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# Fluid Mechanics Test 2

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• ###### Question 1
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When a dolphin glides through air, it experiences an external pressure of 0.75 m of mercury. The absolute pressure on dolphin when it is 5 m below the free surface of the water is
###### Solution
Concept:

Pabs = Pgauge + Patm

Calculation:

Given:

h = 5 m

Patm = 0.75 m of mercury = ρHg × g × 0.75 = 13600 × 9.81 × 0.75 = 100062 Pa

Pgauge = ρwater × g × h

Pgauge = 1000 × 9.81 × 5 = 49050 Pa

Pabs = Pgauge + Patm

Pabs = 49050 + 100062

Pabs = 149112 Pa = 149112 N/m2 = 0.149 N/mm20.15 N/mm2

• ###### Question 2
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Calculate the capillary rise or fall in a glass tube of 2.5 mm diameter when immersed vertically in mercury. Take surface tension σ = 0.52 N/m for mercury in constant with glass and angle of contact = 130°
###### Solution
Concept:

Expression for capillary rise or fall

$$h = \frac{{4\sigma \cos \theta }}{{\rho gd}}$$

Calculation:

Given: θ = 130°, σ = 0.52 N/m, d = 2.5 × 10-3m

$$h = \frac{{4\; \times \;0.52\; \times\; \cos 130^\circ }}{{13600\; \times \;9.81\; \times \;2.5\; \times \;{{10}^{ - 3}}}}$$

h = - 0.4 cm

• ###### Question 3
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Velocity profile over a curved surface is given as : $$\frac{u}{U} = 2{\left( {\frac{y}{\delta }} \right)^2} - {\left( {\frac{y}{\delta }} \right)^3}$$

The flow ______

###### Solution
Concept:

$${\left( {\frac{{\partial u}}{{\partial y}}} \right)_{y = 0}} \Rightarrow + {\rm{ve\;}} \Rightarrow {\rm{\;stick\;to\;surface}}$$

$${\left( {\frac{{\partial u}}{{\partial y}}} \right)_{y = 0}} \Rightarrow 0{\rm{\;}} \Rightarrow {\rm{\;on\;verge\;of\;separation}}$$

$${\left( {\frac{{\partial u}}{{\partial u}}} \right)_{y = 0}} \Rightarrow - {\rm{ve\;}} \Rightarrow flow~is~seperated$$

Calculation:

$$\frac{u}{U} = 2{\left( {\frac{y}{\delta }} \right)^2} - {\left( {\frac{y}{\delta }} \right)^3}$$

$$\frac{{\partial u}}{{\partial y}} = U\left[ {4\left( {\frac{4}{\delta }} \right) - 3{{\left( {\frac{y}{\delta }} \right)}^2}} \right]$$

$${\left( {\frac{{\partial u}}{{\partial y}}} \right)_{y = 0}} = 0$$

∴ Flow is on the verge of separation.

• ###### Question 4
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The value of friction factor is misjudged by 25% in using Darcy-Weisbach equation. The resulting error in the discharge will be?
###### Solution
Concept:

$${h_L} = \frac{{fL{Q^2}}}{{12{D^5}}}$$

$$Q\; \propto {\rm{\;}}\frac{1}{{\sqrt f }}\;\;\; - - - \left( 1 \right)$$

Calculation:

Here f is misjudged by 25 %, it means f can be 1.25f or 0.75f

Case: 1

Let it be 1.25f

$${Q_1} = \frac{k}{{\sqrt f }}$$

$${Q_2} = \frac{k}{{\sqrt {1.25f} }}$$

$$Error = \frac{{{Q_2} - {Q_1}}}{{{Q_1}}} \times 100 = - 11\;\%$$

Case: 2

Let it be 0.75f

$${Q_1} = \frac{k}{{\sqrt f }}$$

$${Q_2} = \frac{k}{{\sqrt {0.75f} }}$$

$$Error = \frac{{{Q_2} - {Q_1}}}{{{Q_1}}} \times 100$$

Error = 15.47%

• ###### Question 5
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A centrifugal pump lifts 0.0125 m3 / s of water from a well with a static lift of 30 m. If the brake power of the driving electric motor is 5 kW, what is the overall efficiency of the pump-set?
###### Solution
Explanation:

Pump is defined as a device which transfer the input mechanical energy of a motor or a of an engine into pressure energy or kinetic energy or both of a liquid.

