text: 5.2 Interference 5. Frequency Wavelength Calculator Frequency (hz) is some sort of vibration occurs during a particular period of time as either in a sound waves or in an electromagnetic field. The yellow light given off by sodium vapor lights has a wavelength of 589 nm. anything would help. Light from a sodium lamp ( wavelength= 589 nm) illuminates a narrow slit and is observed on a screen 75 cm behind the slit. Wavelength of monochromatic light, Î» = 589 nm Speed of light, c = 3 × 10 8 m / s Refractive index, Î¼ = 1. For the light of wavelength 589 nm, calculate the critical angles for the following substances when surrounded by air: (a) fused quartz, (b) polystyrene, and (c) sodium chloride. The dstance between the first and third dark fringes is 7.5 mm. What is the width (in mm) of the Determine the angle of refraction Î¸r. Another source produces a line 3.71 cm from the central peak. (b) What is its wavelength in glass whose index of refraction is 1. What is the angular fringe separation if the entire arrangement is immersed in water (n=1.33)? At What Angle Does It Produce Its Second Minimum 0.3537 What Is The Relationship That Shows The Condition For Wavelength of light of a sodium line, Î» = 589 nm = 589 × 10 â9 m Mass of an electron, me= 9.1 × 10 â31 kg Mass of a neutron, mn= 1.66 × 10 â27 kg Planckâs constant, h = 6.6 × 10 â34 Js (a) For the kinetic energy K It passes through glass (n = 1.52) and then through carbon disulfide (n = 1.63). The wavelength of sodium light in air is 589 nm. A narrow ray of yellow light from glowing sodium (Î»0 = 589 nm) traveling in air strikes a smooth surface of water at an angle of Î¸i = 40.5 . The angle of emergence, theta (), for a ) Question: The Wavelength Of Yellow Sodium Light In Air Is 589 Nm. Determine the angle of refraction Î¸r. Wavelength, Î» = 589 nm = What is its frequency? Light from a sodium lamp ( wavelength= 589 nm) illuminates a narrow slit and is observed on a screen 75 cm behind the slit. This question is part of praxis science iii Asked by Koby , â¦ They are: 1 = 588.995 nm 2 = 589.592 nm These would easily be differentiated on a diffraction grating. A light ray of wavelength 589 nm (produced by a sodium lamp) traveling through air is incident on a smooth, flat slab of crown glass at an angle of 30.0º to the normal. 33 (a) For reflected light, Wavelength is going to remain the same for incident and reflected light. Question: (a) Sodium Vapor Light Averaging 589 Nm In Wavelength Falls On A Single Slit Of Width 3.33 Um. Solution for (a) Sodium vapor light averaging 589 nm in wavelength falls on a single slit of width 5.62 µm. The dstance between the first and third dark fringes is 7.5 mm. Thanks The wavelength of yellow sodium light in air is 589nm. Find The Wavelengths Of The Light (a) In The Water And (b) In The Glass. Hopefully this is not a homework question. Calculate the velocity, frequency, and wavelength of the sodium D line (Î» = 589 nm) as light from this source passes through a species whose refractive index, n D, is 1.09. a. a. Solution for A light of wavelength 589 nm (produced by a sodium m lamp) traveling through air is incident a smooth, flat slab of â¦ When yellow sodium light, Î» = 589 nm, falls on a diffraction grating, its first-order peak on a screen 72.0 cm away falls 3.32 cm from the central peak. The wavelength of yellow sodium light in air is 589 nm. From the results of (a) and (b) find its speed in this glass. What Is The Frequency Of The Yellow Light? Click hereðto get an answer to your question ï¸ Monochromatic light of wavelength 589 nm is incident from air on a water surface. Light of wavelength 624 nm is incident perpendicularly on a soap film (n=1.33) suspended in air.What are the (a) least and (b) second least thicknesses of the film for which the reflections from the film undergo fully A monochromatic beam of light in air has a wavelength of 589 nm in air. Consider a double-slit experiment: * Fringe maxima will occur when $d \sin{\theta} = m \lambda$ * Fringe minima will occur when [math] d \sin{\theta} =( m + 1/2) \lambda What are the first-order â¦ ð Give the gift of Numerade. What is its wavelength in glass whose index of refraction is 1.65? 5 2? The wavelength of Sodium light actually consists of two wavelengths called the D lines. The wavelength of sodium light in glass if it is 589.3 nm in air is (The refractive index of glass is 1.5). What is its frequency? Use the relationship c=ÆA to find the wavelength of the light in air. The wavelength of yellow sodium light in air is 589 nm. A double-slit arrangement produces interference fringes for sodium light (wavelength = 589 nm) that are 0.20 degrees apart. (a) What is its frequency? (b) What is its wavelength in glass whose index of refraction is 1.52? asked by John on April 15, 2018 physics I am so lost! If mu for water is 1.33 , find the wavelength, frequency and speed of the refracted light. (a) What is its frequency? (c) From the results of (a) and (b), find its speed in this glass. Wavelength of incident monochromatic light, Î» = 589 nm = 589 × 10 â9 m Speed of light in air, c = 3 × 10 8 m/s Refractive index of water, Î¼ = 1.33 (a) The ray will reflect back in the same medium as that of incident ray. Click hereðto get an answer to your question ï¸ Monochromatic light of wavelength 589 nm is incident from air on a water surface. At what angle does it â¦ A narrow beam of sodium yellow light, with wavelength 589 nm in vacuum, is incident from air onto a smooth water surface at an angle of incidence of 13.0 . What are the wavelength, frequency and speed of(a) Reflected, and(b) Refracted light?Refractive index of water is 1.33. 361 nm b. (c) From the results of (a) and (b), find its speed in this glass. The wavelength of yellow sodium light in air is 589 nm. Blue-Green light has a frequency of about 6x10^14 Hz. How does this wavelength compare with the size of an atom, â¦ What Is 589 Nm In Scientific Notation? Chemistry (Electrons in Atoms) The wavelength Question: 16 Light Of Wavelength 436 Nm In Air Enters A Fish Bowl Filled With Water, Then Exits Through The Crown-glass Wall Of The Container. Light at wavelength 589 nm from a sodium lamp is incident perpendicularly on a grating with 40000 rulings over width 76 \mathrm{nm} . Determine (c) the angle of refraction and (d) the wavelength of the light in (b) Find its wavelength in water (refractive index = 1.33). (a) Find its frequency in air. What is the frequency of yellow sodium light if wavelenght in air is 589 nm? 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# Basic Examples on Definite Integrals Set 2

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Example - 4

Evaluate \begin{align}\int\limits_{ - \infty }^{ + \infty } {\frac{1}{{\left( {{x^2} + {a^2}} \right)\left( {{x^2} + {b^2}} \right)}}\,\,dx}\end{align}

Solution: \begin{align}\frac{1}{{\left( {{x^2} + {a^2}} \right)\left( {{x^2} + {b^2}} \right)}}\end{align} can be rewritten as \begin{align}\frac{1}{{\left( {{b^2} - {a^2}} \right)}}\left\{ {\frac{1}{{{x^2} + {a^2}}} - \frac{1}{{{x^2} + {b^2}}}} \right\}\end{align}

Therefore, the given integral becomes

\begin{align}&\frac{1}{{\left( {{b^2} - {a^2}} \right)}}\left\{ {\int\limits_{ - \infty }^{ + \infty } {\frac{1}{{\left( {{x^2} + {a^2}} \right)}}dx} - \int\limits_{ - \infty }^{ + \infty } {\frac{1}{{\left( {{x^2} + {b^2}} \right)}}dx} } \right\}\\ &= \frac{1}{{\left( {{b^2} - {a^2}} \right)}}\left\{ {\left. {\frac{1}{a}{{\tan }^{ - 1}}\frac{x}{a}} \right|_{ - \infty }^{ + \infty } - \left. {\frac{1}{b}{{\tan }^{ - 1}}\frac{x}{b}} \right|_{ - \infty }^{ + \infty }} \right\}\\ &= \frac{1}{{\left( {{b^2} - {a^2}} \right)}}\left\{ {\frac{\pi }{a} - \frac{\pi }{b}} \right\}\\ &= \frac{\pi }{{ab\left( {a + b} \right)}} \end{align}

Sometimes, to modify an integral, an appropriate substitution has to be used; the same way we did in the unit on Indefinite Integration. For example, integrals containing the expression (x 2 + a 2 ) can be simplified (or modified) using the substitution $$x = a\tan \theta .$$

For evaluating a definite integral too, we can use the appropriate substitution, provided we change the limits of integration accordingly also. This will become clear in subsequent examples.

