of the downconverted signal becomes $i|Ω=(2m1−14m13)ℜPcJ1(m2)cos ϕ cos Ωt.$

According to Eq. (2), the transfer function of such a link is illustrated as Fig. 1, where the nonlinearity (i.e., harmonic) induces the IMD3 components.

In order to achieve the high linearity of the link, the inverse transfer function shown as in Fig. 1 (i.e., the red line) should be estimated accurately, which is the outstanding ability of an ANN-based equalizer. Different from the previous digital postprocessing approaches, IMD3 compensation based on such an equalizer does not require any information of the transmission link. Moreover, this method does not require any prior information from the input RF signal either when the training process is completed.

The equalization-based inverse link transfer function estimation, which can be considered as a black box, is illustrated as Fig. 2. The architecture is a three-layer multilayer perceptron that consists of an input layer, a hidden layer, and an output layer. In the proposed scheme, only one neuron in the input and output layers is required, while the number of neurons in the hidden layer is 20. The activation functions $f(·)$ and $g(·)$ are a tangent sigmoid function for hidden layer neurons and a linear function for the output layer neurons, respectively. In this case, the outputs of the hidden layer (i.e., $ui$) and the output layer (i.e., $y$) can be expressed as $ui=f(Vix+ci),$$y=g(∑i=1NWiui+b),$where $Vi$ and $Wi$ are weights of the hidden and output layers, $ci$ and $b$ are bias values of the hidden and output layers, and $x$ is the input vectors. Finally, a supervised learning method called backpropagation (BP) is utilized for training of such an intelligent equalizer (i.e., updating weights $Vi$ and $Wi$) by comparing the output vectors $y$ and targets vectors $T$, which are the initial downconverted signals without transmitting DE-MZM/DD link. The training process is completed when the mean square error (MSE) $‖y−T‖2$ achieves the predetermined value. Once the weights are determined, the inverse link transfer function could be obtained according to Eq. (4).

Figure 2.Equalization-based inverse link transfer function estimation. BP, backpropagation.

3. Experimental Setup

The experimental setup is illustrated in Fig. 3. A light source at 1550 nm with the power of 8 dBm is modulated by a DE-MZM that is driven by a two-tone RF signal and an LO signal. On the one hand, those RF signals are downconverted directly by a wideband mixer. Subsequently, such signals as target vectors are sent to an IIE through a low-pass filter and a delay line. On the other hand, the RF signals and LO signal are converted to an optical signal, and then are amplified by an erbium-doped fiber amplifier (EDFA). Afterwards, such an optical signal is detected by a PD with a responsivity of 6 V/W (including an amplifier) and a 15 GHz bandwidth. In the experiment, the frequencies of the two-tone RF and LO signals are set to be 2.11, 2.13, and 2.5 GHz, respectively. In this case, the downconverted signals with the frequencies of 0.37 and 0.39 GHz are generated after the PD. Finally, such signals are digitalized by a real-time oscilloscope running at 40 GS/s, and then processed in an IIE with the target vectors, where the highly linear downconverted link can be achieved. In addition, the DE-MZM needs to be biased at the quadrature point by utilizing a bias feedback controller, which could stabilize the link transfer function.

Figure 3.Experimental setup of proposed linearized microwave downconversion link. LD, laser diode; PC, polarization controller; DE-MZM, dual-electrode Mach–Zehnder modulator; LO, local oscillator; EDFA, erbium-doped fiber amplifier; PD, photodetector; IIE, intelligent impairment equalization.

4. Results and Discussion

In the experiment, the overall data comprising 800,002 photocurrent vectors from the oscilloscope are divided into two distinct subsets, including training and testing data sets. Among them, the length of the training subset is set to be 10,000, as well as that of the target vectors. Once the training phase is completed, the whole link transfer function is determined. In this case, a linearized microwave downconversion link is realized without requiring any prior information.

Figure 4(a) shows the MSE during the training phase as a function of the number of epochs, which are the steps in the ANN training process. Obviously, the MSE decreases with an increase in the number of epochs. As can be seen from the inset of Fig. 4(a), the IMD3 components are observed when the MSE is $10−6$, since the inverse transfer function fitted by the ANN deviates from the theoretical calculation. Therefore, as the predetermined MSE value decreases to $10−7$, the best training performance could achieve $3.2×10−8$ at epoch 5. In this case, the measured training time for 10,000 points is approximately 2.3 s, while the testing time for 200,000 points is only 0.12 s, which indicates that the fast training speed and information processing speed are achieved. Compared to the approach in Ref. [17], the faster convergence speed is obtained by using the proposed IIE.

Figure 4.(a) Dependence of MSE on the number of epochs for the training subsets; inset, the electrical spectrum of two-tone signal with the MSE of 10−6; (b) DE-MZM/DD-based transmission link and calculated inverse link transfer functions; black solid curve, theoretical inverse transfer function; red dotted curve, IIE-based inverse link transfer function; inset, whole downconversion link transfer characteristic by applying IIE.

In order to estimate the effectiveness of IIE, the DE-MZM/DD transmission link and inverse link transfer characteristics are shown in Fig. 4(b), where the theoretical inverse transfer function calculated by Eq. (2) is also presented. It is evident that the IIE-based fitting curve (i.e., the red dotted line) that agrees well with the theoretical calculation (i.e., the black solid line), presents the inverse characteristic of the DE-MZM/DD link. By applying this inverse function, the whole link transfer characteristic after compensation is illustrated in the inset of Fig. 4(b), which indicates that the highly linear microwave downconversion link is realized.

Figures 5(a) and 5(b) show the electrical spectra before and after applying IIE when the input two-tone frequencies are 2.11 and 2.13 GHz. For the first case, strong IMD3 components are observed when the RF input power is 7 dBm per tone. These two components are hardly mitigated by using electrical filters due to the small frequency space between them and the fundamental products. However, the IMD3 suppression of 15 dB is achieved, while the powers of fundamental products keep constant by applying the IIE. In addition, the electrical spectra are measured as shown in Figs. 5(c)5(f) when the frequencies of the input two-tone RF signals are changed to be 2.1, 2.14 GHz and 4.11, 4.13 GHz, respectively. In these cases, the ANN is still trained by using two-tone signals with the frequency of 2.11 and 2.13 GHz. As can be seen, the IMD3 components are also suppressed significantly after compensation with the proposed IIE, which indicates that such an ANN can also work for the two-tone RF signals with other frequencies.

Figure 5.Electrical spectra of the two-tone input tests before and after applying IIE when the frequencies are set to be (a), (b) 2.11 GHz and 2.13 GHz; (c), (d) 2.10 GHz and 2.14 GHz; and (e), (f) 4.11 GHz and 4.13 GHz, respectively.

In order to further verify the performance of the SFDR, the link transfer functions are trained by using the proposed IIE when the input RF power changes from 8 dBm to 12 dBm. In this case, the SFDRs before and after IIE are shown in Fig. 6. As can be seen, the SFDR before linearization is $95.2 dB·Hz2/3$, with a measured noise floor of $−140 dBm/Hz$. After equalization, the SFDR increases to $106.5 dB·Hz2/3$.

Figure 6.Measured SFDR before (solid line) and after (dashed line) IIE.

5. Conclusion

In conclusion, a linearized downconverted microwave photonic link based on fast and IIE is theoretically analyzed and experimentally demonstrated. With the supervised learning, the whole transfer function of the link can be linearized. Experimental results show that an IMD3 suppression of 15 dB is achieved, and the training epochs reduce to only five. Furthermore, the SFDR is improved from $95.2 dB·Hz2/3$ to $106.5 dB·Hz2/3$. The proposed linearization method is highly promising in noncooperative RoF systems.

References

[1] P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, A. Bogoni. A fully photonics-based coherent radar system. Nature, 507, 341(2014).

[2] Z. Y. Chen, T. Zhou, X. Zhong, J. X. Liu, M. W. Wang, J. Ye. Stable downlinks for wideband radio frequencies in distributed noncooperative system. J. Light. Technol., 36, 4514(2018).

[3] J. J. Zhao, T. Liu, B. Wang, W. K. Zhang. Distributed phase synchronization transmission based on fiber. Electr. Inf. Warfare Technol., 34, 64(2019).

[4] H. Mao. Information processing method of electronic warfare events based on communication technology. Secur. Commun. Netw., 2022, 9309710(2022).

[5] Z. Y. Chen, L. S. Yan, W. Pan, B. Luo, X. Zou, Y. H. Guo, H. Y. Jiang, T. Zhou. SFDR enhancement in analog photonic links by simultaneous compensation for dispersion and nonlinearity. Opt. Express, 21, 20999(2013).

[6] Y. Y. Li, A. J. Wen, D. J. Shan. An analog photonic down-conversion link with simultaneous IMD3 distortion suppression and dispersion-induced power fading compensation. J. Light. Technol., 38, 3908(2020).

[7] G. F. Li, T. Shang, T. Gao, Y. L. Zhang, D. Chen. Photonic downconversion link with linearization and full spectrum utilization. Proceedings of Topical Meeting on Microwave Photonics (MWP), THP.2(2017).

[8] X. Zhu, T. Jin, H. Chi, G. C. Tong, D. Li, L. L. Zou, W. M. Liu, Y. Fu. Photonic down-conversion and linearization with tunable IF bandwidth and improved spurious-free dynamic range. Electron. Lett., 55, 105(2019).

[9] Z. Niu, H. Yu, M. Chen, P. Li, H. Chen, S. Xie. High linearity downconverting analog photonic link based on digital signal post-compensation. Optical Fiber Communication Conference, W2A.37(2014).

[10] Y. Pan, L. S. Yan, Z. Y. Chen, W. Pan, B. Luo, X. H. Zou, J. Ye, A. L. Yi. Adaptive linearized microwave downconversion utilizing a single dual-electrode Mach-Zehnder modulator. Opt. Lett., 40, 2649(2015).

[11] P. X. Li, W. Pan, X. H. Zou, W. L. Bai, Y. Pan, X. Y. Han, L. S. Yan. Fast self-adaptive generic digital linearization for analog microwave photonic systems. J. Light. Technol., 39, 7894(2014).