Centrifugal pump is type of dynamic pressure pump.it also known as velocity pump.

$$Q = 0.0125\frac{{{m^3}}}{s}$$

Shaft power or rating of motor (PS) = 5 kW

$$density\;\left( \rho \right) = 1000\frac{{kg}}{{{m^3}}}$$

$$\eta = \;\frac{{Power\;output\;}}{{shaft\;power}}$$

$$\eta = \;\frac{{\rho \times g \times Q \times H}}{{{P_S}}}$$

$$\eta = \;\frac{{1000 \times 9.81 \times 0.0125 \times 30}}{{5000}}$$

η = 0.73575 ≈ 0.7360∴ η = 73.60 %

• ###### Question 6
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Which of the following conditions is true regarding the validity of Pascal’s Law?

I- It is valid for ideal fluid in motion.

II- It is valid when real fluids with rigid body motion.
###### Solution
Concept:

Pascal’s Law states that at any point in a fluid the intensity of pressure is equal in all the directions if and only if shear stresses acting fluid are zero.

Newton’s law of viscosity:

$$\tau = \frac{{\mu du}}{{dy}}$$

Where,

τ is the shear stress on the plate.

μ is dynamic viscosity of fluid.

Explanation:

Statement- I - It is valid for ideal fluid in motion.

Since fluid is in motion, u ≠ 0 but

For ideal fluid, μ = 0 ⇒ τ = 0

Therefore, Pascal’s law is valid for ideal fluids even in state of motion.

Statement II- It is valid when real fluids with rigid body motion

Rigid body motion means every layer of fluid has uniform velocity.

For real fluid in motion mean that μ ≠ 0 and u ≠ 0

But, du = 0 τ = 0∴ For Real fluids Pascal’s law is also valid with rigid body motion.

• ###### Question 7
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A head loss in 100 m length of a 0.1 m diameter pipe (f=0.02) carrying water is 10 m. The boundary shear stress in Pa is ____
###### Solution
Concept:

For any type of flow, the shear stress at wall,

$$\tau = - \left( {\frac{{dp}}{{dx}}} \right)\frac{R}{2}\;$$

$$\therefore \tau = \frac{{\rho g{h_L}}}{L} \times \frac{R}{2}$$

$$\tau = \frac{{1000 \times 9.81 \times 10}}{{100}} \times \frac{{0.1}}{4}$$

∴ τ = 24.525 Pa

• ###### Question 8
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Which of the following represents generalised continuity equation?
###### Solution
Concept:

In generalised equation, density (ρ) is not constant and the flow is three dimensional

$$\frac{{D\rho }}{{Dt}} + \;\rho \left( {\nabla .\vec V} \right) = 0$$

After expansion,

$$\frac{{\partial \rho }}{{\partial t}} + u\frac{{\partial \left( \rho \right)}}{{\partial x}} + \rho \frac{{\partial \left( u \right)}}{{\partial y}} + v\frac{{\partial \left( \rho \right)}}{{\partial y}} + \rho \frac{{\partial \left( v \right)}}{{\partial y}} + w\frac{{\partial \left( \rho \right)}}{{\partial z}} + \rho \frac{{\partial \left( w \right)}}{{\partial z}} = 0$$

$$\frac{{\partial \rho }}{{\partial t}} + \frac{{\partial \left( {\rho u} \right)}}{{\partial x}} + \frac{{\partial \left( {\rho v} \right)}}{{\partial y}} + \frac{{\partial \left( {\rho w} \right)}}{{\partial z}} = 0$$

$$\frac{{\partial \rho }}{{\partial t}} + \;\nabla .\left( {\rho \vec V} \right) = 0$$So, options a, b and c are true.