Example –5

If $${I_n} = \int\limits_0^{x/4} {{{\tan }^n}x\,dx,\,}$$ prove that

(a) $${I_n} + {I_{n - 2}} = \frac{1}{{n - 1}}$$              (b) $$\frac{1}{{n + 1}}\,\,\, < \,\,\,2{I_n} < \frac{1}{{n - 1}}$$

Solution: (a)

\begin{align}&{I_n} + {I_{n - 2}} = \int\limits_0^{\pi /4} {{{\tan }^n}x\,dx} + \int\limits_0^{\pi /4} {{{\tan }^{n - 2}}x\,dx}\\&\qquad\qquad= \int\limits_0^{\pi /4} {\left( {{{\tan }^n}x + {{\tan }^{n - 2}}x} \right)} \,dx\\&\qquad\qquad= \int\limits_0^{\pi /4} {{{\tan }^{n - 2}}x\left( {{{\tan }^2}x + 1} \right)} \,\,dx\\&\qquad\qquad = \int\limits_0^{\pi /4} {{{\tan }^{n - 2}}x\left( {{{\tan }^2}x + 1} \right)} \,\,dx \end{align}

The substitution tan x = t can now be used to simplify this integral. However, we must change the limits of integration according to this substitution:

\begin{align}&\tan x = t\, \quad \Rightarrow \quad {\sec ^2}x\,dx = dt\,\\&{\rm{If}}\,\,\,\,x = 0 \quad\; \Rightarrow \quad t = 0\\& {\rm{If}}\,\,\,\,x = \frac{\pi }{4}\, \quad \Rightarrow \quad t = 1\, \end{align}

Thus, the modified integral (in terms of the new variable t) is:

\begin{align}&{I_n} + {I_{n - 2}} = \int\limits_0^1 {{t^{n - 2}}\,dt} \\&\quad\;\;\left. {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= \frac{{{t^{n - 1}}}}{{n - 1}}} \right|_0^1\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, &\qquad\;\;\;\;\;\;\;\;= \frac{1}{{n - 1}}\end{align}

(b) In the integral that we are considering, the limits of integration are 0 to $$\frac{\pi }{4},\,i.e,\,x \in \left[ {0,\,\,\frac{\pi }{4}} \right]$$

In this interval, tan x < 1. Thus,

${\tan ^{n + 2}}x < {\tan ^n}x < \,{\tan ^{n - 2}}x\,\,\,\,\,\,\forall \,\,\,\,x \in \left( {0,\,\,\frac{\pi }{4}} \right)$

From property (4), we can therefore say that:

$\int\limits_0^{\pi /4} {{{\tan }^{n + 2}}x\,dx < } \int\limits_0^{\pi /4} {{{\tan }^n}x < } \int\limits_0^{\pi /4} {{{\tan }^{n - 2}}x}$

or

\begin{align}&\qquad\;\;\;{I_{n + 2}}\,\, < \,\,{I_n} < {I_{n - 2}}\\ &\Rightarrow\quad {I_n}\,\, + \,\,{I_{n + 2}}\,\, < 2{I_n}\,\,\, < {I_n} + {I_{n - 2}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(1)\end{align}

Using the result of part (a) for the first and third terms in (1), we get our desired result:

$\frac{1}{{n + 1}}\,\, < \,\,2{I_n}\,\,\, < \,\,\,\frac{1}{{n - 1}}$

Definite Integration
Definite Integration
grade 11 | Questions Set 1
Definite Integration
Definite Integration
grade 11 | Questions Set 2


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Contents

# Contents

## Idea

The femtometer is a physical unit of length.

## Definition

A femtometer (abbreviated $fm$) is one $10^{15}$th of a meter:

$fm \coloneqq 10^{-15} m \,.$

The radius of the proton is about $0.8 \, fm$.


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Shear Strength of Soil

 Question 1
The most appropriate triaxial test to assess the long-term stability of an excavated clay slope is
 A consolidated drained test B unconsolidated undrained test C consolidated undrained test D unconfined compression test
GATE CE 2021 SET-2   Geotechnical Engineering
Question 1 Explanation:
To assess the long term stability of clayey soil, the results of consolidated drained (CD) test are used.
 Question 2
Based on drained triaxial shear tests on sands and clays, the representative variations of volumetric strain $(\Delta V / V)$ with the shear strain $(\gamma)$ is shown in the figure.

Choose the CORRECT option regarding the representative behaviour exhibited by Curve P and Curve Q.
 A Curve P represents dense sand and overconsolidated clay, while Curve Q represents loose sand and normally consolidated clay B Curve P represents dense sand and normally consolidated clay, while Curve Q represents loose sand and overconsolidated clay C Curve P represents loose sand and overconsolidated clay, while Curve Q represents dense sand and normally consolidated clay D Curve P represents loose sand and normally consolidated clay, while Curve Q represents dense sand and overconsolidated clay
GATE CE 2021 SET-1   Geotechnical Engineering
Question 2 Explanation:

 Question 3
In a drained tri-axial compression test, a sample of sand fails at deviator stress of 150 kPa under confining pressure of 50 kPa. The angle of internal friction (in degree, round off to the nearest integer) of the sample, is ________
 A 78 B 54 C 37 D 44
GATE CE 2020 SET-1   Geotechnical Engineering
Question 3 Explanation:
$\text{Sand }(C=0); \; \sigma _d=150;\; \sigma _3=50;\; \sigma _1=200$
$\sigma _1=\sigma _3 \tan^2\left ( 45+\frac{\phi }{2} \right )+2c\tan \left ( 45+\frac{\phi }{2} \right )$
$200=50 \tan^2 \left ( 45+\frac{\phi }{2} \right )$
$\phi =36.87^{\circ}$
So, the angle of internal friction to the nearest integer is $37^{\circ}$.
 Question 4
The total horizontal and vertical stresses at a point X in a saturated sandy medium are 170 kPa and 300 kPa, respectively. The static pore-water pressure is 30 kPa. At failure, the excess pore-water pressure is measured to be 94.50 kPa, and the shear stresses on the vertical and horizontal planes passing through the point X are zero. Effective cohesion is 0 kPa and effective angle of internal friction is $36^{\circ}$. The shear strength (in kPa, up to two decimal places) at point X is ______
 A 42.23 B 59.45 C 78.82 D 52.52
GATE CE 2018 SET-2   Geotechnical Engineering
Question 4 Explanation:
\begin{aligned} \because\quad \sigma_{n} &=\frac{300+170}{2}+\frac{300-170}{2} \cos 2 \alpha \\ \alpha &=45^{\circ}+\frac{36^{\circ}}{2}=63^{\circ} \\ \Rightarrow\quad \sigma_{n} &=196.79 \mathrm{kPa} \\ \sigma_{n}^{\prime} &=\sigma_{\mathrm{n}}-u=196.79-(30+94.5) \\ \sigma_{n}^{\prime} &=72.29 \mathrm{kPa} \\ \Rightarrow\quad \tau &=c+\sigma_{\mathrm{n}}^{\prime} \tan \phi \\ &=72.29 \text { tan } 36=52.52 \mathrm{kPa} \end{aligned}
 Question 5
A conventional drained triaxial compression test was conducted on a normally consolidated clay sample under an effective confining pressure of 200 kPa. The deviator stress at failure was found to be 400 kPa. An identical specimen of the same clay sample is isotropically consolidated to a confining pressure of 200 kPa and subjected to standard undrained triaxial compression test. If the deviator stress at failure is 150 kPa, the pore pressure developed (in kPa, up to one decimal place) is ______
 A 100 B 125 C 25 D 75
GATE CE 2018 SET-1   Geotechnical Engineering
Question 5 Explanation:
$I^{st}$ Specimen : Drained condition
$\bar{\sigma}_{3}=200 \mathrm{kPa}: \sigma_{\mathrm{d}}=400 \mathrm{kPa}: \bar{\sigma}_{1}=600 \mathrm{kPa}$
$II^{nd}$Specimen : Undrained condition
\begin{aligned} \sigma_{3} &=200 \mathrm{kPa} \\ \sigma_{\mathrm{d}} &=150 \mathrm{kPa} \\ \sigma_{1} &=\sigma_{3}+\sigma_{\mathrm{d}}=350 \mathrm{kPa} \end{aligned}
Let pore pressure developed is u
\begin{aligned} \therefore \quad \bar{\sigma}_{3}&=(200-u)\\ \bar{\sigma}_{1}&=(350-u)\\ \end{aligned}
From stress relationship
$\bar{\sigma}_{1}=\bar{\sigma}_{3} \tan ^{2}\left(45^{\circ}+\frac{\phi}{2}\right)+2 c\tan\left(45^{\circ}+\frac{\phi}{2}\right)$
For clay under drained condition c=0
\begin{aligned} \therefore \quad \bar{\sigma}_{1} &=\bar{\sigma}_{3} \tan ^{2}\left(45^{\circ}+\frac{\phi}{2}\right) \\ 600 &=200 \tan ^{2}\left(45^{\circ}+\frac{\phi}{2}\right) \\ \phi &=30^{\circ} \end{aligned}
For second specimen,
\begin{aligned} \bar{\sigma}_{1}&=\bar{\sigma}_{3} \tan ^{2}\left(45^{\circ}+\frac{\phi}{2}\right)\\ (350-u) &=(200-u) \tan ^{2}\left(45^{\circ}+\frac{30^{\circ}}{2}\right) \\ u &=125 \mathrm{kPa} \end{aligned}
 Question 6
Following are the statements related to the stress paths in a triaxial testing of soils:
P. If $\sigma _{1}=\sigma _{3}$ , the stress point lies at the origin of the p-q plot.
Q. If, $\sigma _{1}=\sigma _{3}$ the stress point lies on the p-axis of the p-q plot.
R. If $\sigma _{1} \gt \sigma _{3}$ , both the stress points p and q are positive.
For the above statements, the correct combination is
 A P - False; Q - True; R - True B P - False; Q - True; R - True C P - False; Q - True; R - False D P - True; Q - False; R - False
GATE CE 2017 SET-2   Geotechnical Engineering
Question 6 Explanation:
On stress path in terms of p-q plot
\begin{aligned} p=\frac{\sigma_{1}+\sigma_{3}}{2} \\ q=\frac{\sigma_{1}-\sigma_{3}}{2} \end{aligned}
If $\sigma_{1}=\sigma_{3}$ then $q=0$ and $p \neq 0$ hence point does not lie at origin, point will lies on the p-axis.
If $\sigma_{1}>\sigma_{3}$, both the stress point will be positive.
 Question 7
A consolidated undrained $\bar{CU}$ triaxial compression test is conducted on a normally consolidated clay at a confining pressure of 100 kPa. The deviator stress at failure is 80 kPa, and the pore-water pressure measured at failure is 50 kPa. The effective angle of internalfriction (in degrees, up to one decimal place) of the soil is _________
 A 26.4 B 48.2 C 58.1 D 45.6
GATE CE 2017 SET-1   Geotechnical Engineering
Question 7 Explanation:
CU test is conducted.
\begin{aligned} \sigma_{3}= 100 &\mathrm{kPa}, \sigma_{a}=80 \mathrm{kPa}, u=50 \mathrm{kPa} \\ \sigma_{1}&=\sigma_{3}+\sigma_{d} =100+80=180 \mathrm{kPa} \\ \bar{\sigma}_{1}&=\sigma_{1}-u= 180-50=130 \mathrm{kPa}\\ \bar{\sigma}_{3}&=\sigma_{3}-u=100-50=50 \mathrm{kPa} \\ \bar{\sigma}_{1}&=\bar{\sigma}_{3} \tan ^{2}\left(45+\frac{\phi^{\prime}}{2}\right)+2 C^{\prime} \tan \left(45+\frac{\phi^{\prime}}{2}\right) \\ \text{Assume }\quad C^{\prime}&=0 \\ \bar{\sigma}_{1}&=\bar{\sigma}_{3} \tan ^{2}\left(45+\frac{\phi^{\prime}}{2}\right) \\ 130&=50 \tan ^{2}\left(45+\frac{\phi^{\prime}}{2}\right) \\ \phi^{\prime}&=26.4^{\circ} \end{aligned}
 Question 8
A soil sample is subjected to a hydrostatic pressure, $\sigma$. The Mohr circle for any point in the soil sample would be
 A a circle of radius $\sigma$ and center at the origin B a circle of radius $\sigma$ and center at a distance $\sigma$ from the origin C a point at a distance $\sigma$ from the origin D a circle of diameter $\sigma$ and center at the origin
GATE CE 2017 SET-1   Geotechnical Engineering
Question 8 Explanation:
Hydrostatic pressure acts equally in all directions.

 Question 9
A drained triaxial compression test on a saturated clay yielded the effective shear strength parameters as ${c}'=15\, kPa \: \, and \: {\phi }'=22^{\circ}.$ Consolidated Undrained triaxial test on an identical sample of this clay at a cell pressure of 200 kPa developed a pore water pressure of 150kPa at failure. The deviator stress (expressed in kPa) at failure is _________ .
 A 104.38 B 50.25 C 154.39 D 160.22
GATE CE 2016 SET-1   Geotechnical Engineering
Question 9 Explanation:
Cell pressure,
$\sigma_{3}=200 \mathrm{kPa}$
Pore water pressure.
\begin{aligned} u &=150 \mathrm{kPa} \\ \therefore \bar{\sigma}_{3} &=\sigma_{3}-U=200-150=50 \mathrm{kPa} \end{aligned}
Effective shear parameters $\left(c^{\prime}=150 \mathrm{kPa}\text{ and } \phi^{\circ}=22^{\circ}\right)$ is given
\begin{aligned} \therefore \quad \bar{\sigma}_{1} &=\bar{\sigma}_{3} \tan ^{2}\left(45+\frac{\varphi^{\prime}}{2}\right)+2 c^{\prime} \tan \left(45+\frac{\phi^{\prime}}{2}\right) \\ \bar{\sigma}_{1} &=50+\tan ^{2}\left(45+\frac{22}{2}\right)+2 \times 15 \tan \left(45+\frac{22}{2}\right) \end{aligned}
$\bar{\sigma}_{1}=154.376 \mathrm{kPa}$
Now
$\begin{array}{l} \vec{\sigma}_{1}=\bar{\sigma}_{3}+\sigma_{\sigma} \\ \sigma_{d}=\bar{\sigma}_{1}-\bar{\sigma}_{3}=154.376-50 \end{array}$
$\text{Deviator stress, }\sigma_{\sigma}=104.376 \mathrm{kPa}$
 Question 10
In the consolidated undrained triaxial test on a saturated soil sample, the pore water pressure is zero
 A during shearing stage only B at the end of consolidation stage only C both at the end of consolidation and during shearing stages D under none of the above conditions
GATE CE 2016 SET-1   Geotechnical Engineering
There are 10 questions to complete.

1 thought on “Shear Strength of Soil”