[12] Y. LeCun, Y. Bengio, G. Hinton. Deep learning. Nature, 521, 436(2015).

[13] M. Lonardi, J. Pesic, P. Jenneve, P. Ramantanis, N. Rossi, A. Ghazisaeidi, S. Bigo. Optical nonlinearity monitoring and launch power optimization by artificial neural networks. J. Light. Technol., 38, 2637(2020).

[14] A. M. Hammond, R. M. Camacho. Designing integrated photonic devices using artificial neural networks. Opt. Express, 27, 29620(2019).

[15] F. N. Khan, K. Zhong, X. Zhou, W. H. Al-Arashi, C. Y. Yu, C. Liu, A. P. T. Lau. Joint OSNR monitoring and modulation format identification in digital coherent receivers using deep neural networks. Opt. Express, 25, 17767(2017).

[16] M. Hao, L. S. Yan, A. L. Yi, L. Jiang, Y. Pan, W. Pan, B. Luo. Stokes space modulation format identification for optical signals using probabilistic neural network. IEEE Photon. J., 10, 7202213(2018).

[17] E. J. Liu, Z. M. Yu, Z. Q. Wan, L. Shu, K. X. Sun, L. L. Gui, K. Xu. Linearized wideband and multi-carrier link based on TL-ANN. Chin. Opt. Lett., 19, 113901(2021).

Zhiyu Chen, Xin Zhong, Lin Jiang, Jiaxin Xu, Jingxian Liu, Yan Pan, Tao Zhou. Linearized microwave downconversion link based on fast and intelligent impairment equalization for noncooperative systems[J]. Chinese Optics Letters, 2023, 21(2): 023902


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# Is CMAC vulnerable to length extension attacks?

Given $m_1, m_2$ such that $MAC_k(m_1) = MAC_k(m_2)$ is it possible to construct more collisions with pairs of the form $m_1|x, m_2|x$?

Here is the CMAC picture from Wikipedia where it is only part of the one-key MAC article:

I don't see how it could be because it treats the last block of the message differently but this was mentioned in the comments of another question so I wanted to clarify.

Here is the quote from poncho:

CMAC and XCBC are similar to the UMAC case (even though a universal hash is not involved); if you find two messages that have the same CMAC value, it is likely that a simple length-extension to the two messages will allow the attacker to create other pairs of messages with the same MAC.

Bonus question: Are there other ways to exploit a collision?

format: plain


text:
It's stated that the gradient of:

$\frac{1}{2}x^TAx - B^Tx +C$

is

$\frac{1}{2}A^Tx + \frac{1}{2}Ax - b$

How do you grind out this equation? Or specifically, how do you get from $x^TAx$ to $A^Tx + Ax$?

-

The only thing you need to remember/know is that $$\dfrac{\partial (x^Ty)}{\partial x} = y$$ and the chain rule, which goes as $$\dfrac{d(f(x,y))}{d x} = \dfrac{\partial (f(x,y))}{\partial x} + \dfrac{\partial y^T}{\partial x} \dfrac{\partial (f(x,y))}{\partial y}$$ Hence, $$\dfrac{\partial (b^Tx)}{\partial x} = \dfrac{\partial (x^Tb)}{\partial x} = b$$
$$\dfrac{\partial (x^TAx)}{\partial x} = \dfrac{\partial (x^Ty)}{\partial x} + \dfrac{\partial y^T}{\partial x} \dfrac{\partial (x^Ty)}{\partial y}$$ where $y = Ax$.
$$\dfrac{\partial (x^TAx)}{\partial x} = \dfrac{\partial (x^Ty)}{\partial x} + \dfrac{\partial y^T}{\partial x} \dfrac{\partial (x^Ty)}{\partial y} = y + \dfrac{\partial (x^TA^T)}{\partial x} \dfrac{\partial (x^Ty)}{\partial y} = y + A^Tx = (A+A^T)x$$
To help future generations: the full specification of the chain rule used here is $$\frac{df(g,h)}{dx} = \frac{d(g(x)^T)}{dx} \frac{\partial f(g,h)}{\partial g} + \frac{d(h(x)^T)}{dx} \frac{\partial f(g,h)}{\partial h}$$ The order of multiplication is very important since we're dealing with vectors! –  Neil Traft Sep 23 '14 at 9:58


text:
# Tex2hs

This tool takes a Latex document, usually intended for consumption with lhs2tex, and attempts to type check it. While literate Haskell is great for writing a document which is a complete piece of code, when writing fragments of code interspersed with alternative definitions it can be tricky. Tex2hs takes a .tex file, and automatically tries to splice the fragments together in a way that type checks.

## An Example

Let's imagine we are writing a paper:

We have developed a method for short-cut fusion using |foldr|/|build|. By introducing the definition:

\begin{code}
build :: ((a -> [a] -> [a]) -> [b] -> c) -> c
build g = g (:) []
\end{code}

The standard definition of |map| is:

\begin{code}
map :: (a -> b) -> [a] -> [b]
map f []      = []
map f (x:xs)  = f x : map f xs
\end{code}

But we redefine it as:

\begin{code}
map :: (a -> b) -> [a] -> [b]
map f = foldr ((:) . f) []
\end{code}

Which now allows us to have the property:

\begin{code}
propEq g k z = foldr k z (build g) == g k z
\end{code}

This code won't actually type check in a literate document -- for a start map has been defined once in the Prelude, and twice in the paper. We also need to link the build into the propEq. If we try running Tex2hs we get:

$tex2hs sample.tex Checking line 1... success Checking line 1... success Checking line 3... success Checking line 8... success Checking line 10... FAILURE ERROR "temp.hs":4 - Ambiguous variable occurrence "map" *** Could refer to: Main.map Hugs.Prelude.map  We can fix this error easily by creating a file Include.hs in the same directory as the paper with the text import Prelude hiding (map). This file is a great place to put definitions and imports which are necessary. Running the checker again, this time specifying to go from line 10: $ tex2hs sample.tex 10..
Checking line 10... success
Checking line 18... FAILURE
ERROR "temp.hs":5 - Type error in explicitly typed binding
*** Term           : map
*** Type           : (([a] -> [a]) -> [b] -> [b]) -> [a] -> [b]
*** Does not match : (([a] -> [a]) -> [b] -> [b]) -> [[a] -> [a]] -> [[b] -> [b]]
*** Because        : unification would give infinite type

We can now spot the bug, it should have been (:) . f, not the other way round. We fix this bug, and now the entire document type checks.


text:
My Math Forum > Math what this operator stands for ?

 Math General Math Forum - For general math related discussion and news

 August 14th, 2018, 01:52 AM #1 Newbie   Joined: Aug 2018 From: czech repiblic Posts: 1 Thanks: 1 what this operator stands for ? Hello, can't find this operator on wikipedia. It has < on top and under it is _ Thanks. Thanks from EBTERTTBT
 August 14th, 2018, 01:53 AM #2 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,634 Thanks: 567 Math Focus: Yet to find out. You mean this: $\le$? edit: It stands for "less than or equal to" by the way. Thanks from EBTERTTBT Last edited by Joppy; August 14th, 2018 at 02:43 AM.
August 15th, 2018, 12:40 AM   #3
Global Moderator

Joined: Dec 2006

Posts: 19,505
Thanks: 1741

Quote:
 Originally Posted by empleat Hello, can't find this operator on wikipedia
It is included in this list of mathematical symbols.
The symbol $\leqslant$ is also in use and has the same meaning.

 Tags operator, stands

 Thread Tools Display Modes Linear Mode

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text:
Wrox Programmer Forums PHP ( Cannot modify header information)
 |
 Beginning PHP Beginning-level PHP discussions. More advanced coders should post to the Pro PHP forum.
 Welcome to the p2p.wrox.com Forums. You are currently viewing the Beginning PHP section of the Wrox Programmer to Programmer discussions. This is a community of software programmers and website developers including Wrox book authors and readers. New member registration was closed in 2019. New posts were shut off and the site was archived into this static format as of October 1, 2020. If you require technical support for a Wrox book please contact http://hub.wiley.com

August 12th, 2004, 05:12 AM
 squared_ring Registered User Join Date: Aug 2004 Posts: 4 Thanks: 0 Thanked 0 Times in 0 Posts
PHP ( Cannot modify header information)

hello ppl.. well i was learing to use cookies, but i am stuck cus the below posted script generates

Warning: Cannot modify header information - headers already sent by (output started at D:\xtams\xtams-php\pm.php:1) in D:\xtams\xtams-php\pm.php on line 15

error occors at the red indicated lines..

I tried to check if this script is sending ne info or white space to the browser.. but doesnt seem to be.. i mean before setcookie() funcn.

pls help..
thanks

<?php
$dbhost = 'localhost';$dbusername = '';
$dbpasswd = '';$database_name = 'xpm_db';
$conn = mysql_pconnect("$dbhost","$dbusername","$dbpasswd" ) or die ("Couldn't connect to server.");
$db = mysql_select_db("$database_name", $conn) or die("Couldn't select database.");$username=$_REQUEST["form_username"];$password=$_REQUEST["form_password"];$sql = "SELECT * FROM users WHERE user='" . $username. "' AND pass='".$password."'";
$result = mysql_query($sql,$conn) or die(mysql_error());$RecTotal= mysql_num_rows($result); if ($RecTotal != 0)
{
setcookie("xpm_user", $row["user"], time()+60, "/", ".localhost.", 0) or die("cannot set cookie"); } else { Header ("location: \"login.php\""); exit; } ?> August 12th, 2004, 10:12 AM  Daniel Walker Friend of Wrox Join Date: Jun 2003 Posts: 256 Thanks: 0 Thanked 0 Times in 0 Posts Check for white space outside of your <?php ?> tags. White space counts as content, so the webserver will wrap a header around it automatically. Then when you try to modify the headers, the PHP engine correctly reports that the headers have benn and gone and been sent, and so lie beyond the realms of modifyability. Take it easy, Dan August 13th, 2004, 02:29 AM  squared_ring Registered User Join Date: Aug 2004 Posts: 4 Thanks: 0 Thanked 0 Times in 0 Posts but dan, tat script is my whole file... and it doesnt have ne added html stuffs... ??? :( August 13th, 2004, 04:22 AM  Daniel Walker Friend of Wrox Join Date: Jun 2003 Posts: 256 Thanks: 0 Thanked 0 Times in 0 Posts Yes, but I'm not talking and about a HTML <head /> header: I mean a hypertext MIME header - stuff like: Expires: Mon, 26 Jul 1997 05:00:00 GMT" Last-Modified: Monday,11th May 2003 19:06:13 GMT" Cache-Control: no-cache, must-revalidate" Pragma: no-cache Content-type: application/x-msexcel Content-Disposition: attachment; filename=$file
Content-Description: PHP Generated Data

The error is saying that something has caused the server to generate a header for you already, so you cannot modify it. The most likely cause is whitespace.