• ###### Question 9
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In an orifice meter, coefficient of discharge is 0.58 and coefficient of vena contracta is 0.61, what is coefficient of velocity?
###### Solution
Concept:

Cd = Cc × Cv

Where, Cd = Coefficient of discharge, Cc = Coefficient of vena contracta, Cv = Coefficient of velocity

Calculation:

Cd = Cc × Cv

0.58 = 0.61 × Cv

∴ Cv = 0.95

• ###### Question 10
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In a flow over a flat plate, laminar boundary layer exists where velocity distribution is

$$\frac{u}{U} = 0.5\frac{y}{\delta }$$

Ratio of momentum thickness to boundary layer thickness is____
###### Solution
Concept:

θ = momentum thickness

$${\rm{\theta \;}} = \mathop \smallint \limits_0^\delta \frac{u}{U}\left[ {1 - \frac{u}{U}} \right]dy$$

Calculation:

$$\theta = \mathop \smallint \limits_0^\delta \frac{u}{U}\left[ {1 - \frac{u}{U}} \right]dy$$

$$\theta = \;\mathop \smallint \limits_0^\delta \frac{y}{{2\delta }}\left[ {1 - \frac{y}{{2\delta }}} \right]dy$$

$$\theta = \;\mathop \smallint \limits_0^\delta \left[ {\frac{y}{{2\delta }} - \frac{{{y^2}}}{{4{\delta ^2}}}} \right]dy$$

$$\theta = \left[ {\frac{{{y^2}}}{{4\delta }} - \frac{{{y^3}}}{{12{\delta ^2}}}} \right]_0^\delta$$

$$\theta = \frac{\delta }{4} - \frac{\delta }{{12}}$$

$$\therefore \frac{\theta }{\delta } = \frac{1}{6}$$

• ###### Question 11
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Match the following:

 P Reynold’s model i. Cavitation Q Froude model ii. Capillary rise R Euler model iii. Pipe flow S Weber model iv. Flow over spillways
###### Solution
Explanation:

Reynold’s model law is applicable for:

(i) Pipe flow

(ii) Resistance experienced by sub-marines, airplanes, etc.

Froude model law is applicable for:

(i) free surface flows such as spillways, channels, etc.

(ii) Sea waves

Euler model law is applicable for:

Where pressure forces are dominant like flow taking place in closed pipe, cavitation phenomenon etc.

Weber model is applicable for:

(i) Capillary rise in narrow passage

(ii) Capillary movement of water in soil

• ###### Question 12
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A 1 : 6 scale model of a passenger car is tested in a wind tunnel. The prototype velocity is 60 km/hr. The model drag is 250 N. The air in the model and prototype can be assumed to have the same properties. Which of the following statements are true?
###### Solution
Concept:

Reynolds number will be same for both model and prototype

Calculation:

Given:

$$\frac{{{L_M}}}{{{L_P}}} = \frac{1}{6}$$

$${V_p} = 60 \times \frac{5}{{18}}\;\frac{m}{s}$$

FM = 250 N

$$Re = \;\frac{{\rho {V_M}{L_M}}}{\mu } = \frac{{\rho {V_P}{L_P}}}{\mu }\; - - - \left( 1 \right)$$

Now,

From (1)

$${V_M} = \frac{{{L_P}}}{{{L_M}}}{V_P} = 100\;\frac{m}{s}$$

Inertial force = ρL2V2

$$\frac{{{F_P}}}{{{F_M}}} = \frac{{{\rho _P}}}{{{\rho _M}}}{\left( {\frac{{{L_P}}}{{{L_M}}}} \right)^2}{\left( {\frac{{{V_P}}}{{{V_M}}}} \right)^2}\;\; - - - \left( 2 \right)$$

Density is same for model and prototype (Given)

From (2),

FP = 250 N

Power on the prototype

P = FPVP

P = 4.167 kW

• ###### Question 13
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A smooth flat plate of total length 2 m is in parallel flow stream of water. The water has uniform free stream velocity of 1 m/s parallel to plate. The critical Reynolds number for a flat plate is 500000. Assume density of water is 1000 kg/m3 and dynamic viscosity of water 1 centipoises. What is the maximum distance measured (in m, up to two decimal places) from leading edge of plate, up-to which the boundary layer formed to be laminar?
###### Solution
Concept:

The Reynolds number for flow over plate can be given as:

$${R_e} = \frac{{\rho UX}}{\mu }$$

Where

X is the distance from leading edge.

U is the free mean velocity.

μ is the dynamic viscosity of fluid flowing.

ρ is the density of fluid flowing.