1. Question 17 is not of soil but of environmental
Thank u for ur efforts!


text:
Reverse chain rule introduction More free lessons at: http://www.khanacademy.org/video?v=X36GTLhw3Gw Need to review Calculating Derivatives that don’t require the Chain Rule? To interpolate the y 2 value: x 1, x 3, y 1 and y 3 need to be entered/copied from the table. If we want to integrate a function that contains both the sum and difference of a number of terms, the main points to remember are that we must integrate each term separately, and be careful to conserve the order in which the terms appear. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. Skip to navigation (Press Enter) Skip to main content (Press Enter) Home; Threads; Index; About; Math Insight. The Power Rule of Derivatives applies to find the power of more than two functions. Free partial derivative calculator - partial differentiation solver step-by-step Nonetheless, we must ask for what kind of transactions the reverse charge is actually designed. by the Chain Rule, dy/dx = dy/dt × dt/dx so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x = 6x(1 + x²) ². In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Instead, the derivatives have to be calculated manually step by step. So all we need to do is to multiply dy /du by du/ dx. Reverse charges apply to all shipments or services supplied to UK-based companies by suppliers located overseas (also prescribed to some cases of domestic trade). To get chain rules for integration, one can take differentiation rules that result in derivatives that contain a composition and integrate this rules once or multiple times and rearrange then. T3 Net Payable The County Council should pay Revenue €135,000 VAT. The Reverse Chain Rule. This is f prime of x so I can just apply the reverse chain rule. In examples such as the above one, with practise it should be possible for you to be able to simply write down the answer without having to let t = 1 + x² etc. The above Calculator computes a derivative of a given function with respect to a variable x using analytical differentiation. Bureau Veritas develops Rules, Rule Notes and Guidance Notes to benefit our clients, partners, and the marine industry overall. Step 3: Apply the power function rule for integration: Which gives you: Step 4: Substitute “u” back in and simplify: Step 5: Add a “C”: Back to Top. We use the vast experience of our engineers, gained through years of service in the marine industry, combined with advanced research into structure and hydrodynamics, to… The County Council should calculate the VAT at 13.5%, €135,000 and show this as VAT on sales (T1) in its January/February 2019 VAT return. This figure also helps us understand why these equations are called “the lever rule.” Sometimes it is also known as the “reverse arm rule,” because for the calculation of α L (liquid) you use the “arm” within the (y A-x A) segment closest to the vapor, and for the vapor calculation (α … According to experts, doing so should be in anyone’s “essential skills” checklist. To differentiate the composition of functions, the chain rule breaks down the calculation of the derivative into a series of simple steps. y 2 is the interpolated value and solution. The plus or minus sign in front of each term does not change. If you learned your derivatives well, this technique of integration won't be a stretch for you. Example 1; Example 2; Example 3; Example 4; Example 5; Example 6; Example 7; Example 8 ; In threads. €135,000.00 T4 Net Repayable 4. This is going to be, we deserve a little mini drum roll here, this is going to be equal to the natural log of the absolute value of our f of x, which is going to be cosine of x. Derivative Rules. The inner function is the one inside the parentheses: x 4-37. €135,000.00 T2 VAT on Purchases The County Council is not entitled to any VAT deduction, so it should show VAT on purchases as "Nil". The sum and difference rules are essentially the same rule. Chain rule for conditional probability: Let us write the formula for conditional probability in the following format $$\hspace{100pt} P(A \cap B)=P(A)P(B|A)=P(B)P(A|B) \hspace{100pt} (1.5)$$ This format is particularly useful in situations when we know the conditional probability, but we are interested in the probability of the intersection. Top; Examples. The chain rule says that. Instead of using the reverse power rule, you should just memorize that the antiderivative of (rule 3 in the table). So when using the chain rule: Derivative calculator finds derivative of sin, cos and tan. Our inverse function calculator uses derivative formula to solve derivative of trig functions. Linear Interpolation Equation Calculator Engineering - Interpolator Formula. Answer to 2: Differentiate y = sin 5x. Solution: y 2 = NOT CALCULATED. Product Rule Calculator is a free online tool that displays the differentiation of a given function when one function is multiplied with another function. How to Use the Product Rule Calculator? U Substitution for Trigonometric Functions. That material is here. In this form, the chain rule could be applied repeatedly to every input variable $$u$$, ... allowing the tape / expression graph to be re-used for multiple reverse-mode AD calculations (in case we have multiple output variables). That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite f ∘ g — the function which maps x to (()) — in terms of the derivatives of f and g and the product of functions as follows: (∘) ′ = (′ ∘) ⋅ ′. Submit it here! In calculus, the chain rule is a formula to compute the derivative of a composite function. Identifying the Change of Variables for U-Substitution. Get the free "Composite Function Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. x 1: y 1: x 2: y 2: x 3: y 3: Solving for y 2. If you don’t understand why, you can find intermediate steps with algebraic formulas on Symbolab’s integral calculator. Show Instructions. Step 1: Identify the inner and outer functions. For an example, let the composite function be y = √(x 4 – 37). The Chain Rule mc-TY-chain-2009-1 A special rule, thechainrule, exists for diﬀerentiating a function of another function. As part of the established rules, EMU member-nations could no longer use the inverse of the officially published rates. Topic 6: Calculus Differentiation Integration Further methods of integration Kinematics There must be an element of labour involved, but then the reverse charge extends to the whole invoice, including the cost of materials supplied. Well, U-Sub is nothing more than the reverse of the chain rule! Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Find more Mathematics widgets in Wolfram|Alpha. Solve your math problems using our free math solver with step-by-step solutions. The reverse charge doesn’t apply when the supply consists only of materials. Learning mathematics is definitely one of the most important things to do in life. Free online calculator that allows you to dynamically calculate the differential equation. It doesn’t apply to work that is zero-rated. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. This unit illustrates this rule. Please note that the letters "W" and "F" denote the constant values truth and falsehood and that the lower-case letter "v" denotes the disjunction. Calculus Calculator: Learn Limits Without a Limit! Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself. x 2 defines the point to perform the interpolation. There are rules we can follow to find many derivatives.. For example: The slope of a constant value (like 3) is always 0; The slope of a line like 2x is 2, or 3x is 3 etc; and so on. Chain Rule: Problems and Solutions. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). But how do we do this? Chain Rule Examples: General Steps. However, the mechanism applies only if the beneficiary is either an entrepreneur or a corporate entity. The JD Edwards World term for this requirement is the no inverse rule. Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. Here are useful rules to help you work out the derivatives of many functions (with examples below). Great! 2. As u = 3x − 2, du/ dx = 3, so. The reverse power rule works for all powers (including negative and decimal powers) except for a power of negative one. An "impossible problem"? The no inverse rule minimizes the possibility of rounding differences that can sometimes occur when using the divisor or multiplier method of exchange rate calculation. The derivs array contains all the adjoints / derivatives at the same index as its associated node. The Derivative tells us the slope of a function at any point.. Integration by substitution is just the reverse chain rule. What? We will need to make a change of variables. Displaying the steps of calculation is a bit more involved, because the Derivative Calculator can't completely depend on Maxima for this task. Page Navigation. BYJU’S online product rule calculator tool makes the calculation faster, and it displays the differentiation of the function in a fraction of seconds. Solve your math problems using our free math solver with step-by-step solutions. Find more Mathematics widgets in Wolfram|Alpha. The reverse charge rules only apply to work that is subject to VAT at either 5% or 20%. Logic calculator: Server-side Processing Help on syntax - Help on tasks - Other programs - Feedback - Deutsche Fassung Examples and information on the input syntax. Get the free "Double Integral Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. According to Paul’s Online notes, the essence of the substitution rule is to take an integral in terms of X’s and transform or change it into terms of U’s. Let u = 5x (therefore, y = sin u) so using the chain rule. How? Inputs: x 1. x 2. x 3. y 1. y 3. Counting is crucial, and The only problem is that we want dy / dx, not dy /du, and this is where we use the chain rule. Do you have a question or doubt about this topic? Test your antiderivatives by differentiating them. The Reverse Chain Rule. For some kinds of integrands, this special chain rules of integration could give … The calculator will find the gradient of the given function (at the given point if needed), with steps shown. Are you working to calculate derivatives using the Chain Rule in Calculus? Subject to VAT at either 5 % or 20 % is vital that you undertake plenty of practice exercises that... Substitution is just the reverse charge doesn ’ t apply to work that is.... 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# Models and neuron groups¶

## Model equations¶

The core of every simulation is a NeuronGroup, a group of neurons that share the same equations defining their properties. The minimum NeuronGroup specification contains the number of neurons and the model description in the form of equations:

G = NeuronGroup(10, 'dv/dt = -v/(10*ms) : volt')

This defines a group of 10 leaky integrators. The model description can be directly given as a (possibly multi-line) string as above, or as an Equations object. For more details on the form of equations, see Equations. Brian needs the model to be given in the form of differential equations, but you might see the integrated form of synapses in some textbooks and papers. See Converting from integrated form to ODEs for details on how to convert between these representations.

Note that model descriptions can make reference to physical units, but also to scalar variables declared outside of the model description itself:

tau = 10*ms
G = NeuronGroup(10, 'dv/dt = -v/tau : volt')

If a variable should be taken as a parameter of the neurons, i.e. if it should be possible to vary its value across neurons, it has to be declared as part of the model description:

G = NeuronGroup(10, '''dv/dt = -v/tau : volt
tau : second''')

To make complex model descriptions more readable, named subexpressions can be used:

G = NeuronGroup(10, '''dv/dt = I_leak / Cm : volt
I_leak = g_L*(E_L - v) : amp''')

For a list of some standard model equations, see Neural models (Brian 1 –> 2 conversion).

## Noise¶

In addition to ordinary differential equations, Brian allows you to introduce random noise by specifying a stochastic differential equation. Brian uses the physicists’ notation used in the Langevin equation, representing the “noise” as a term $$\xi(t)$$, rather than the mathematicians’ stochastic differential $$\mathrm{d}W_t$$. The following is an example of the Ornstein-Uhlenbeck process that is often used to model a leaky integrate-and-fire neuron with a stochastic current:

G = NeuronGroup(10, 'dv/dt = -v/tau + sigma*xi*tau**-0.5 : volt')

You can start by thinking of xi as just a Gaussian random variable with mean 0 and standard deviation 1. However, it scales in an unusual way with time and this gives it units of 1/sqrt(second). You don’t necessarily need to understand why this is, but it is possible to get a reasonably simple intuition for it by thinking about numerical integration: see below.

## Threshold and reset¶

To emit spikes, neurons need a threshold. Threshold and reset are given as strings in the NeuronGroup constructor:

tau = 10*ms
G = NeuronGroup(10, 'dv/dt = -v/tau : volt', threshold='v > -50*mV',
reset='v = -70*mV')

Whenever the threshold condition is fulfilled, the reset statements will be executed. Again, both threshold and reset can refer to physical units, external variables and parameters, in the same way as model descriptions:

v_r = -70*mV  # reset potential
G = NeuronGroup(10, '''dv/dt = -v/tau : volt
v_th : volt  # neuron-specific threshold''',
threshold='v > v_th', reset='v = v_r')

You can also create non-spike events. See Custom events for more details.