Take it easy,
Dan

August 13th, 2004, 05:51 AM
 squared_ring Registered User Join Date: Aug 2004 Posts: 4 Thanks: 0 Thanked 0 Times in 0 Posts

hi again.. well i get exactly what u mean... but the thing is how am i to go bout hunting for that header created ("that irritating white space")....

August 13th, 2004, 06:06 AM
 Daniel Walker Friend of Wrox Join Date: Jun 2003 Posts: 256 Thanks: 0 Thanked 0 Times in 0 Posts

Well, check if there is a newline, space characters, or anythin else before that opening <?php. In other words, the "<" of "<?php" should be on line one, character one of your file.

Whenever you get this error message, this is the first thing to check.

Of course, if it's not that, then I've led you astray, so let me know, and I'll have another go... :)

August 14th, 2004, 05:29 AM
 squared_ring Registered User Join Date: Aug 2004 Posts: 4 Thanks: 0 Thanked 0 Times in 0 Posts

ATLAST.... got tat white space... :D well thanks a lot dan.. if nething i get back to u if u don mind.. neway thanks once again.....

 Similar Threads Thread Thread Starter Forum Replies Last Post How to modify flash so it pulls php url every 15 s method Flash (all versions) 4 February 27th, 2007 06:19 PM Yet another Cannot modify header question TimB Beginning PHP 2 November 26th, 2005 11:43 PM Modify header and footer of word file in ASP Andraw Classic ASP Basics 1 January 14th, 2005 08:04 PM [Question]Warning: Cannot modify header informatio unknownu Beginning PHP 1 August 19th, 2004 05:16 AM Cannot add header information.... cmiller Beginning PHP 1 September 2nd, 2003 07:02 PM


text:
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# Polar cod in jeopardy under the retreating Arctic sea ice

## Abstract

The Arctic amplification of global warming is causing the Arctic-Atlantic ice edge to retreat at unprecedented rates. Here we show how variability and change in sea ice cover in the Barents Sea, the largest shelf sea of the Arctic, affect the population dynamics of a keystone species of the ice-associated food web, the polar cod (Boreogadus saida). The data-driven biophysical model of polar cod early life stages assembled here predicts a strong mechanistic link between survival and variation in ice cover and temperature, suggesting imminent recruitment collapse should the observed ice-reduction and heating continue. Backtracking of drifting eggs and larvae from observations also demonstrates a northward retreat of one of two clearly defined spawning assemblages, possibly in response to warming. With annual to decadal ice-predictions under development the mechanistic physical-biological links presented here represent a powerful tool for making long-term predictions for the propagation of polar cod stocks.

## Introduction

The Arctic Ocean winter sea ice cover has steadily declined since the 1970s, and the majority of this reduction has been observed in the Barents Sea1,2,3,4. With a warmer and increasingly ice-free Arctic Ocean, the border between the Boreal and Arctic biomes is predicted to move north5,6. While the present warming has allowed an opportunistic northwards expansion of Boreal species into the northern Barents Sea Arctic ecosystem7, little attention has been given to the effect of the recent ice retreat and variability on the ice-associated fish community. A keystone species in the ice-associated Arctic marine food web is the polar cod, one of few species linking the lower (i.e. zooplankton) and higher (e.g. other fish, mammals, seabirds) trophic levels8,9,10. The polar cod is endemic to the Arctic and their early life history is strongly adapted to the presence of ice: from spawning of eggs under the ice11; the ability of eggs to develop in sub-freezing temperatures12; larvae feeding on zooplankton specific to the seasonal ice-melt-water blooms13,14; and low mortality of larvae in the close to freezing temperatures typical of Arctic water masses15. At the same time observations made by Soviet-era researchers in the early 1960s11 remain the most complete descriptions of polar cod spawning along the Eurasian shelf, with a historical stronghold in spawning activity in the south-eastern corner of the Barents Sea (also known as the Pechora Sea) and a suggested spawning east of Svalbard. The Barents Sea polar cod stock have been monitored annually by a joint Norwegian-Russian survey since 198616, and the total stock biomass (TSB) has varied vastly in the past three decades, between a minimum of 127,000 t in 1990, up to a maximum of 1,941,000 t in 2006 yet with no clear trend17. The spatial distribution and abundance of 0-group polar cod (~6 months old) has also varied considerably in this period18, suggesting high recruitment variability both in space and time.

Given the tightly linked early life history of the polar cod with ice, a natural candidate for the large variation in polar cod recruitment and biomass is the inter-annual variation in ice cover18. The two main factors driving the observed variability in ice cover in the Barents Sea is the inflow of warm Atlantic water from the west and the inflow of cold, less saline Arctic water from the north and east4,19,20. The inflow of Atlantic water from the west and Arctic water from the northeast set up a strong temperature and salinity front across the entire Barents Sea, here termed the Polar Front21,22 (Fig. 1). North of the Polar Front the less dense Arctic water masses stabilizes the water column sufficiently to prevent upwards flux of the warmer Atlantic water, allowing ice to form4. In March-April the extent of the seasonal ice cover in the Barents Sea is at its largest, and at this time period the Polar Front and the ice edge usually coincide. During 1990–2017 the winter maximum ice cover in the Barents Sea varied between 632,304 km2 and 1,129,719 km2, with an average of 869,961 km2.

Here we investigate possible links between inter-annual variability in the Barents Sea ice cover and the variability in spawning distribution and recruitment of polar cod. To locate the spawning locations of polar cod under the ice we “back-tracked” larval drift trajectories from observed 0-group polar cod in autumn to the most probable spawning locations in spring. This was done by Lagrangian particle advection simulations within a 3D dynamical ocean model. Drifting particles were “spawned” uniformly over the parts of the Barents Sea that was encompassed by the historical marginal ice zone (i.e. where ice concentration had been at least 15% at any time in period 1990–2017), and subsequently an objective search algorithm identified drift trajectories that intersected the 0-group observations of the autumn survey. The coupled biophysical model was simulated over 28 spawning seasons (1990–2017) and was compared to the catch of 0-group polar cod in 8302 pelagic trawls distributed throughout the Barents Sea. The mechanistic interpretation of the scores from the objective search algorithm is twofold: first the backtracking from observations indicates spawning at a given release point for a given year (viz. inter-annual change in spawning location); and second the frequency of links between spawning location and the observed 0-group abundance reflects larval survival during the drift phase and indicates supply of recruits from a given spawning location (viz. recruitment variability in response to environmental conditions). We report a positive relationship between year class strength of polar cod to the inter-annual variability of ice cover, and a negative effect of maximum summer temperatures encountered in late larval phase, where a common driver is the heating of the system. We also observed a clear northward retreat of one the two identified spawning assemblages towards the end of the study period that may also be a response to warming. Understanding the physical-biological interactions during recent decades of both varying and changing ice conditions allow predictions of future development of the polar cod stock in the Barents Sea.

## Results

### Inter-annual change in spawning location

The objective search algorithm revealed two clearly separated maxima in spawning location probability, with one spawning assemblage in the Pechora Sea and one east of Svalbard. Moreover, the location of the two main spawning areas identified coincided with the high probability of 0-group occurrence downstream to the two areas (Generalized additive model (GAM) explaining 40% of the variation in 0-group presence/absence, Fig. 2). At the same time there was considerable inter-annual variability in the location of the center of the back-calculated main spawning areas. For example, in 1990 when the observed spatial distribution of 0-group polar cod was strongest correlated with spawning east of Svalbard, most of the eggs and larvae drifted with the East Spitsbergen Current into the Storfjord, further along the west coast of Svalbard, or onto the Svalbard Bank, where most of the 0-group polar cod was found that year (Fig. 3a, and see Fig. 1 for references to location). Recruitment in 1995 was very low, indicated by a record low abundance of 0-group polar cod found in the autumn cruise, coinciding with the Pechora Sea largely being ice-free during most of winter and spring (Fig. 3b) and maximum temperatures encountered in late summer exceeding 10 °C. This in contrast to 1998 when the ice cover in the south-eastern Barents Sea was at its highest in the study period, and the release points in the Pechora Sea was strongly correlated with the high abundance of 0-group polar cod found in a wide swath along Novaya Zemlya (Fig. 3c). Perhaps the most anomalous year in terms of drift patterns was in 2013 when the most probable spawning location of the eastern Svalbard spawning assemblage was situated north of Kvitøya, close to the shelf edge towards the Nansen Basin–yielding a qualitatively different dispersal pattern than previous years, where the majority of larvae initiated from this particular location were advected along the margins of the continental shelf instead of into the Barents Sea (Fig. 3d). This northern displacement of the eastern Svalbard spawning assemblage in 2013 was not an isolated instance, with a similar displacement also in 2009, 2015, and 2017. The predicted western spawning center was also located on the Great Bank in many of the years in the middle of the study period (2000s), and even near the Central Bank in some years. The average northwards displacement towards the end of the study period (2015–2017) constituted 2° of latitude (approx. 220 km) compared to the yearly 2000s (2001–2005), culminating a clear decadal northward trend (Fig. 4).