If Reynolds number is less than or equal to critical Reynold’s number than boundary layer formed over flat plate is laminar i.e.Re ≤ ReCr

Calculation:

Given:

U = 1 m/s ; ρ = 1000 kg/m3 ; μ = 1 centipoise = 10-3 N s/m2

Re cr = 5 × 105

Let ‘x’ is the distance from leading edge up to which Boundary layer formed is laminar,

Now,

Rex ≤ Re cr

$$\frac{{\rho Ux}}{\mu } \le 5 \times {10^5}$$

$$\frac{{{{10}^3} \times 1 \times x}}{{{{10}^{ - 3}}}} \le 5 \times {10^5}$$

x ≤ 0.5 m

∴ The maximum distance from leading edge of the plate up to which boundary layer is laminar is 0.50 m.

• ###### Question 14
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In a small pipe of 0.5mm diameter and length 2m, water flows at 48cm3/minute. Viscosity is 1.92 centipoise at low temperature flow condition. Pressure drop in MPa is______
###### Solution
Concept:

$$u = \left( {\frac{{ - \partial P}}{{\partial x}}} \right)\left[ {\frac{{{R^2} - {r^2}}}{{4\mu }}} \right]$$

Calculation:

Given: D = 0.5 mm ⇒ R = 0.25 mm, L = 2m, Q = 48 cm3/min = 8 × 10-7 m3/sec,

μ = 1.92 centipoise = 1.92 × 10-3 Ns/m2

Now,

Q = A × Vavg

$$8 \times {10^{ - 7}} = \frac{\pi }{4} \times {\left( {0.5 \times {{10}^{ - 3}}} \right)^2} \times {V_{avg}}$$

∴ Vavg = 4.074 m/s

$$\therefore {u_{max}} = \left( {\frac{{ - \partial P}}{{\partial x}}} \right)\left[ {\frac{{{R^2}}}{{4\mu }}} \right]$$

And umean = Vavg umax

$$\therefore {V_{avg}} = \left( {\frac{{ - \partial P}}{{\partial x}}} \right)\left[ {\frac{{{R^2}}}{{8\mu }}} \right]$$

$$\therefore {\rm{\Delta }}P = \frac{{{V_{avg}} \times L \times 8\mu }}{{{R^2}}}$$

∴ ΔP = 2 MPa

Points to Remember:

$${\rm{\Delta }}P = \frac{{32\;\mu \;{V_{avg}}\;L}}{{{D^2}}}$$∴ ΔP = 2 MPa

• ###### Question 15
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The velocity potential function in a two-dimensional flow field is given by Ø = x2 – y2. Which of the following statements are true?
###### Solution
Concept:

$$U = \; - \frac{{\partial \emptyset }}{{\partial x}} = \; - \frac{{\partial \varphi }}{{\partial y}}\;\; - - - \left( 1 \right)$$

$$V = \; - \frac{{\partial \emptyset }}{{\partial y}} = \;\frac{{\partial \varphi }}{{\partial x}}\;\;\; - - - \left( 2 \right)$$

$$Q = \;{\varphi _2} - {\varphi _1}\;\;\; - - - \left( 3 \right)$$

Q is flow per unit width

Now,

From (1),

$$U = \; - 2x = \; - \frac{{\partial \varphi }}{{\partial y}}\; \Rightarrow \;\varphi \left( {x,y} \right) = 2xy + f\left( x \right)\; - - - \;\left( 4 \right)$$

$$V = \;2y = \;\frac{{\partial \varphi }}{{\partial x}}\; \Rightarrow \;\varphi \left( {x,y} \right) = 2xy + f\left( y \right)\;\; - - - \;\left( 5 \right)$$

At (1, 2)

U = -2 m/s

V = 4 m/s

$$Velocity = \;\sqrt {{{\left( { - 2} \right)}^2} + {4^2}}$$

Velocity = 4.47 m/s

Now,

From (4) and (5)

φ(x, y) = 2xy

φ(1, 3) = 6

φ(3, 4) = 24

Q = φ(3, 4) - φ(1, 3) = 24 - 6

Q = 18 units

• ###### Question 16
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For a two dimensional, unsteady flow of an incompressible fluid, the velocity field is given by V(2x2 t + y) î - 4xyt ĵ  Then which of the following statements are true?