## Refractoriness¶

To make a neuron non-excitable for a certain time period after a spike, the refractory keyword can be used:

G = NeuronGroup(10, 'dv/dt = -v/tau : volt', threshold='v > -50*mV',
reset='v = -70*mV', refractory=5*ms)

This will not allow any threshold crossing for a neuron for 5ms after a spike. The refractory keyword allows for more flexible refractoriness specifications, see Refractoriness for details.

## State variables¶

Differential equations and parameters in model descriptions are stored as state variables of the NeuronGroup. They can be accessed and set as an attribute of the group. To get the values without physical units (e.g. for analysing data with external tools), use an underscore after the name:

>>> G = NeuronGroup(10, '''dv/dt = -v/tau : volt
...                        tau : second''')
>>> G.v = -70*mV
>>> G.v
<neurongroup.v: array([-70., -70., -70., -70., -70., -70., -70., -70., -70., -70.]) * mvolt>
>>> G.v_  # values without units
<neurongroup.v_: array([-0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07, -0.07])>

The value of state variables can also be set using string expressions that can refer to units and external variables, other state variables, mathematical functions, and a special variable i, the index of the neuron:

>>> G.tau = '5*ms + (1.0*i/N)*5*ms'
>>> G.tau
<neurongroup.tau: array([ 5. ,  5.5,  6. ,  6.5,  7. ,  7.5,  8. ,  8.5,  9. ,  9.5]) * msecond>

You can also set the value only if a condition holds, for example:

>>> G.v['tau>7.25*ms'] = -60*mV
>>> G.v
<neurongroup.v: array([-70., -70., -70., -70., -70., -60., -60., -60., -60., -60.]) * mvolt>

## Subgroups¶

It is often useful to refer to a subset of neurons, this can be achieved using Python’s slicing syntax:

G = NeuronGroup(10, '''dv/dt = -v/tau : volt
tau : second''',
threshold='v > -50*mV',
reset='v = -70*mV')
# Create subgroups
G1 = G[:5]
G2 = G[5:]

# This will set the values in the main group, subgroups are just "views"
G1.tau = 10*ms
G2.tau = 20*ms

Here G1 refers to the first 5 neurons in G, and G2 to the second 5 neurons. In general G[i:j] refers to the neurons with indices from i to j-1, as in general in Python. Subgroups can be used in most places where regular groups are used, e.g. their state variables or spiking activity can be recorded using monitors, they can be connected via Synapses, etc. In such situations, indices (e.g. the indices of the neurons to record from in a StateMonitor) are relative to the subgroup, not to the main group

The following topics are not essential for beginners.

## Shared variables¶

Sometimes it can also be useful to introduce shared variables or subexpressions, i.e. variables that have a common value for all neurons. In contrast to external variables (such as Cm above), such variables can change during a run, e.g. by using run_regularly(). This can be for example used for an external stimulus that changes in the course of a run:

G = NeuronGroup(10, '''shared_input : volt (shared)
dv/dt = (-v + shared_input)/tau : volt
tau : second''')

Note that there are several restrictions around the use of shared variables: they cannot be written to in contexts where statements apply only to a subset of neurons (e.g. reset statements, see below). If a code block mixes statements writing to shared and vector variables, then the shared statements have to come first.

By default, subexpressions are re-evaluated whenever they are used, i.e. using a subexpression is completely equivalent to substituting it. Sometimes it is useful to instead only evaluate a subexpression once and then use this value for the rest of the time step. This can be achieved by using the (constant over dt) flag. This flag is mandatory for subexpressions that refer to stateful functions like rand() which notably allows them to be recorded with a StateMonitor – otherwise the monitor would record a different instance of the random number than the one that was used in the equations.

For shared variables, setting by string expressions can only refer to shared values:

>>> G.shared_input = '(4.0/N)*mV'
>>> G.shared_input
<neurongroup.shared_input: 0.4 * mvolt>

## Storing state variables¶

Sometimes it can be convenient to access multiple state variables at once, e.g. to set initial values from a dictionary of values or to store all the values of a group on disk. This can be done with the get_states() and set_states() methods:

>>> group = NeuronGroup(5, '''dv/dt = -v/tau : 1
...                           tau : second''')
>>> initial_values = {'v': [0, 1, 2, 3, 4],
...                   'tau': [10, 20, 10, 20, 10]*ms}
>>> group.set_states(initial_values)
>>> group.v[:]
array([ 0.,  1.,  2.,  3.,  4.])
>>> group.tau[:]
array([ 10.,  20.,  10.,  20.,  10.]) * msecond
>>> states = group.get_states()
>>> states['v']
array([ 0.,  1.,  2.,  3.,  4.])

The data (without physical units) can also be exported/imported to/from Pandas data frames (needs an installation of pandas):

>>> df = group.get_states(units=False, format='pandas')
>>> df
N      dt  i    t   tau    v
0  5  0.0001  0  0.0  0.01  0.0
1  5  0.0001  1  0.0  0.02  1.0
2  5  0.0001  2  0.0  0.01  2.0
3  5  0.0001  3  0.0  0.02  3.0
4  5  0.0001  4  0.0  0.01  4.0
>>> df['tau']
0    0.01
1    0.02
2    0.01
3    0.02
4    0.01
Name: tau, dtype: float64
>>> df['tau'] *= 2
>>> group.set_states(df[['tau']], units=False, format='pandas')
>>> group.tau
<neurongroup.tau: array([ 20.,  40.,  20.,  40.,  20.]) * msecond>

A NeuronGroup can define parameters that are not stored in this group, but are instead a reference to a state variable in another group. For this, a group defines a parameter as linked and then uses linked_var() to specify the linking. This can for example be useful to model shared noise between cells:

inp = NeuronGroup(1, 'dnoise/dt = -noise/tau + tau**-0.5*xi : 1')

neurons = NeuronGroup(100, '''noise : 1 (linked)
dv/dt = (-v + noise_strength*noise)/tau : volt''')
neurons.noise = linked_var(inp, 'noise')

If the two groups have the same size, the linking will be done in a 1-to-1 fashion. If the source group has the size one (as in the above example) or if the source parameter is a shared variable, then the linking will be done as 1-to-all. In all other cases, you have to specify the indices to use for the linking explicitly:

# two inputs with different phases
inp = NeuronGroup(2, '''phase : 1
dx/dt = 1*mV/ms*sin(2*pi*100*Hz*t-phase) : volt''')
inp.phase = [0, pi/2]

neurons = NeuronGroup(100, '''inp : volt (linked)
dv/dt = (-v + inp) / tau : volt''')
# Half of the cells get the first input, other half gets the second
neurons.inp = linked_var(inp, 'x', index=repeat([0, 1], 50))

## Time scaling of noise¶

Suppose we just had the differential equation

$$dx/dt=\xi$$

To solve this numerically, we could compute

$$x(t+\mathrm{d}t)=x(t)+\xi_1$$

where $$\xi_1$$ is a normally distributed random number with mean 0 and standard deviation 1. However, what happens if we change the time step? Suppose we used a value of $$\mathrm{d}t/2$$ instead of $$\mathrm{d}t$$. Now, we compute

$$x(t+\mathrm{d}t)=x(t+\mathrm{d}t/2)+\xi_1=x(t)+\xi_2+\xi_1$$

The mean value of $$x(t+\mathrm{d}t)$$ is 0 in both cases, but the standard deviations are different. The first method $$x(t+\mathrm{d}t)=x(t)+\xi_1$$ gives $$x(t+\mathrm{d}t)$$ a standard deviation of 1, whereas the second method $$x(t+\mathrm{d}t)=x(t+\mathrm{d}/2)+\xi_1=x(t)+\xi_2+\xi_1$$ gives $$x(t)$$ a variance of 1+1=2 and therefore a standard deviation of $$\sqrt{2}$$.

In order to solve this problem, we use the rule $$x(t+\mathrm{d}t)=x(t)+\sqrt{\mathrm{d}t}\xi_1$$, which makes the mean and standard deviation of the value at time $$t$$ independent of $$\mathrm{d}t$$. For this to make sense dimensionally, $$\xi$$ must have units of 1/sqrt(second).

For further details, refer to a textbook on stochastic differential equations.