### Recruitment variability in response to ice loss and heating

Recruitment strength in the eastern Barents Sea/Pechora Sea had a significant correlation with maximum yearly ice cover (r = 0.69, t = −4.49, df = 26, p < 0.001, see Fig. 5), reiterating the importance of ice for early pelagic stages. The recruitment in the Pechora Sea was also significantly correlated with the estimated TSB two and a half years later (r = 0.61, t = 3.58, df = 22, p < 0.001), confirming the Pechora Sea spawning assemblage as the main supplier of recruits to the Barents Sea stock monitored during autumn surveys. Recruitment east of Svalbard had no correlation with either ice cover, temperature, or TSB at any lag, indicating other effects at play than those tested within the scope of this study. A linear regression of maximum ice cover over the study period showed a weak, yet significant yearly decrease of 1.1% (F = 4.93 on 1 and 26 df, p = 0.035, R2 = 0.16). Moreover, the 10-day mean-filtered maximum temperature encountered by larvae in late summer was significantly different between the two main spawning areas (t = −7.46, df = 47, p< 0.001), with maximum temperatures encountered in the eastern Barents Sea on average between 2.1 °C and 3.6 °C degrees warmer (95% CI). The average yearly maximum temperature encountered in the Pechora/eastern Barents Sea was 5.7 °C but in extreme years more than 9 °C (95% CI: 2.4–9.0 °C), well beyond the thermal tolerance limit of the larvae (≈100% mortality at extended periods above 7 °C15). A linear regression model of eastern Barents Sea recruitment strength as function of covariates Barents Sea ice cover, maximum temperature encountered by larvae, and TSB, explained 72% of the variation in recruitment strength (F = 20.7 on 3 and 23 df, p < 0.001). Ice and temperature explained the majority of the variance, and a smaller portion explained by TSB $$({\mathrm{R}}_{{\mathrm{ice}}\,{\mathrm{cover}}}^2 = 48\% ,{\mathrm{R}}_{{\mathrm{max}}\,{\mathrm{temp}}}^2 = 44\% ,{\mathrm{R}}_{{\mathrm{TSB}}}^2 = 16\% )$$. However, the two variables ice cover and temperature where found to share a high portion of the explained variance $$({\mathrm{R}}_{{\mathrm{ice}}\,{\mathrm{cover}}}^2 \cap {\mathrm{R}}_{{\mathrm{max}}\,{\mathrm{temp}}}^2 = 25\% )$$, albeit being significantly negatively correlated (r = −0.38, t = −2.09, df = 26, p = 0.045).

## Discussion

We suggest two plausible mechanisms driving the large recruitment variability observed in the south-eastern parts of the Barents Sea, mediated in turn by both heat and ice. First, the presence/absence of ice likely causes a “match/mis-match” scenario23,24 specific to the seasonally ice-covered Arctic food web. The primary food item for the first-feeding larval phase is the naupliar stages of Arctic copepod Calanus glacialis that are nourished by the algae bloom confined to the meltwater plume13,25,26. A mis-match most likely arises in years with reduced or no ice cover in the Pechora Sea when the phenology of the spring bloom follows Atlantic-like bloom dynamics initiated by thermal stratification rather than ice-melt25,27,28. However, if larvae indeed survive that first critical ice-associated phase all three major Calanus species found in the Barents Sea are eaten indiscriminately13. This includes the highly abundant C. finmarchicus and the larger (but rarer) C. hyperboreus, both advected from the west with the warmer Atlantic water masses, and all three Calanus species having exceptionally high energy content29. The ample food supply available to the later larval stages in the Atlantic-influenced water masses found in the south-eastern parts of the Barents Sea thus represents a near ideal nursery habitat. At the same time, the high summer temperatures sometimes encountered there may also be associated with high risk by pushing larvae beyond thermal thresholds15. For example, the seven warmest years in terms of maximum summer temperatures encountered (as observed in 1990, 1995, 2000, 2012–2013, 2016–2017), resulted in the five years of lowest recruitment. However, due to the collinearity of ice cover and high temperatures encountered in south-eastern Barents Sea during drift in late summer, where a common driver is the wind-driven advection of warmer water masses from the west20, we were not able to disentangle which of the two identified mechanisms that affected recruitment in the Barents Sea the most.

Regional downscaling of future climate change scenarios predicts an increased frequency of years with reduced or no ice cover in the south-eastern Barents Sea towards 205030,31. However, due to the strong topographic steering of the incoming Atlantic water masses away from the northern parts of the Barents Sea (cf. Figure 1), the cold and icy winter-conditions currently found east of Svalbard is likely to persist into the foreseeable future19,32,33. The predicted deterioration of the south-eastern Barents Sea nursery habitat will thus increase the relative importance of the supply of recruits coming from the eastern Svalbard spawning assemblage in the future. At the same time the observed northward displacement of the eastern Svalbard spawning assemblage towards the end of the study period may qualitatively alter the dispersal pathway of eggs and larvae. The advection of the surface layer in the far northern Barents Sea area is mainly influenced by the prevalent wind field, where the meltwater and pack ice generally follow a net westward direction towards the Fram Strait19,20. Other potential dispersal pathways that may be realized in the northernmost Barents Sea include getting entrained with the Atlantic boundary current flowing along the Nansen Basin34, or lateral transport towards the Arctic Ocean interior with eddies detaching from the boundary current itself35. In any case, polar cod larvae and juveniles drifting with the pack ice in the Arctic Ocean interior is not an uncommon phenomenon36, although the fate of these drifting individuals is not known.

The biophysical links highlighted here represent fundamental knowledge of how a keystone species of the Arctic food web respond to contemporary climatic forcing. As the frequency of years with low or no ice cover in the south-eastern Barents Sea is predicted to increase towards the middle of the millennium, we expect recruitment of polar cod to the south-eastern Barents Sea to become even more variable or diminished, most likely having dramatic consequences for the entire food web. And ultimately if summer temperatures continue to exceed the thermal tolerance of larvae, the south-eastern Barents Sea as a nursery area will be unsuitable altogether. Conversely, the topographical steering of Atlantic water masses away from the north-western Barents Sea will most likely facilitate ice and spawning east of Svalbard even in a warmer climate than today.

## Methods

### Recruitment and spawning stock biomass indices

The 0-group polar cod data were sampled on annual surveys run between late August and early October in the period 1990 and 2017, covering almost the entire Barents Sea within a regular grid of ~65 km. At each station the upper water layer (0–60 m) was sampled by three pelagic trawls with a 20 × 20 m opening, keeping the headlines at 0 m, 20 m, and 40 m. The pelagic trawls were towed at a speed of 3 knots over a time interval of 10 min, corresponding to a tow length of 0.5 nautical miles (≈0.93 km). If dense concentrations of fish appeared on the echo-sounder deeper than 40 m, additional tows were performed at 60 and 80 m. During the study period of 27 years 8302 of these depth-integrated trawl hauls were done. Due to the selectivity of the gear37, the catches were adjusted for capture efficiency using a stratified sample mean method38,39. As a proxy for total stock biomass (TSB) we estimated the total mass of polar cod found in echo-sounder transects and pelagic trawls throughout the Barents Sea17, identical to the method used for estimating capelin (Mallotus villosus) stock size in the Barents Sea40.

### Ocean circulation model and ice module

The hydrodynamic model used to represent the currents and oceanographic conditions (i.e. temperature, salinity, and ice concentration/cover) in the study area was based on the Regional Ocean Modeling System (ROMS, http://myroms.org), a free-surface, hydrostatic, primitive equation ocean general circulation model41,42. The ROMS model was run with a horizontal resolution of 4 × 4 km in an orthogonal, curvilinear grid covering parts of the North Atlantic and all the Nordic and Barents seas (see inset in Fig. 1 for extent of ROMS model) over the time period 1960–201720,43,44,45. The output from ROMS contained velocity fields, ice concentration, temperature, and salinity in 32 terrain following vertical layers, and a temporal resolution of 24 h.

### Drift simulations and search algorithm

The advection of particles in the horizontal plane was modelled by the Runge-Kutta fourth order scheme LADIM46,47. As early life stages of polar cod are usually found close to the surface13, particles were uniformly distributed in the upper 10 meters with a fixed depth throughout the drift phase from 1 January to 30 September. In an exhaustive search for potential spawning areas of polar cod in the Barents Sea, particles were released in a regular grid (≈40 km equidistance, 537 positions in total) across the entire Barents Sea shelf shallower than 400 m that had been covered by an ice concentration of more than 15% in the period 1990–2017 (see extent of release grid in Fig. 2). A new ensemble of 100 particles were released at every point in the grid, every day from 1 January to 30 April, repeated for every year between 1990 and 2017 (yielding a total of 639,030 particles each year). Subsequently, an objective search algorithm identified drift trajectories that intersected the 0-group observations of the autumn survey within a three-week period of the surveys. The ability of the drift trajectories to explain the observed 0-group abundance and distribution was thus interpreted as a confirmation of spawning at a given release point and a high larval survival integrated over the drift phase. To allow a direct comparison between number of simulated drift intersections and 0-group abundance, both indices were log-transformed and scaled between 0 and 1. In line with the hypothesis of ice as a prerequisite for spawning, the drift trajectories’ ability to predict the observed 0-group abundance was weighted by ice concentration at drift start point (i.e. at spawning area). To elucidate on the possible effects of heating on recruitment we extracted temperature profiles from all individual drift trajectories, and to decrease the effect of minor cold spells or heat waves on the subsequent analysis we applied a 10-day moving average filter on the temperature profiles.