###### Solution
Concept: For a fluid flow to be possible, it should obey the continuity equation which is given by $$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0~$$ (Incompressible, 2D) The convective acceleration is given by $${{\alpha }_{xC}}=u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}~~;~~{{\alpha }_{yC}}=u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}$$ The local acceleration is given by $${{\alpha }_{xL}}=\frac{\partial u}{\partial t}~~;~~{{\alpha }_{yL}}=\frac{\partial v}{\partial t}$$ The acceleration in a direction is given by α = αC + αL   Calculation: Given V = (2x2t + y) î - 4xyt ĵ ⇒ u = 2 x2 t + y ; v = - 4 x y t ; $$\frac{\partial \left( 2{{x}^{2}}t+y \right)}{\partial x}+\frac{\partial \left( -4xyt \right)}{\partial y}=4xt-4xt=0~~$$  (Option 1 is wrong)  Convective acceleration in the x-direction is $${{\alpha }_{xC}}=\left( 2{{x}^{2}}t+y \right)\frac{\partial \left( 2{{x}^{2}}t+y \right)}{\partial x}+\left( -4xyt \right)\frac{\partial \left( 2{{x}^{2}}t+y \right)}{\partial y}$$ ⇒ αxc = [(2 x2 t + y) × 4 x t] + [(- 4 x y t) × 1] = 8 x3 t2 ∴ αxc at (1, 2) and t = 3 sec is =  8 × 13 × 9 = 72 m/s2 (Option 2) Now, Local acceleration in y-direction is $${{\alpha }_{yL}}=\frac{\partial \left( -4xyt \right)}{\partial t}=-4xy$$ ∴ αyL = -4 × 1 × 2 = - 8 m/s2 (Option 3 is wrongNow, Local acceleration in x-direction is $${{\alpha }_{x}}=\frac{\partial \left( 2{{x}^{2}}t+y \right)}{\partial t}=2{{x}^{2}}=2\times {{1}^{2}}=2$$ ∴ Acceleration in x-direction is αx = 72 + 2 = 74 m/s2 ; Now, Convective acceleration in y-direction is $${{\alpha }_{yC}}=\left( 2{{x}^{2}}t+y \right)\frac{\partial \left( -4xyt \right)}{\partial x}+\left( -4xyt \right)\frac{\partial \left( -4xyt \right)}{\partial y}$$ ⇒ αyc = [(2 x2 t + y) × - 4 y t] + [(- 4 x y t) × - 4 x t] = 16 x2 y t2 – 4 y2 t – 8 x2 y t2 ; ⇒ αyc = 8 x2 y t2 – 4 y2 t αyc at (1, 2) and t = 3 sec is = 96 m/s2 ; ∴ Acceleration in y-direction is αy = 96 – 8 = 88 m/s2; Total acceleration = (262 + 882)0.5 ∴ αtotal = (262 + 882)0.5 = 91.76 m/s2 (Option 4 wrong)
• ###### Question 17
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A Fluid flow is given by

$$V = \left( {2x + \frac{{{y^3}}}{3} - {x^2}y} \right)\hat i + \left( {x{y^2} - \frac{{{x^3}}}{3} - 2y} \right)\hat j$$

Check whether the given flow is possible case of fluid flow or not. If possible, determine angular velocity about z-axis.
###### Solution
Concept:

To check possibility of fluid flow, continuity equilibrium should be verified, i.e.

$$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0$$

$${\omega _z} = \frac{1}{2}\left( {\frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}}} \right)$$

Calculation:

$$u = 2x + \frac{{{y^3}}}{3} - {x^2}y$$

$$v = x{y^2} - 2y - \frac{{{x^3}}}{3}$$

$$\frac{{\partial u}}{{\partial x}} = 2 - 2xy\;\;\;;\;\frac{{\partial u}}{{\partial y}} = {y^2} - {x^2}$$

$$\frac{{\partial v}}{{\partial x}} = {y^2} - \frac{{3{x^2}}}{3} = {y^2} - {x^2}$$

$$\frac{{\partial v}}{{\partial y}} = 2xy - 2$$

For 2-D flow, continuity equation,

$$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0$$

(2 – 2xy) + (2xy - 2) = 0

Hence, it represents possible case of fluid flow.