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IMR Press / CEOG / Volume 49 / Issue 1 / DOI: 10.31083/j.ceog4901017
Open Access Case Report
Fertile window and biophysical biomarkers of cervical secretion in subfertile cycles: a look at biotechnology applied to NaProTechnology
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1 Unit of Restaurative Roproductive Medicine, Clinical Consulting G&E, 26002 Logroño, La Rioja, Spain
2 Computer Science, Engineering Department, University of La Rioja, 26005 Logroño, La Rioja, Spain
3 Le Moyne College, Biology Department, Syracuse University, New York, NYC 10007, USA
4 Materials Science Area, Engineering Department, University of La Rioja, 26005 Logroño, Spain
5 Engineering Department, University of La Rioja, 26005 Logroño, Spain
6 Engineering Department, University Central of Colombia, 110311 Bogotá, Colombia
7 Physics and Applied Mathematics Department, University of Navarra Pamplona, 31003 Pamplona, Spain
8 Department of Obstetrics and Gynecology, Clínica Universidad de Navarra, 31003 Pamplona, Spain

Clin. Exp. Obstet. Gynecol. 2022 , 49(1), 1; https://doi.org/10.31083/j.ceog4901017
Submitted: 21 January 2021 | Revised: 9 March 2021 | Accepted: 13 April 2021 | Published: 13 January 2022
This is an open access article under the CC BY 4.0 license.
Abstract

Background: The principal objective of this study was to correlate biophysical properties of vaginal discharge present in the cervical mucus with the timing of the fertile window. In particular, we produce measures of the viscoelasticity of the cervical secretion using two methods. The first uses only the elasticity extracted from the Creighton Model Fertility Care System (CrMs) scale, calculated P-6 ovulation estimated day (OED) with respect to the peak day of the CrMs. The second uses a numerical method that takes into account the changes in viscoelasticity, but without reference to the peak day calculated using the CrMs model. Using both methods, twelve records were obtained from a single female subject. Methods: The methodology used to evaluate the viscoelasticity factor was by measuring the approximate length in centimeters (cm) of the vaginal discharge of cervical discharge. For this, the scale of the stretching graph established by observing the stretching of CrMS was used, taking into account the previous 6 days at peak day P-6. The first method, which we termed CFW (Clinical Fertile Window), uses a measure based on the approximate length (cm) of the maximal stretchiness of the vaginal discharge. The second method we termed SFW (Software-CrMS/strectching) (Software-based Fertile Window). Results:The fertile window was detected correctly in 100% of the cases using either method, and a correlation value of 0.71 was observed between the two methods. Conclusions: We conclude that the assessment of viscoelasticity using SFW algorithm allowed in this pilot study to detect the fertile window and to describe the evolution pattern of cervical discharge throughout the fertile window. Our study provides support for the use of computational methods in detecting the fertile window, taking only into account the time evolution of the cervical discharge throughout the menstrual cycle.

Keywords
Fertile window
NaProTechnology
Cervical secretion
Ovulation
Fertility awareness
Subfertility
Biophysical biomarkers
Biotechnology
1. Introduction

Determination of the fertile window through the observation of the length of the menstrual cycle is an already established procedure. Each month, the ovary has a cohort of antral follicles sensitive to a certain concentration of follicle-stimulating hormone (FSH) [1]. At the follicular stage, the antral follicles need to be sensitive to the action of FSH, in order to be rescued from the follicular pool, and have receptors to FSH for adequate aromatization to occur in the antral follicle [2]. These physiological changes are decisive for an increase in the production of estrogens. Thus, around 5 to 12 days of the menstrual cycle, the preantral follicle presents a significant increase in the number of stromal cells [3-5]. This event is essential for endocervical cells to become stimulated to produce type-L and type-S secretions. As a result, a significant increase in the biophysical properties of cervical secretion occurs around the estimated day of ovulation (OED), changes which correlate with the progressive elevation of steroid derivatives in blood and urine, in synchrony with follicular development [6-9].

Herein we review the main concepts relevant to understanding the changes observed in the cervical secretion, focusing on the production of estrogen by endocervical cells. In particular, we review the basic physiology of ovulation concerning the evolution of the biophysical characteristics of the cervical secretion with the fertile window, and its impact for the processes of follicular growth, selection, recruitment and maturation.

The purpose of this work is to review the role of the viscoelasticity and transparency of cervical mucus can play as signals and biophysical markers of changes in cervical secretion properties in order to recognize the occurrence of the fertile window. Viscoelasticity often refers to many terms that describe the rheological properties of cervical mucus, including viscosity, elasticity, plasticity, shrinkage, adherence, spinbarkheit and stretchiness, among others. The ability to lengthen or stretch the cervical discharge has been one of the classical parameters used to assess the progression of cervical discharge throughout the menstrual cycle. That the elasticity is greater on the days preceding the peak day, and during the peak day, has been reported in previous studies [10-17]. Accordingly, assessing the biophysical properties through the CrMs method [10] is well supported by the literature. Thus, it has been possible to recognize the viscoelasticity of the cervical discharge as a clinical sign to detect the fertile window [17-19].

This allows us to systematically track the progressive changes related to the fertile window, from the left kurtosis phase to the postovulatory phase, after the peak day, in both normal and subfertile cycles [10]. Using this scale as a normalizing artifact, we are able to gauge the effectiveness of our detection methods as applied to ovulatory cycles in subfertile populations.

Therefore, it is possible to apply concepts derived from ovulatory cycles in subfertile patients. It is estimated that around 12% to 15% of couples may experience infertility. The fertile window in subfertile patients has proven efficacy in spontaneous pregnancies [14]. The changes observed in the biophysical parameters of the cervical secretion: as well as the volume, the viscoelasticity or spinnbarkheit, the transparency, and the crystallization of the cervical secretion, are likely to be used to identify the fertile window in subfertile patients [20,21].

2. Materials and methods
2.1 Menstrual cycles

We analyzed biophysical parameters obtained from observations of a total of 12 menstrual cycles that occurred in a 29-year old female patient between 2017 and 2018. The subject gave their informed consent for inclusion before they participated in the study, which was conducted in strict accordance with the Declaration of Helsinki, and whose protocol was approved by the Ethics Committee of CEImLAR, Center for Biomedical Research of La Rioja (CIBIR) (approval number P.I.339).

The shortest cycle was 29 days, and the longest was 39 days. The first day of the cycle was defined as the first day of menstruation, and the last day of the cycle, the day before the start of the next menstruation. Twelve records of the typical evolution of cervical mucus were documented by recording the charts of the observation of vaginal discharge from cervical secretion with the Creighton Model Fertility Care System (CrMS).

A single individual carried out extraction and analysis of data obtained from the survey records. This was done to reduce inter-observer variability because the clinical evidence taken into account in the measurements carried out by the patient were then extracted and corroborate by the same single individual who performed the clinical study.

The patient received a clinical diagnosis of infertility after more than one year of regular sexual intercourse without contraception. During initial diagnosis, we ruled out various gynecological pathologies that could affect follicular development, such as hipertiroidism, hipotiroidism, hiperprolactinemia, hirsutism, ehiperandroegenemia, and ovarian polycystic syndrome, as well as other functional causes as anorexia and obesity.

A basic infertility study was performed, in which no detectable cause of female sterility was documented, including the exploration of the genital tract, transvaginal echography, Pap smear, colposcopy, histerosalpinografy and analytical functional test in which no detectable causes of female sterility were detected. The presence of male factor of treated oligoastenozoospermia was confirmed without significant improvement despite an effective sperm count (ESC) of 627.000 sperm.

The medical examination also included a follicular follow-up study, which was carried out at the beginning of the clinical evaluation.

Signs compatible with ultrasound ovulation were documented by observing the sign of the double peri-follicular contour, follicular rupture and emptying registered by the presence of irregular walls and mixed echo images inside the follicle, and postovulatory mixed echo image compatible with corpus luteum was verified.

2.2 Criteria for establishing clinical fertile window and software-based fertile window

The objective of the study was to model the evolution and progression of the viscoelasticity of the cervical secretion in the fertile window of ovulatory menstrual cycles using a computational method. We validate the method by means of the elasticity of the cervical mucus, as measured in cm according to the CrMs scale, taking into account the data calculated from day P-6 as estimated day of ovulation (OED) retrospectively. The computational method accounts for changes in the measurement of viscoelasticity of the vaginal discharge of cervical secretion throughout the menstrual cycle. The method also uses the CrMs scale, but does not take into account the peak day calculated from the CrMS model. The Spinbarkheit assessment of cervical discharge was coded using the following guidelines: A: Sticky: 0.5 cm (Less than 0.65 cm, equivalent to ¼ inch). B: Tacky: 1 cm, equivalent to the lower value of the CrMS classification interval between (1–2 cm); (1.27–1.905 cm), which corresponds to (½–¾ inch for the CrMS system). C: Stretchy: 2.5 cm was recorded when the yarn was equal to or greater than 1 inch. The days on which there were dryness and no secretion were tabulated as 0. Table 1 summarizes the scale used to evaluate the parameter of viscoelasticity. Fig. 1 describes the distribution of the mean viscoelasticity values in cm throughout the menstrual cycle. Within the methodology used by the algorithm, the aforementioned scale was chosen, because this scale according to CrMS describes better than other scales, the variation that occurs from the follicular phase of cervical secretion. In this way, it was possible to record the progressive evolutionary change in a systematic way from the left kurtosis of the fertile window to the postovulatory phase after the peak day as seen in Graph 1. The clinical window in this article was established as CFW: (CrMS/stretching) by means of the retrospective assessment of the mentioned scale taking into account the 6 days prior to the peak day: P-6. The methodology used consisted in comparing the two fertile windows with ODE. To check the evolution in the characterization of the viscoelasticity parameter of the cervical secretion, the last day of maximum fertility was established, with the letter “P” for the peak day, according to the recognition rules for the Peak day of CrMS. The correlation test was performed to calculate the percentage of coincidence of the intervals, and the points of change (POC) between both windows. The interval between both windows was evaluated CFW: (CrMS/stretching) and SFW: (Soft-CrMS/stretching) by means of a graph of middles points, and correlation test of overlap of intervals. The correlation test was performed to calculate the percentage of coincidence of the intervals, and the points of change (POC) between both windows.