### Statistical modelling

A probabilistic map of 0-group polar cod presence was calculated by using a two-dimensional binomial GAM smoother, based on the geographical coordinates of pelagic trawls, presence-absence of polar cod larvae in the pelagic trawls, and using the logit-link function as implemented in R-package “mgvc”48. Moreover, to quantify the effect of environmental conditions on larval survival/recruitment strength, we fitted a linear regression model with recruitment strength as independent variable with the covariates Barents Sea ice cover (area of the Barents Sea covered by ice concentration higher than 15%, extracted from the ROMS model), maximum temperature encountered by larvae (10-day mean-filtered over 100 larvae released from the most likely spawning area for a given year), and estimated TSB. This regression model was fitted separately for the north-western (Svalbard) and south-eastern (Pechora Sea) spawning areas as implemented in the base R-package “stats”49. In the model selection phase, we applied a stepwise model selection scheme with the initial inclusion of all relevant variables, where only the variables deemed significant was included in the final model. Due to the high degree of collinearity between some of the variables, we also did a variance partitioning analysis to disentangle the separate and/or common effects of the variables50.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

The datasets generated during and/or analysed during the current study are not publicly available due to large file sizes but are available from the corresponding author on reasonable request.

## Code availability

Custom code and algorithms used for analyses in this study will be made available on request to the authors.

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## Author information

Authors

### Contributions

M.B.O.H., E.E., H.G., and F.V. conceived the study, interpreted the data and wrote the paper. M.B.O.H designed and performed the model simulations and statistical analyses.

### Corresponding author

Correspondence to Mats Brockstedt Olsen Huserbråten.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Huserbråten, M.B.O., Eriksen, E., Gjøsæter, H. et al. Polar cod in jeopardy under the retreating Arctic sea ice. Commun Biol 2, 407 (2019). https://doi.org/10.1038/s42003-019-0649-2

• Accepted:

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• Randi B. Ingvaldsen
• Karen M. Assmann
• Andrey V. Dolgov

Nature Reviews Earth & Environment (2021)

• ### Distinct genetic clustering in the weakly differentiated polar cod, Boreogadus saida Lepechin, 1774 from East Siberian Sea to Svalbard

• María Quintela
• Shripathi Bhat
• Torild Johansen

Polar Biology (2021)

• ### Key processes regulating the early life history of Barents Sea polar cod

• Harald Gjøsæter
• Mats Huserbråten
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• ### Arctic gadids in a rapidly changing environment

• Franz Mueter
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Polar Biology (2020)


text:
# Aerosols measurement

It is possible to determine the distribution in number and size of the particles constituting the aerosols. These particles, whose diameter is between 10-3 et 100 micrometers, are particularly concentrated over the industrialized regions of the Northern Hemisphere.

The Angström formalism allows to characterize the aerosols by means of two coefficients: the wavelength exponent ('alpha') and the turbidity coefficient of Angström ('beta').

The wavelength exponent is an index sensitive to the aerosol size distribution. It is inversely related to the average particle size of the aerosols: the smaller the particles, the higher the exponent.

For example, for the year 2007, there were variations from 0.2 in January to about 1.0 in August in the middle of the Atlantic (in the Azores), while it varied from 0.75 in January To about 1.5 in July over Paris (in beaches similar to those measured in the middle of the Amazonian equatorial forest). It is much more stable in the middle of the Pacific in the equatorial zone: for example, in Nauru, the variations of this coefficient vary from 0.3 to 0.65 on average, with peaks not exceeding 0.8 during periods of stronger activity cyclonic. Major events such as the huge bush and forest fires and volcanic eruptions increase the coefficient observed over periods longer than the usual seasonal peaks. In the south of the Algerian plateau, at Tamanrasset at 1400 meters altitude, it is even lower and varies even less, around 0.1 and 0.35 (with its peak in the middle of winter).

This coefficient is also a good indicator of the proportion of precipitated atmospheric water, where the concentration of aerosols plays a role now recognized as very important. It allows to anticipate the volume of precipitation expected in a season. Depending on the concentration of water in the atmosphere, a higher coefficient favors the concentration of clouds and higher precipitation.

Aerosols also have complex effects on albedo (*), and longer-term climate change, their influence is still difficult to take into account due to the multiplicity of sources, the complexity of their transport in the High atmosphere and many other partially correlated factors.

Global measurement campaigns should be used to establish these correlations and better understand and anticipate climate change, but still suffer from insufficient data (outside North America and Western Europe where sensor networks are well developed).

(*) Albedo: relation between the incident energy and the energy reflected by a surface.

The turbidity coefficient of Angström is characteristic of the amount of these particles in the atmosphere.

To calculate these coefficients, measurements of the optical thickness must be made by evaluating the intensity of the sunlight, targeting the sun with a photometer. We also need to perform these measurements in at least two different wavelengths.

We will call tau_a, the measurement of the optical thickness realized in a known wavelength.

tau_a can be converted into a solar light transmission rate tau_x, which is more easily appreciable : tau_x = 100 * e^[-tau_a]

Light wavelength, optical depth and atmospheric turbidity are related by the Angström turbidity coefficient : tau_a = beta * lambda^-alpha

where beta is the Angström turbidity coefficient, 'lambda the wavelength (in micrometers) and α, the Angstrom wavelength exponent.

Attention, beta and α are independent of the wavelength.

Calculation of the wavelength exponent (bb alpha)

The exponent of Angström alpha is calculated from optical thickness data ( tau_ [alpha_n] ) taken at two different wavelengths' lambda_1 and lambda_2:

((bb tau_[a_1] = beta * lambda_1^-alpha),(bb tau_[a_2] = beta * lambda_2^-alpha)) <=> bb tau_[a_1] / bb tau_[a_2] = lambda_1^-alpha / lambda_2^-alpha <=> ln(bb tau_[a_1]/bb tau_[a_2])=-alpha * ln(lambda_1 / lambda_2) <=>

ln(bb tau_[a_1]/bb tau_[a_2])=alpha * ln(lambda_2/lambda_1)<=> alpha=ln(bb tau_[a1]/bb tau_[a_2])/ln(lambda_2/lambda_1)

The typical value range of alpha is 0.5 to 2.5 with an average for the natural atmosphere, of the order of 1.3.

Calculation of the Angström turbidity coefficient (bb beta)
beta = bb tau_[a_1] * lambda_1^alpha = bb tau_[a_2] * lambda_2^alpha

Lambda is expressed in micrometers.beta values less than 0.1 correspond to a relatively clear atmosphere and values greater than 0.2 are associated with a disturbed atmosphere.

Example

Seysses,  September 1st 2010 at 12:11:19 TU

lambda_1 = 0,675 mu m        bb tau_[a_1]= 0,100

lambda_2 = 0,532 mu m        bb tau_[a_2]= 0,135

Calculation of alpha :

alpha = ln([0,100] / [0,135]) / ln([0,532] / [0,675]) = bb [1,126]

Calculation of beta :

beta = 0,10 * 0,675^[1,126] = bb[ 0,064]


text:
## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 902312, 14 pages.

### General Common Fixed Point Theorems and Applications

#### Abstract

The main result is a common fixed point theorem for a pair of multivalued maps on a complete metric space extending a recent result of $\text{Đ}$orić and Lazović (2011) for a multivalued map on a metric space satisfying Ćirić-Suzuki-type-generalized contraction. Further, as a special case, we obtain a generalization of an important common fixed point theorem of Ćirić (1974). Existence of a common solution for a class of functional equations arising in dynamic programming is also discussed.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 902312, 14 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357180325

Digital Object Identifier
doi:10.1155/2012/902312

Mathematical Reviews number (MathSciNet)
MR2880848

Zentralblatt MATH identifier
1276.54040

#### Citation

Singh, Shyam Lal; Mishra, Swami Nath; Chugh, Renu; Kamal, Raj. General Common Fixed Point Theorems and Applications. J. Appl. Math. 2012, Special Issue (2012), Article ID 902312, 14 pages. doi:10.1155/2012/902312. https://projecteuclid.org/euclid.jam/1357180325

#### References

• S. B. Nadler Jr., Hyperspaces of Sets: A Text with Research Questions, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49, Marcel Dekker, New York, NY, USA, 1978.
• T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008.
• M. Kikkawa and T. Suzuki, “Three fixed point theorems for generalized contractions with constants in complete metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2942–2949, 2008.
• S. B. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
• G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no. 4, pp. 261–263, 1976.
• A. Meir and E. Keeler, “A theorem on contraction mappings,” Journal of Mathematical Analysis and Applications, vol. 28, pp. 326–329, 1969.
• A. Abkar and M. Eslamian, “Fixed point theorems for Suzuki generalized nonexpansive multivalued mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 457935, 10 pages, 2010.
• S. Dhompongsa and H. Yingtaweesittikul, “Fixed points for multivalued mappings and the metric completeness,” Fixed Point Theory and Applications, vol. 2009, Article ID 972395, 15 pages, 2009.
• D. $\DJ$orić and R. Lazović, “Some Suzuki-type fixed point theorems for generalized multivalued mappings and applications,” Fixed Point Theory and Applications, vol. 2011, article 40, 2011.
• G. Moţ and A. Petruşel, “Fixed point theory for a new type of contractive multivalued operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3371–3377, 2009.
• O. Popescu, “Two fixed point theorems for generalized contractions with constants in complete metric space,” Central European Journal of Mathematics, vol. 7, no. 3, pp. 529–538, 2009.
• S. L. Singh and S. N. Mishra, “Coincidence theorems for certain classes of hybrid contractions,” Fixed Point Theory and Applications, vol. 2010, Article ID 898109, 14 pages, 2010.
• S. L. Singh and S. N. Mishra, “Remarks on recent fixed point theorems,” Fixed Point Theory and Applications, vol. 2010, Article ID 452905, 18 pages, 2010.
• S. L. Singh, H. K. Pathak, and S. N. Mishra, “On a Suzuki type general fixed point theorem with applications,” Fixed Point Theory and Applications, vol. 2010, Article ID 234717, 15 pages, 2010.
• L. B. Ćirić, “Fixed points for generalized multi-valued contractions,” Matematički Vesnik, vol. 9(24), pp. 265–272, 1972.
• S. Reich, “Fixed points of multi-valued functions,” Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, vol. 51, pp. 32–35, 1971.
• I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001.
• L. B. Ćirić, “On a family of contractive maps and fixed points,” Institut Mathématique. Publications. Nouvelle Série, vol. 17(31), pp. 45–51, 1974.
• K. P. R. Sastry and S. V. R. Naidu, “Fixed point theorems for generalised contraction mappings,” Yokohama Mathematical Journal, vol. 28, no. 1-2, pp. 15–29, 1980.
• L. B. Ćirić, “A generalization of Banach's contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974.
• R. Baskaran and P. V. Subrahmanyam, “A note on the solution of a class of functional equations,” Applicable Analysis, vol. 22, no. 3-4, pp. 235–241, 1986.
• R. Bellman, Methods of Nonliner Analysis. Vol. II, Mathematics in Science and Engineering, Vol. 61-II, Academic Press, New York, NY, USA, 1973.
• R. Bellman and E. S. Lee, “Functional equations in dynamic programming,” Aequationes Mathematicae, vol. 17, no. 1, pp. 1–18, 1978.
• P. C. Bhakta and S. Mitra, “Some existence theorems for functional equations arising in dynamic pro-gramming,” Journal of Mathematical Analysis and Applications, vol. 98, no. 2, pp. 348–362, 1984.
• N. A. Assad and W. A. Kirk, “Fixed point theorems for set-valued mappings of contractive type,” Pacific Journal of Mathematics, vol. 43, pp. 553–562, 1972.
• B. Djafari Rouhani and S. Moradi, “Common fixed point of multivalued generalized $\phi$-weak contractive mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 708984, 13 pages, 2010.
• I. A. Rus, “Fixed point theorems for multi-valued mappings in complete metric spaces,” Mathematica Japonica, vol. 20, pp. 21–24, 1975.
• H. K. Pathak, Y. J. Cho, S. M. Kang, and B. S. Lee, “Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming,” Le Matematiche, vol. 50, no. 1, pp. 15–33, 1995.
• S. L. Singh and S. N. Mishra, “On a Ljubomir Ćirić fixed point theorem for nonexpansive type maps with applications,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 4, pp. 531–542, 2002.