Now,

$${\rm{Angular\;velocity\;}}\left( {{\omega _z}} \right) = \frac{1}{2}\left( {\frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}}} \right)$$

$${\omega _z} = \frac{1}{2}\left[ {\left( {{y^2} - {x^2}} \right) - \left( {{y^2} - {x^2}} \right)} \right] = 0$$∴ ωz = 0

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text: Title Heuristic treatments of tricritical behaviour as observed in $^{4}He-^{3}He$ mixtures and in polymeric assembliesHeuristic treatments of tricritical behaviour as observed in $^{4}He-^{3}He$ mixtures and in polymeric assemblies Author March, N.H. Zhang, Z.D. Faculty/Department Faculty of Sciences. Physics Research group Department of Physics Publication type article Publication 2010London, 2010 Subject Physics Source (journal) Physics and chemistry of liquids. - London Volume/pages 48(2010):2, p. 279-284 ISSN 0031-9104 ISI 000277647300010 Carrier E Target language English (eng) Full text (Publishers DOI) Affiliation University of Antwerp Abstract Tricritical behaviour in three dimensions is accessible to experiment in, for example, 4He-3He mixtures and some polymeric assemblies. Here, by largely heuristic arguments, which appeal however to some specific model Hamiltonians, we indicate how one can understand the gist of the experimental results near the tricritical point in the systems cited above. We point out that the spin fluctuations at the tricritical point are much weaker than those at the critical point, which leads to the conclusion that the tricritical behaviours can be described well by the mean-field exponents with logarithmic corrections, at d = 3. E-info http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000277647300010&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000277647300010&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000277647300010&DestLinkType=CitingArticles&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 Handle

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### Still have math questions?

Algebra
Question

1. TRUE or FALSE: Write T if the statement is true. Otherwise, write $$F$$ . Write your answers before each number:

$$U = \{ x / x$$ is a natural number less than $$11 \}$$ For numbers $$1$$ to $$5$$ consider the following:

$$A = \{ 1,5,7,8 \}$$

$$B = \{ x / x$$ is a composite number less than $$8 \}$$

$$C = \{ 4,8,9,10 \}$$

1. $$\{ 9 \} \in A$$

2. $$A = C$$ 3. $$n ( B ) = 3$$

4. $$A \cap B$$ is a singleton

5. The power set of $$A$$ has $$15$$ non-empty elements.

II. Consider the following sets:

$$U = \{ x / x$$ is a natural number less than $$11 \}$$

$$A = \{ 1,5,7,8 \}$$

$$B = \{ x / x$$ is a composite number less than $$8 \}$$

$$C = \{ 4,8,9,10 \}$$

Find the following:

1. $$A \cap B$$

2. $$( A \cup C ) ^ { \prime }$$

3. $$C - B$$ 4. $$A \times D$$ if $$D = \{ a , b \}$$

5. $$( A \cup B ) \cap C$$


text:
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text:
For an algebraic curve $V$ in $\C^k \times \C^n$ it is investigated when it satisfies the Phragm\'en-Lindel\"{o}f condition $\PL (\omega)$ of evolution in certain classes of ultradifferentiable functions. Necessary as well as sufficient conditions are obtained which lead to a complete characterization for curves in $\C \times \C^n$. Some examples illustrate how these results can be applied.

### Characterizing the Phragmen-Lindeloef condition for evolution on algebraic curves.

#### Abstract

For an algebraic curve $V$ in $\C^k \times \C^n$ it is investigated when it satisfies the Phragm\'en-Lindel\"{o}f condition $\PL (\omega)$ of evolution in certain classes of ultradifferentiable functions. Necessary as well as sufficient conditions are obtained which lead to a complete characterization for curves in $\C \times \C^n$. Some examples illustrate how these results can be applied.
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## Electronic Communications in Probability

### A counter example to central limit theorem in Hilbert spaces under a strong mixing condition

#### Abstract

We show that in a separable infinite dimensional Hilbert space, uniform integrability of the square of the norm of normalized partial sums of a strictly stationary sequence, together with a strong mixing condition, does not guarantee the central limit theorem.

#### Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 62, 12 pp.