Table 1.Spinbarkheit and cervical secretion scale extracted from creighton model fertility care system (CrMS)* for CFW: (CrMS/stretching).
 A: Sticky: 0.5 cm CrMS: (Less than 0.65 cm / ¼ inch) B: Tacky: 1 cm CrMS: (1.27–1.905 cm / ½–¾ inch) C: Stretchy: 2.5 cm CrMS (2.5 or greater / 1 inch or greater) * Hilgers TW. The Medical & Surgical Practice of NaProTECHNOLOGY. First Edition July 2004. Published by: Pope Paul VI Institute Press 6901 Mercy Road Omaha, Nebraska 68106 USA.
Fig. 1.

The mean values of the stretching data of each day of the menstrual cycle were represented. The evolution of the stretching according to A: Sticky: 0.5 cm is recorded. Tacky: 1 cm. Strechy 2.5 cm: The typical course of cervical mucus is documented by observing the vaginal discharge of cervical discharge (VDRS) based on the classification system of the Creighton Model Fertility Care System (CrMS).

The maximum transparency was registered with the letter “k” extracted from the CrMS registry, which was not taken into account to define, nor was the clinical fertile window CFW: (CrMS/strectching), nor ODE nor was it considered in counts in the SFW calculation: (Soft-CrMS/stretching). The k “clear” is the maximum transparency standard on the CrMs scale. To assess the contribution of the variants of the peak day, Last Max Spinbarkheit (LMS) was considered, for which the estimation of the variations of Peak Day was taken into account in a descriptive way through the concept of: LMS & Transparency by “K” from CrMS. Different from the Peak Day concept, which is defined as any type of discharge that is clear, stretches or has isolated lubricative properties, or in combination with each other, according to CrMS.

2.3 Software implementation phase and statistical study

The software to calculate SFW was made up of two paired applications following an object-orientation design, and programmed in C# and Java. Unit tests were done using JUnit and NUnit. The first application, running under Windows 10, was used to collect the measurements and store them in a database. The second application followed a three-tier design. The first tier supported data access and included mechanisms for data query, modification and insertion given the database constructed by the first application.

The second tier implements all necessary functions to manipulate the logical entities abstracted from the data model. The third layer implemented the presentation logic, which allows the end user to interface with the system. Finally, an exhaustive study was carried out to find out the necessary algorithms, also carrying out a formal verification. A normality test was carried out to describe the study sample with the Shapiro-Wilk test. The correlation study was performed with Excel in conjunction with “R”; environment and programming language focused on statistical analysis of the correlation test. An intuitive graphical interface was developed.

3. Results

The fertile window using CFW: (CrMS/stretching) it was possible to detect them in 100% of the cycles. Using the present series, it was possible to evaluate the elasticity of the vaginal discharge by estimating the length in cm. The CFW: (CrMS/stretching); P-6 interval was distributed between days 10 and 25 of the menstrual cycle as shown in Fig. 1. In Fig. 2, the records that make up each cycle with the two fertile windows are observed. The days in blue are those referring to the stretching described in the clinical scale of Table 1. This figure describes the days before the beginning of the fertile window, the fertile window and the stretching the days after the end of the fertile window of CFW (CrMS/stretching). The clinical window of the SFW software (Soft-CrMS/stretching) presented a normal distribution with a mean of 5.08 days (SD $\pm{}$ 2.87 days). The days detected by the algorithm are graphed in yellow as SFW: (Soft-CrMS/stretching). It was possible to identify the days of the fertile window in yellow, both at the beginning and at the end of it, identified as points of change (POC). A midpoint concordance is displayed between each cycle of the two windows equivalent to the correlation coefficient of 0.71, as observed in Fig. 3. The concordance between the two fertile windows coincides in a percentage greater than 50% in 75% of the cycles, as shown in Table 2. In Fig. 4, a graph of the middle points of each cycle of average stretching is made for both clinical fertile window, as for the fertile window of the algorithm.

Fig. 2.

Example of one record. Stretching in centimeters (cm) is made for both fertile windows; for clinical fertile windows CFW: (CrMS/stretching) as for the fertile window of the algorithm SFW: (Soft-CrMS/stretching). Mean value of SFW: (Soft-CrMS/stretching) was: 5.08 days ($\pm{}$ 2.87 SD). The days in blue are those referring to the stretching described in the clinical scale of Table 1, it describes the days before the beginning of the clinical fertile window, the fertile window and the stretching the days after the end of the clinical fertile window of CFW (CrMS/stretching). The days detected by the algorithm are graphed in yellow as SFW (Soft-CrMS/stretching). It was possible to identify the days of the fertile window in both at the beginning and at the end of it, identified as points of change (POC).

Fig. 3.

Graph plotted showing the correlation coefficient between SFW and CFW (r = 0.71).

Table 2.Fertile windows in 12 cycles according to SFW and CFW.
 Cycle Inicial Day SFW End Day SFW Inicial Day CFW End Day CFW Precision Test Overlap % 1 16 20 14 20 0.67 66.67 2 13 17 10 16 0.50 50.00 3 15 21 14 20 0.83 83.33 4 14 18 16 22 0.33 33.33 5 17 20 17 23 0.50 50.00 6 14 18 13 19 0.67 66.67 7 12 26 19 25 1 100 8 13 19 12 18 0.83 83.33 9 15 18 11 17 0.33 33.33 10 16 21 15 21 0.83 83.33 11 8 12 10 16 0.33 33.33 12 14 17 14 20 0.50 50.00
Fig. 4.

A graph of the middle points of each cycle of average stretching is made for both clinical fertile windows CFW: (CrMS/stretching), as for the fertile window of the algorithm SFW: (Soft-CrMS/stretching).

Last Max Spinbarkheit (LMS) is related to the maximum progression of, stretching and transparency by “K” according to CrMS. A systematic concordance between the K and the relationship with P was found. Apparently, the characteristics of the stretching and lubrication capacity are possible thanks to the physical properties of the endocervical crypts type L and S. In this series, all clinical windows registered this gradual rise in relation to LMS. LMS and K was observed in 41.6%, and LMS + (K $\leq$1 Day) was found in 4 cycles (33.3 %), and Spinbarkheit $\leq$ LMS + (K $\leq$1 Day) = 3 cycles (25%). In less percentage, in three cases did not coincide with the peak day, or with the day of greatest transparency or greatest elasticity, but was very near (-1 day OED).

The fertile window was detected in 100% of the cases with both methods. A correlation coefficient of 0.71 was obtained, between the clinical fertile window detected by means of the parameter of viscoelasticity of the cervical secretion, and the prediction of viscoelasticity by means of a software computer tool.

In this context, it is evident that as soon as the cervical mucus stretches more, and the elasticity increases, both the days before the peak day and the days after the viscoelasticity parameter. It was possible to record it. Within another of the biophysical parameters, the characteristics of the last days of maximum fertility were taken into account through the concept of LMS & Transparency by “K” of CrMS. Twelve records of the typical evolution of cervical mucus were documented by recording the observation graphs of the vaginal discharge of cervical discharge, with CrMS. In this way, it was possible to record the progressive evolutionary change in a systematic way from the left kurtosis of the fertile window to the postovulatory phase after the peak day as seen in this series. The purpose of the present work consisted in comparing the biophysical variable of the viscoelasticity of the cervical secretion, by comparing the clinical window of the CrMS model that uses the peak day according to the CrMs model. Thus, the main of outcome is chosen as the peak day of the CrMs scale. And there are two ways of measuring, two, valuations; one uses only the elasticity extracted from the CrMs scale, as described in detail in the manuscript, which refers to the peak day of the CrMS, and the other is the computer tool. A computer tool is used that determines the progression of viscoelasticity without any reference point, by means of a Neperian algorithm that takes into account the means of the previous values, it evaluates each new value that is presented, until a significant lag is detected, with a change between 25–50% in the measurements between the previous values, and the consecutive ones, in this way it detects the POC at the beginning and at the end to indicate the fertile window called by the software. The scale of the CrMs model was chosen for several reasons, because it is reproducible, because in patients trained in the CrMS method, and effectively allows the reproducibility of clinical assessments, and is recognized worldwide.