text:
Area between Two Curves Calculator

### Area under the Curve Calculator

 Area between Two Curves area under a curve angle between two curves area under standard normal curve Calculate Area of a Quadrilateral Calculate Area of Polygon
 Area under a Curve Calculator Normal Distribution Curve Calculator Difference of Two Squares Calculator How to Calculate Distance between Two Points Multiplying Two Fractions Calculator Two Step Equation Calculator


text:
University Physics Volume 1

# Key Equations

### Key Equations

 Wave speed $v=λT=λfv=λT=λf$ Linear mass density $μ=mass of the stringlength of the stringμ=mass of the stringlength of the string$ Speed of a wave or pulse on a string undertension $|v|=FTμ|v|=FTμ$ Speed of a compression wave in a fluid $v=Βρv=Βρ$ Resultant wave from superposition of twosinusoidal waves that are identical except for aphase shift $yR(x,t)=[2Acos(ϕ2)]sin(kx−ωt+ϕ2)yR(x,t)=[2Acos(ϕ2)]sin(kx−ωt+ϕ2)$ Wave number $k≡2πλk≡2πλ$ Wave speed $v=ωkv=ωk$ A periodic wave $y(x,t)=Asin(kx∓ωt+ϕ)y(x,t)=Asin(kx∓ωt+ϕ)$ Phase of a wave $kx∓ωt+ϕkx∓ωt+ϕ$ The linear wave equation $∂2y(x,t)∂x2=1vw2∂2y(x,t)∂t2∂2y(x,t)∂x2=1vw2∂2y(x,t)∂t2$ Power averaged over a wavelength $Pave=EλT=12μA2ω2λT=12μA2ω2vPave=EλT=12μA2ω2λT=12μA2ω2v$ Intensity $I=PAI=PA$ Intensity for a spherical wave $I=P4πr2I=P4πr2$ Equation of a standing wave $y(x,t)=[2Asin(kx)]cos(ωt)y(x,t)=[2Asin(kx)]cos(ωt)$ Wavelength for symmetric boundaryconditions $λn=2nL,n=1,2,3,4,5...λn=2nL,n=1,2,3,4,5...$ Frequency for symmetric boundary conditions $fn=nv2L=nf1,n=1,2,3,4,5...fn=nv2L=nf1,n=1,2,3,4,5...$

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text:
Thinking Mathematically (6th Edition)

Published by Pearson

Chapter 4 - Number Representation and Calculation - 4.3 Computation in Positional Systems - Exercise Set 4.3: 21

Answer

$10_{two}$

Work Step by Step

Start with the right-most column. $1-1=0$ Record: $0_{two}$ $\begin{array}{lllll} & 1 & 0 & 0 & 1_{two}\\ - & & 1 & 1 & 1_{two}\\ -- & -- & -- & -- &-- \\ & & & & 0_{two} \end{array}$ Next column: $0$ is less than $1$... borrow 1 $two$ from the $10_{two},$ because we can't borrow from zero. $(0+2)-1=1$ Record the $1.$ $\begin{array}{lllll} & 1 & 0 & 0 & 1_{two}\\ - & & 1 & 1 & 1_{two}\\ -- & -- & -- & -- &-- \\ & & & 1 & 0_{two} \end{array}$ We are left with $10_{two}-1_{two}$, but since we borrowed a two from the previous column, the remaining subtraction is $1_{two}-1_{two}=0$, which we do not record. Result: $10_{two}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# 1. NAME▲

tex, virtex, initex - text formatting and typesetting

# 2. SYNOPSIS ▲

tex [ options ] [& format ] [ file | \ commands ]

# 3. DESCRIPTION ▲

Run the \*(TX typesetter on file , usually creating file.dvi . If the file argument has no extension, ".tex" will be appended to it. Instead of a filename, a set of \*(TX commands can be given, the first of which must start with a backslash. With a & format argument \*(TX uses a different set of precompiled commands, contained in format.fmt ; it is usually better to use the

\*(TX formats the interspersed text and commands contained in the named files and outputs a typesetter independent file (called DVI , which is short for "DeVice Independent" ). \*(TX's capabilities and language are described in "The \*(OXbook" . \*(TX is normally used with a large body of precompiled macros, and there are several specific formatting systems, such as \*(LX, which require the support of several macro files.

This version of \*(TX looks at its command line to see what name it was called under. If they exist, then both

initex and

virtex are symbolic links to the

tex executable. When called as

R initex (or when the

-ini option is given) it can be used to precompile macros into a .fmt file. When called as

virtex it will use the plain format. When called under any other name, \*(TX will use that name as the name of the format to use. For example, when called as

tex the tex format is used, which is identical to the plain format. The commands defined by the plain format are documented in "The \*(OXbook" . Other formats that are often available include latex and amstex .

The non-option command line arguments to the \*(TX program are passed to it as the first input line. (But it is often easier to type extended arguments as the first input line, since UNIX shells tend to gobble up or misinterpret \*(TX's favorite symbols, like backslashes, unless you quote them.) As described in "The \*(OXbook" , that first line should begin with a filename, a \controlsequence , or a &formatname .

The normal usage is to say

Sélectionnez
``tex paper``

to start processing paper.tex . The name paper will be the ``jobname'', and is used in forming output filenames. If \*(TX doesn't get a filename in the first line, the jobname is texput . When looking for a file, \*(TX looks for the name with and without the default extension ( .tex ) appended, unless the name already contains that extension. If paper is the ``jobname'', a log of error messages, with rather more detail than normally appears on the screen, will appear in paper.log , and the output file will be in paper.dvi .

This version of \*(TX can look in the first line of the file paper.tex to see if it begins with the magic sequence %& . If the first line begins with %& format -translate-file  tcxname then \*(TX will use the named format and translation table tcxname to process the source file. Either the format name or the

-translate-file specification may be omitted, but not both. This overrides the format selection based on the name by which the program is invoked. The

-parse-first-line option or the

parse_first_line configuration variable controls whether this behaviour is enabled.

The e response to \*(TX's error prompt causes the system default editor to start up at the current line of the current file. The environment variable TEXEDIT can be used to change the editor used. It may contain a string with "%s" indicating where the filename goes and "%d" indicating where the decimal line number (if any) goes. For example, a TEXEDIT string for

emacs can be set with the

sh command

Sélectionnez
``TEXEDIT="emacs +%d %s"; export TEXEDIT``

A convenient file in the library is null.tex , containing nothing. When \*(TX can't find a file it thinks you want to input, it keeps asking you for another filename; responding `null' gets you out of the loop if you don't want to input anything. You can also type your EOF character (usually control-D).

# 4. OPTIONS ▲

This version of \*(TX understands the following command line options.

-enc

.rb Enable the enc\*(TX extensions. This option is only effective in combination with

R -ini . For documentation of the enc\*(TX extensions see

-file-line-error

.rb Print error messages in the form file:line:error which is similar to the way many compilers format them.

-no-file-line-error

.rb Disable printing error messages in the file:line:error style.

-file-line-error-style

.rb This is the old name of the

-file-line-error option.

• -fmt  format
.rb Use format as the name of the format to be used, instead of the name by which \*(TX was called or a %& line.

-halt-on-error

.rb Exit with an error code when an error is encountered during processing.

-help

.rb Print help message and exit.

-ini

.rb Start in INI mode, which is used to dump formats. The INI mode can be used for typesetting, but no format is preloaded, and basic initializations like setting catcodes may be required.

• -interaction  mode
.rb Sets the interaction mode. The mode can be either batchmode , nonstopmode , scrollmode , and errorstopmode . The meaning of these modes is the same as that of the corresponding \commands.

-ipc

.rb Send DVI output to a socket as well as the usual output file. Whether this option is available is the choice of the installer.

-ipc-start

.rb As

R -ipc , and starts the server at the other end as well. Whether this option is available is the choice of the installer.

• -jobname  name
.rb Use name for the job name, instead of deriving it from the name of the input file.
.rb Sets path searching debugging flags according to the bitmask. See the Kpathsea manual for details.
• -mktex  fmt
.rb Enable mktex fmt , where fmt must be either tex or tfm .