Dates
Accepted: 29 August 2014
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465316764

Digital Object Identifier
doi:10.1214/ECP.v19-3249

Mathematical Reviews number (MathSciNet)
MR3254741

Zentralblatt MATH identifier
1334.60020

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes

Rights

#### Citation

Giraudo, Davide; Volny, Dalibor. A counter example to central limit theorem in Hilbert spaces under a strong mixing condition. Electron. Commun. Probab. 19 (2014), paper no. 62, 12 pp. doi:10.1214/ECP.v19-3249. https://projecteuclid.org/euclid.ecp/1465316764

#### References

• Aliprantis, Charalambos D.; Border, Kim C. Infinite dimensional analysis. A hitchhiker's guide. Third edition. Springer, Berlin, 2006. xxii+703 pp. ISBN: 978-3-540-32696-0; 3-540-32696-0
• Araujo, Aloisio; Giné, Evarist. The central limit theorem for real and Banach valued random variables. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-Chichester-Brisbane, 1980. xiv+233 pp. ISBN: 0-471-05304-X
• Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
• R. C. Bradley, phIntroduction to strong mixing conditions. Vol. 1, Kendrick Press, Heber City, UT, 2007.
• Dedecker, Jérôme; Merlevède, Florence. On the almost sure invariance principle for stationary sequences of Hilbert-valued random variables. Dependence in probability, analysis and number theory, 157–175, Kendrick Press, Heber City, UT, 2010.
• Dehling, Herold. Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahrsch. Verw. Gebiete 63 (1983), no. 3, 393–432.
• Denker, Manfred. Uniform integrability and the central limit theorem for strongly mixing processes. Dependence in probability and statistics (Oberwolfach, 1985), 269–289, Progr. Probab. Statist., 11, Birkhäuser Boston, Boston, MA, 1986.
• Doob, J. L. Stochastic processes. John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. viii+654 pp.
• Doukhan, Paul; Massart, Pascal; Rio, Emmanuel. The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 1, 63–82.
• Giraudo, Davide; VolnÃ½, Dalibor. A strictly stationary $\beta$-mixing process satisfying the central limit theorem but not the weak invariance principle. Stochastic Process. Appl. 124 (2014), no. 11, 3769–3781.
• Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9
• Merlevède, Florence; Peligrad, Magda. On the weak invariance principle for stationary sequences under projective criteria. J. Theoret. Probab. 19 (2006), no. 3, 647–689.
• Merlevède, Florence; Peligrad, Magda; Utev, Sergey. Sharp conditions for the CLT of linear processes in a Hilbert space. J. Theoret. Probab. 10 (1997), no. 3, 681–693.
• Mori, Toshio; Yoshihara, Ken-ichi. A note on the central limit theorem for stationary strong-mixing sequences. Yokohama Math. J. 34 (1986), no. 1-2, 143–146.
• Politis, Dimitris N.; Romano, Joseph P. Limit theorems for weakly dependent Hilbert space valued random variables with application to the stationary bootstrap. Statist. Sinica 4 (1994), no. 2, 461–476.
• Rio, Emmanuel. Covariance inequalities for strongly mixing processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), no. 4, 587–597.
• Rio, Emmanuel. Théorie asymptotique des processus aléatoires faiblement dépendants. (French) [Asymptotic theory of weakly dependent random processes] Mathématiques & Applications (Berlin) [Mathematics & Applications], 31. Springer-Verlag, Berlin, 2000. x+169 pp. ISBN: 3-540-65979-X
• Rosenblatt, M. A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U. S. A. 42 (1956), 43–47.
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• Tone, Cristina. Central limit theorems for Hilbert-space valued random fields satisfying a strong mixing condition. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 77–94.
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• VolnÃ½, Dalibor. Approximation of stationary processes and the central limit problem. Probability theory and mathematical statistics (Kyoto, 1986), 532–540, Lecture Notes in Math., 1299, Springer, Berlin, 1988.


text:
Fibonnacci sequence

• September 14th 2012, 02:36 AM
mastermin346
Fibonnacci sequence
Can anyone explain if the Fibonnacci sequence can be placed into a parabolic or hyperbolic function?
• September 14th 2012, 06:29 PM
johnsomeone
Re: Fibonnacci sequence
If you're asking if there's a function $\phi:\mathbb{R}^+ \rightarrow \mathbb{R}$ such that the graph of $\phi$ is a parabola or hyperbola,

and that $\phi$ satistfies, for all positive integers $n, \ \phi(n) = F_n$, the $n^{th}$ Fibonnacci number, then the answer is no.

Parabolas and hyperbolas only have a few "free parameters", and the requirement $\phi(n) = F_n \forall n \in \mathbb{Z}^+$ would quickly exhaust them.