The fertile window recorded by the length of the menstrual cycle is a procedure already studied by Wilcox [22]. In Wilcox’s study [22], he identifies in his series of regular cycles the possibility of finding the self-reported fertile window between days 7 and 21 of the menstrual cycle [22]. The possibility of being in a fertile window on the 4th day of the cycle was 2%, by interpreting the duration of the menstrual cycle, 17% on the 7th day of the cycle, and 54% between the 12th and 13th of the cycle [22]. However, in his study, the self-reported possibility of the menstrual cycle to be in a fertile window has a correlation index (CI) of 0.55, which makes it possible for the fertile window to be between days 7 and 21 of the cycle in 50 % of cases in the fertile window. The CrMs scale allows recording the progression and variability of the elasticity and transparency of the cervical secretion [10,12-14].

Within the clinical classifications, Scarpa [23] describes the symptom of vaginal discharge mucus in four ranges. He manages to identify a mucus of greater fertile characteristics (a mucus 4) within the fertile window range between days 10 and 17 days. He finds in 50% of cases the presence of this type of mucus around the 13th day of the cycle, and outside this interval less than 20%. Likewise, he associates a 30% probability of pregnancy, when he is within the range of greater probability of finding type 4 mucus. Fehring manages to frame by assessing the cervical mucus gradient (MMG) the detection of the peak of cervical mucus in 97% between the days (–4 and +4) around ovulation [24]. These results are similar to those described by Hilgers [25] in 1978 where he shows evidences a peak day between (–3 and +3) in relation to the peak day of ovulation. Bigelow describes the characteristics of cervical discharge on a fourth scales [26]. He found a relationship between the quality of cervical mucus and the probability of pregnancy, being the day –3 OED higher for a type 4 mucus, which decreased as he moved away from OED. Although the highest amount of type 4 mucus was evident in –2 OED. These findings have also been corroborated with the Creighton Model Fertility Care System. In such a way it has been possible through cervical secretion to identify and describe cervical secretion from the early follicular phase to the peak day of cervical mucus [27].

The restorative and restorative methodology of the menstrual cycle makes it possible to recover the natural biological approach related to the pathophysiological process of ovulation, in the field of fertility recognition [14]. The estimated day of ovulation (OED) has been recognized as one of the fundamental strategies to determine the signs that define or frame the fertile period. And the peak day of cervical secretion has been considered a point of reference that allows locating the evolutionary clinical changes of the fertile window [28].

It was observed that the peak day in some cycles did not coincide with the day of maximum elasticity, which was taken in counts according to the standards of the CsMS model. These mentioned changes are well correlated with the dynamics of follicular growth, and the elevation of the determinations in urine of estrone-3-glucuronide (E3G). The difference between the number of estrogens produced at the beginning of the cycle and at the ovulatory level is significant, which makes up the aforementioned physiological findings [4,5].

The E3G determination increases from 20–30 ng/mL from day -6 relative to the estimated day of ovulation (OED), until the day of ovulation, at which point the LH surge is normally detected greater than 30 IU/L [6-10]. This peak generally occurs 24 to 48 hours before ovulation but is never after it. The basal body temperature (BBT) curve was not taken into account in this serie, but a high correlation is known to detect the fertile window [28].

One outcome of the validation of a computational method for measuring the biophysical properties of cervical secretion is its possible inclusion as an essential component in a clinical apparatus, as part of a more comprehensive mechanism that also takes into account sperm swimming properties [29,30].

Currently several biomarkers are known which assist in the determination of the fertile window, which open the door for the development of future biotechnological applications based on Brown’s theory [1,21]. Aldecrewtz reports a difference in the production of estrogen that exists between the initial follicular phase and the middle of the cycle, and describes a POC which indicates the start of the fertile window [6]. Similarly, the peak of LH [7,8,10] has been used to determine the end of the fertile window, the LH peak Is found between 24 and 48 hours before ovulation, but never before, and therefore can be used as a predictor. Those two well-established points in time bound the interval -6 of the fertile window with respect to the Expected Day of Ovulation (OED). Ecochard [17] relates detection of the fertile window to basal temperature changes by echography [17] and report a detection accuracy of the fertile window on the order of 98 to 99%. Currently there exist several scales [10,15,23,24] which provide a clinical assessment of the changes in the properties of the cervical secretion all throughout the menstrual cycle, which were initially described by Billings in his 1972 publication in The Lancet [12].

The mucin filaments tend to align longitudinally within the cervical canal, creating aqueous channels between the filaments. It seems that the effect of the parallel fibers of the glycoprotein bands can exert a driving effect on the gametes themselves. Natural swimming channels are formed from the interaction between the non-ciliated cylindrical epithelial cells which secrete mucin granules, and the hair cells that drive cervical mucus from the crypt of origin to the external cervical orifice. This process is considered to be dependent on the rheological forces associated with mucus flow from the cervical crypts, which tend to align the mucin filaments longitudinally within the cervical canal.

The integration of sperm swimming in the fertile window approach allows the development of new technologies that integrate the joint assessment of cervical factor and sperm swimming. It is necessary to consider more advances in biotechnological devices that can contribute to giving information to the study on the total interaction of the transport factor and the capacity for sperm rise.

Further studies are needed to further evaluate the efficacy of the method, by introducing not only more data points, but by also incorporating other biophysical characteristics for the cervical secretion, such as measures of fluidity or elasticity, as well as other factors, such as sperm rise [29,30].

The k “clear” is the maximum pattern of clarity, on the CrMs scale, its evidence is constant and may improves the result of the algorithm, which has not been calculated in this study. The use of K rating in the Creighton scale may also improve the algorithm, and we plan to conduct such studies in the future. The incorporation of transparency as part of the biophysical assessment, would also be beneficial [28-30].

Finally, given the increasing use of Machine Learning techniques in the sciences, all across the board, it would be useful to determine whether one could extend the computational method used herein, to include other observed quantities, in order to produce biophysical feature vectors that could be used to construct training data. Such data could be used to train a neural network that could be used to predict the fertile window.

5. Conclusions

The rating scale used in the CrMs model allowed us to use the elasticity and transparency parameters to help identify and characterize the peak day. We found a strong correlation between the fertile windows obtained through clinical measures compared to those obtained from the computational method. Therefore, since our method is solely based on changes observed at the vulvar level in the elasticity of the cervical secretion, we expect it could be used to the beginning of the fertility window and the end of the fertile interval in both fertile and sub-fertile patients.

The assessment of the midpoint of the clinical window in the series allowed to extract the elasticity and transparency parameters, to identify and characterize POCs, at the beginning, at the midpoint, and at the end of the fertility window in conjunction with a computer tool.

The evaluation of the term “last value of maximum elasticity” (LMS), was a parameter that allowed to characterize the progression of viscoelasticity in conjunction with the “K” of transparency of cervical secretion of CrMs, in relation to the peak day of the fertile window.

Author contributions

JMML developed study concept, data acquisition, study design, methodology, project administration, performed data acquisition and data interpretation, and wrote manuscript and draft. OMM, MMC, AFA all contributed to software development, draft elaboration, stadistic validataion, and writing original draft. JEM Preparation including pre-publication stages. DS and JS preparation, creation and presentation of the published work, commentary and revision-including pre-publication stages. JLA reviewed the manuscript and final topics to editing. All authors read and approved the final manuscript.

Ethics approval and consent to participate

The subject gave their informed consent for inclusion before they participated in the study, which was conducted in strict accordance with the Declaration of Helsinki, and whose protocol was approved by the Ethics Committee of CEImLAR, Center for Biomedical Research of La Rioja (CIBIR) (approval number P.I.339).

Acknowledgment

We would like to thank all those who helped us during the construction of this manuscript, including the reviewers, for their suggestions. Kathleen Copeland’s help and Tracey Parnell’s enthusiastic communication support have been greatly appreciated. Finally, we would like to thank Phil Boyle and Joseph Stanford of the board committee of the International Institute of Restorative Reproductive Medicine (IIRRM) for their interest and advice.

Funding

This research received no external funding.

Conflict of interest

The authors declare no conflict of interest.

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