-mltex

.rb Enable ML\*(TX extensions. Only effective in combination with

R -ini .

• -no-mktex  fmt
.rb Disable mktex fmt , where fmt must be either tex or tfm .
• -output-comment  string
.rb Use string for the DVI file comment instead of the date.
• -output-directory  directory
.rb Write output files in directory instead of the current directory. Look up input files in directory first, then along the normal search path. See also description of the TEXMFOUTPUT environment variable.

-parse-first-line

.rb If the first line of the main input file begins with %& parse it to look for a dump name or a

-translate-file option.

-no-parse-first-line

.rb Disable parsing of the first line of the main input file.

• -progname  name
.rb Pretend to be program name . This affects both the format used and the search paths.

-recorder

.rb Enable the filename recorder. This leaves a trace of the files opened for input and output in a file with extension .fls .

-shell-escape

.rb Enable the \write18{ command } construct. The command can be any shell command. This construct is normally disallowed for security reasons.

-no-shell-escape

.rb Disable the \write18{ command } construct, even if it is enabled in the texmf.cnf file.

-src-specials

.rb Insert source specials into the DVI file.

• -src-specials  where
.rb Insert source specials in certain places of the DVI file. where is a comma-separated value list: cr , display , hbox , math , par , parent , or vbox .
• -translate-file  tcxname
.rb Use the tcxname translation table to set the mapping of input characters and re-mapping of output characters.
• -default-translate-file  tcxname
.rb Like

-translate-file except that a %& line can overrule this setting.

-version

.rb Print version information and exit.

# 5. ENVIRONMENT ▲

See the Kpathsearch library documentation (the `Path specifications' node) for precise details of how the environment variables are used. The

kpsewhich utility can be used to query the values of the variables.

One caveat: In most \*(TX formats, you cannot use ~ in a filename you give directly to \*(TX, because ~ is an active character, and hence is expanded, not taken as part of the filename. Other programs, such as \*(MF, do not have this problem.

• TEXMFOUTPUT
Normally, \*(TX puts its output files in the current directory. If any output file cannot be opened there, it tries to open it in the directory specified in the environment variable TEXMFOUTPUT. There is no default value for that variable. For example, if you say tex paper and the current directory is not writable, if TEXMFOUTPUT has the value /tmp , \*(TX attempts to create /tmp/paper.log (and /tmp/paper.dvi , if any output is produced.)
• TEXINPUTS
Search path for \input and \openin files. This should probably start with ``.'', so that user files are found before system files. An empty path component will be replaced with the paths defined in the texmf.cnf file. For example, set TEXINPUTS to ".:/home/usr/tex:" to prepend the current directory and ``/home/user/tex'' to the standard search path.
• TEXFORMATS
Search path for format files.
• TEXPOOL
search path for

tex internal strings.

• TEXEDIT
Command template for switching to editor. The default, usually

R vi , is set when \*(TX is compiled.

• TFMFONTS
Search path for font metric ( .tfm ) files.

# 6. FILES ▲

The location of the files mentioned below varies from system to system. Use the

kpsewhich utility to find their locations.

• texmf.cnf
Configuration file. This contains definitions of search paths as well as other configuration parameters like

R parse_first_line .

• tex.pool
Text file containing \*(TX's internal strings.
• texfonts.map
Filename mapping definitions.
• *.tfm
Metric files for \*(TX's fonts.
• *.fmt
Predigested \*(TX format (. fmt) files.
• \$TEXMFMAIN/tex/plain/base/plain.tex
The basic macro package described in the \*(OXbook.

# 7. NOTES ▲

This manual page is not meant to be exhaustive. The complete documentation for this version of \*(TX can be found in the info manual "Web2C: A TeX implementation" .

# 8. BUGS ▲

This version of \*(TX implements a number of optional extensions. In fact, many of these extensions conflict to a greater or lesser extent with the definition of \*(TX. When such extensions are enabled, the banner printed when \*(TX starts is changed to print

R TeX .

This version of \*(TX fails to trap arithmetic overflow when dimensions are added or subtracted. Cases where this occurs are rare, but when it does the generated DVI file will be invalid.

R mf (1),
Donald E. Knuth, "The \*(OXbook" , Addison-Wesley, 1986, ISBN 0-201-13447-0.
Leslie Lamport, "\*(LX - A Document Preparation System" , Addison-Wesley, 1985, ISBN 0-201-15790-X.
K. Berry, "Eplain: Expanded plain \*(TX" , ftp://ftp.cs.umb.edu/pub/tex/eplain/doc.
Michael Spivak, "The Joy of \*(OX" , 2nd edition, Addison-Wesley, 1990, ISBN 0-8218-2997-1.
TUGboat (the journal of the \*(TX Users Group).

# 10. TRIVIA ▲

\*(TX, pronounced properly, rhymes with ``blecchhh.'' The proper spelling in typewriter-like fonts is ``TeX'' and not ``TEX'' or ``tex.''

# 11. AUTHORS ▲

\*(TX was designed by Donald E. Knuth, who implemented it using his \*(WB system for Pascal programs. It was ported to Unix at Stanford by Howard Trickey, and at Cornell by Pavel Curtis. The version now offered with the Unix \*(TX distribution is that generated by the \*(WB to C system ( web2c ), originally written by Tomas Rokicki and Tim Morgan.

The enc\*(TX extensions were written by Petr Olsak.


text:
# Publication of Research Results in Radiocarbon

In January 2023, the article “Traces of 14C emissions for the operation period of two Ukrainian NPPs: Rivne and Chornobyl” was the State Institution ” O. M. Marzieiev Institute for Public Health” of the NAMS of Ukraine”, the Institute for Safety Problems of Nuclear Power Plants of the NAS of Ukraine, the Rivne NPP of the NNEGC “Energoatom”, as well as the State Institution “Dobrov Research Institute on Scientific and Technological Potential and Science History” of the NAS of Ukraine. The co-author of the article is the Head of the Geoecological Researches Sector, Ph.D. in Tech. Sc. Mykola Panasiuk.

The aim of this study was a comparative retrospective assessment of radiocarbon (14C) as a tracer, caused by operational emissions of Rivne and Chornobyl nuclear power plants (NPPs), which are equipped with different types of nuclear reactors. For this purpose, 14C was studied in annual tree rings of pine taken at a distance of 1.5 km southwest of the Rivne NPP and at a distance of 3.5 km west-northwest of the Chornobyl NPP, near the Yaniv railway station.

It was found that the 14C excess in the annual tree-ring samples of pine near the Chornobyl NPP during the observed operation period (1984–2000) was 3.0–13.0 pMC, except for the 1986, the year of the Chornobyl accident, when the 14C value rose sharply to 182.7 pMC (14C excess 62 pMC). After 2000, the content of 14C in the air near the Chornobyl NPP did not exceed the background values within the uncertainty of the measured data. The concentration of 14C in the samples of annual tree rings of pine near the Rivne NPP for the observation period (1986–2019) corresponded to the background levels within the uncertainty of the measured data. The study of environmental traces of 14C emissions from two NPPs equipped with different types of reactors showed significantly lower emissions of Rivne NPP with VVER compared with emissions from Chornobyl NPP with RBMK reactors.

Online link to the article: https://www.doi.org/10.1017/RDC.2023.3

This post is also available in: Ukrainian

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Insert math as
$${}$$


text:
# Thermochemistry, heat calculation.

1. Jan 10, 2012

### AerospaceEng

1. The problem statement, all variables and given/known data

When 10.00 mL of a solution of strong acid is mixed with 100.00 mL of a solution of weak base in a coffee-cup calorimeter, the temperature rises from 22.8 oC to 26.8 oC. Determine q for the acid-base reaction, assuming that the liquids have densities of 1.00 g/mL and the same heat capacities as pure water.

2. Relevant equations

total volume V=110.0mL
change in temperature delta T = 4
mass m = 110.0g
heat capacity for water C=75.291J/(mol$\circ$C

q = nC$\Delta$T
q = mC$\Delta$T

3. The attempt at a solution

so I used the formula

q = mC$\Delta$T

because with the information given i couldn't find the number of moles and because only the molar heat capacity is given to use i had to find the mass based heat capacity and I found it to be C = 4181.3J/KgK

and then found q = 460J

which then the computer told me it was wrong.

It seemed odd that i had to go find the heat capacity rather than use the one provided. So I feel already there's my first mistake. But i know the formula is right so I can't see why it wouldn't work. However clearly my chemistry isn't that strong so perhaps I'm missing something. Any guidance would be appreciated! Thanks

2. Jan 18, 2012

### ehild

How did you get 460 J for q=mcΔT when m=0.11 kg, ΔT=4 and c=4181.3 J/kgK?

ehild


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### Abstract

The present study estimates the probability density function of the Federal Risk Management Agency's (RMA) net income from reinsuring crop insurance for corn, wheat, and soybeans. Based on 1997 data, it is estimated that there is a 5 percent probability that RMA will need to reimburse at least $1 billion to insurance companies, and that the fair value of RMA's reinsurance services to insurance firms equals$78.7 million. In addition, various hedging strategies are examined for their potential to reduce RMA's reinsurance risk. The risk reduction achievable by hedging is appreciable, but use of derivative contracts alone is clearly no panacea.


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# How do you solve the heat equation?

Mar 23, 2017

$q = m c \Delta T$
(to remember: q equals m-cat)

#### Explanation:

$q =$ heat energy (Joules)
$m =$ mass (grams)
$c =$ specific heat ($\left(\text{Joules")/("Celsius x grams}\right)$ )
$\Delta T =$change in temperature (Celsius)

specific heat is the amount of heat required to raise the temperature of one gram of substance by one degree Celsius. It is different for every element/compound and is usually given.

$\Delta T = {T}_{\text{final" - T_"initial}}$

Mar 23, 2017

Just in case this was a partial differential equations question instead... the heat equation as I know it is:

$k {\nabla}^{2} u - \frac{\partial u}{\partial t} = 0$ $\text{ "" "" }$ $0 < x < L$, $t > 0$
$u \left(0 , t\right) = u \left(L , t\right) = 0$ $\text{ "" }$ $t > 0$
$u \left(x , 0\right) = f \left(x\right)$ $\text{ "" "" "" }$ $0 < x < L$

where:

• ${\nabla}^{2}$ is the second order derivative in all dimensions considered. Its form depends on what coordinates you are in.
• $k$ is a constant describing the extent of heat conduction.
• $u$ is the function describing the current temperature.
• $\frac{\partial u}{\partial t}$ is the change in temperature over time.
• ${\nabla}^{2} u$ is the change in temperature throughout space.

In one dimension cartesian, it would be:

$\boldsymbol{k \frac{{\partial}^{2} u}{\partial {x}^{2}} - \frac{\partial u}{\partial t} = 0}$

To solve this equation, we would apply separation of variables. Since there is a change in temperature, $\frac{\partial u}{\partial t} \ne 0$. By substituting $u \left(x , t\right) = \phi \left(x\right) G \left(t\right)$, we obtain:

$k \frac{{\partial}^{2} \left(\phi \left(x\right) G \left(t\right)\right)}{\partial {x}^{2}} = \frac{\partial \left(\phi \left(x\right) G \left(t\right)\right)}{\partial t}$

$k G \left(t\right) \frac{{d}^{2} \phi \left(x\right)}{{\mathrm{dx}}^{2}} = \phi \left(x\right) \frac{\mathrm{dG} \left(t\right)}{\mathrm{dt}}$

$\frac{1}{\phi \left(x\right)} \frac{{d}^{2} \phi \left(x\right)}{{\mathrm{dx}}^{2}} = \frac{1}{k G \left(t\right)} \frac{\mathrm{dG} \left(t\right)}{\mathrm{dt}} = - \lambda$

Since the two sides of the equation each depend on only one variabe now, they are each equal to the same constant, known as the "separation constant". We call it $- \lambda$ since $\lambda > 0$ can be shown to yield meaningful answers.

Now we have two ordinary differential equations to solve.

$\frac{1}{\phi \left(x\right)} \frac{{d}^{2} \phi \left(x\right)}{{\mathrm{dx}}^{2}} = - \lambda$

$\frac{1}{k G \left(t\right)} \frac{\mathrm{dG} \left(t\right)}{\mathrm{dt}} = - \lambda$

The first:

$\frac{{d}^{2} \phi \left(x\right)}{{\mathrm{dx}}^{2}} + \lambda \phi = 0$

We assume the solution $\phi = {e}^{r x}$ to get that ${r}^{2} + \lambda = 0$ and thus $r = \pm i \sqrt{\lambda}$ and $\phi = {c}_{1} {e}^{i \sqrt{\lambda}} + {c}_{2} {e}^{- i \sqrt{\lambda}}$. By Euler's formula to get real numbers, we get:

$\textcolor{g r e e n}{\phi \left(x\right) = A \cos \left(\sqrt{\lambda} x\right) + B \sin \left(\sqrt{\lambda} x\right)}$

The second equation gives:

$\frac{\mathrm{dG} \left(t\right)}{\mathrm{dt}} = - \lambda k G$

$\int \frac{1}{G} \mathrm{dG} = - \int \lambda k \mathrm{dt}$

$\ln G \left(t\right) = - \lambda k t + C$

$\textcolor{g r e e n}{G \left(t\right) = A {e}^{- \lambda k t}}$

where $A = {e}^{C}$.

Now that we have two ordinary differential equations solutions, we apply boundary conditions to the space equation to get:

$u \left(0 , t\right) = u \left(L , t\right) = 0$

$\implies \phi \left(0\right) = A \cos \left(\sqrt{\lambda} 0\right) + B \sin \left(\sqrt{\lambda} 0\right) = 0$

Since $\sin \left(0\right) = 0$, to satisfy this condition we must have $A = 0$ so that:

$\phi \left(x\right) = B \sin \left(\sqrt{\lambda} x\right)$

The other boundary condition gives:

$\phi \left(L\right) = B \sin \left(\sqrt{\lambda} L\right) = 0$

To get this, $\sqrt{\lambda} L = n \pi$, so that $\lambda = {\left(\frac{n \pi}{L}\right)}^{2}$. Finally, to satisfy the initial conditions so that $u \left(x , 0\right) = f \left(x\right)$ with no time dependence (the stationary state), we would write a Fourier series as:

$\textcolor{b l u e}{{u}_{n} \left(x , t\right) = {\sum}_{n = 1}^{\infty} {B}_{n} \sin \left(\frac{n \pi x}{L}\right) {e}^{- {\left(\frac{n \pi x}{L}\right)}^{2} k t}}$

where ${B}_{n} = \frac{2}{L} {\int}_{0}^{L} \phi \left(x\right) \sin \left(\frac{n \pi x}{L}\right)$.


text:
OpenGraph DrawingFramework

v. 2022.02 (Dogwood)

ogdf::ShellingOrderModule Class Referenceabstract

Base class for modules that compute a shelling order of a graph. More...

#include <ogdf/planarlayout/ShellingOrderModule.h>

Inheritance diagram for ogdf::ShellingOrderModule:

Public Member Functions

virtual ~ShellingOrderModule ()

double baseRatio () const
Returns the current setting of the option base ratio. More...

void baseRatio (double x)
Sets the option base ratio to x. More...

Computes a shelling order of an embedded graph G such that adj lies on the external face. More...

Computes a lefmost shelling order of an embedded graph G such that adj lies on the external face. More...

Protected Member Functions

virtual void doCall (const Graph &G, adjEntry adj, List< ShellingOrderSet > &partition)=0
This pure virtual function does the actual computation. More...

Protected Attributes

double m_baseRatio

Detailed Description

Base class for modules that compute a shelling order of a graph.

Definition at line 45 of file ShellingOrderModule.h.

◆ ~ShellingOrderModule()

 virtual ogdf::ShellingOrderModule::~ShellingOrderModule ( )
inlinevirtual

Definition at line 72 of file ShellingOrderModule.h.

◆ baseRatio() [1/2]

 double ogdf::ShellingOrderModule::baseRatio ( ) const
inline

Returns the current setting of the option base ratio.

Definition at line 70 of file ShellingOrderModule.h.

◆ baseRatio() [2/2]

 void ogdf::ShellingOrderModule::baseRatio ( double x )
inline

Sets the option base ratio to x.

Definition at line 67 of file ShellingOrderModule.h.

◆ call()

 void ogdf::ShellingOrderModule::call ( const Graph & G, ShellingOrder & order, adjEntry adj = nullptr )

Computes a shelling order of an embedded graph G such that adj lies on the external face.

Parameters
 G is the input graph; G must represent a combinatorial embedding. order is assigned the shelling order. adj is an adjacency entry on the external face; if adj is 0, a suitable external face is chosen.

◆ callLeftmost()

 void ogdf::ShellingOrderModule::callLeftmost ( const Graph & G, ShellingOrder & order, adjEntry adj = nullptr )

Computes a lefmost shelling order of an embedded graph G such that adj lies on the external face.

Parameters
 G is the input graph; G must represent a combinatorial embedding. order is assigned the shelling order. adj is an adjacency entry on the external face; if adj is 0, a suitable external face is chosen.

◆ doCall()

 virtual void ogdf::ShellingOrderModule::doCall ( const Graph & G, adjEntry adj, List< ShellingOrderSet > & partition )
protectedpure virtual

This pure virtual function does the actual computation.

A derived class must implement this method. It is called with the embedded graph and an adjacency entry describing the external face, and must return the computed order in partition.

Parameters
 G is the embedded input graph. adj is an adjacency entry on the external face. partition returns the coputed shelling order.

Implemented in ogdf::BiconnectedShellingOrder, and ogdf::TriconnectedShellingOrder.

◆ m_baseRatio

 double ogdf::ShellingOrderModule::m_baseRatio
protected

Definition at line 88 of file ShellingOrderModule.h.

The documentation for this class was generated from the following file:


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# Find the coorddinates of the point of intersection of the straight lines 2x-3y=1 and 5y-x=3 and determine the angle between them

2080 Points
2 years ago
to find the point of intersection simply solve the two eqns together. 2x-3y=1 and 5y-x=3 gives x= 2 and y= 1.
so coordinates of the point of intersection= (2, 1)
now, the formula for angle between 2 lines is
theta= arctan[|(m1 – m2)/(1+m1*m2)|], where m1 and m2 are the slopes of the lines.
here, m1= 2/3 for 2x-3y=1 and m2= 1/5 for 5y-x=3.
so, theta= arctan[|(m1 – m2)/(1+m1*m2)|]= arctan(7/17) which is approx 22.38 degrees.
kindly approve :)
Vikas TU
14149 Points
2 years ago
Dear student
in equation 1 , a1 = 2 , b1 = -3 ,
in equation 2 a2 = 5 , b2 = -1
a1/a2 is not equal to b1 / b2
so, this is the condition of line to intersect ,
and angle between them = 22.38 degree.
Pawan Kumar Karela
14 Points
2 years ago
Given equation    2x-3y=1   (1)
5y-x=3     (2)
we use the elimination method to get the intersection point.
therefore multiply equation (2) by 2 and then add it to equation (1) to eliminate variable y.
2x-3y=1     (3)
-2x+10y=6  (4)
adding (3) and (4) we get
7y=7
y=1      (*)
use value of y to get x
2x-3*1=1
2x=1+3
x=2
therefore point of intersection is (2,1)

to find angle between these two lines we use formula
$tan\Theta =\left | \frac{m1-m2}{1+m1*m2} \right |$
from equation (1)  2x-3y=1...... implies y=(2/3)x-1
therefore m1=2/3
similarly for equation (2)  y=1/5+3/5
therefore m2=1/5
hence $tan\Theta =\left | \frac{(2/3)-(1/5)}{1+(2/3)*(1/5)} \right |$
therefore on solving we get
$tan\Theta =\left | \frac{7}{17}\right |$
$\Theta =tan^-1\left | \frac{7}{17}\right |$