It would be possible that it algebraically worked out, but it doesn't. I'll do the parabola case, and leave to you the hyperbola ( $\phi(x) = b\sqrt{1+(x/a)^2}$).

If a parabola is the graph of a function $\phi$ that satisfies $\phi(n) = F_n$, then let $\phi(x) = ax^2 + bx + c$ for some $a, b, c \in \mathbb{R}$.

Then $\forall n \in \mathbb{Z}^+, \phi(n+2) = \phi(n+1) + \phi(n)$, so $a(n+2)^2 + b(n+2) + c = ( a(n+1)^2 + b (n+1) + c ) + ( an^2 + bn + c )$.

Thus $\forall n \in \mathbb{Z}^+, a $(n+2)^2 - (n+1)^2 - n^2$ + b$(n+2) - (n+1) - (n)$ + $c - c - c$ = 0$, so

$a(- n^2 -2n + 3 )+ b(-n + 1) - c = 0$, so $(-a)n^2 + (-2a-b)n + (3a+b-c) = 0$

The only way that could be true for all n is if each of those n-coefficients was 0. Thus $-a=0, -2a-b = 0, 3a+b-c = 0$.

That system of equations (3 equations in the 3 unknowns $a, b$, and $c$) has only one solution: $a = b = c = 0$.

That means the only quadratic that has a prayer of working is the 0 quadratic $\phi(x) = 0x^2+0x+0 = 0$.

But that obviously doesn't satisfy $\phi(n) = F_n \forall n \in \mathbb{Z}^+$, and it was the only one that had a chance.

-----

I suppose I should note that even if you consider rotated parabolas/hyperbolas, you're still going to quickly run out of free parameters.
(The standard equation for the graph of a generic conic section in the plane has 6 free parameters. That's more than the 3 a quadratic
polynomial has, but it's still less than the infinity required.).


text:
# Borel system of sets

family of Borel sets, or $B$-system, generated by a family of sets $M$
It is seldom used for Borel field of sets or $\sigma$-algebra generated by the set $M$, i.e. the smallest $\sigma$-algebra of subsets of a given set $X$ containing a given family $M$. It is also sometimes used by other authors to denote the smallest family $\mathcal{A}$ containing $M$ which is closed under countable unions and countable intersections (since it is not required that $\mathcal{A}$ be closed under taking complements, $\mathcal{A}$ might be strict subfamily of the $\sigma$-algebra generated by $M$).


text:
# The large deviation principle for the Erdős-Rényi random graph

@article{Chatterjee2011TheLD,
title={The large deviation principle for the Erdős-R{\'e}nyi random graph},
author={Sourav Chatterjee and S. R. S. Varadhan},
journal={Eur. J. Comb.},
year={2011},
volume={32},
pages={1000-1017}
}
• Published 1 October 2011
• Mathematics
• Eur. J. Comb.
168 Citations
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• Mathematics
• 2018
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Moderate deviations of subgraph counts in the Erdős-Rényi random graphs 𝐺(𝑛,𝑚) and 𝐺(𝑛,𝑝)
• Mathematics
Transactions of the American Mathematical Society
• 2020
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Sparse random graphs with many triangles
• Computer Science, Mathematics
• 2021
This paper derives asymptotically sharp bounds on the probability that the Erdős-Rényi random graph contains a large number of vertices that are part of a triangle, and shows that, conditionally on this event, with high probability the graph contains an almost complete subgraph, i.e., the triangles form a near-clique, and has the same local limit as the original Erd� Hungarian random graph.
Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs
• Mathematics
Journal of Theoretical Probability
• 2021
We consider an inhomogeneous Erdős-Rényi random graph GN with vertex set [N ] = {1, . . . , N} for which the pair of vertices i, j ∈ [N ], i 6= j, is connected by an edge with probability r( i N , j
Large Deviations for Dense Random Graphs
A dense graph is a graph whose number of edges is comparable to the square of the number of vertices. The main result of this chapter is the formulation and proof of the large deviation principle for
The upper tail problem for induced 4-cycles in sparse random graphs
Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of
SPECTRAL EDGE IN SPARSE RANDOM GRAPHS: UPPER AND LOWER TAIL LARGE DEVIATIONS BY BHASWAR
• Mathematics
• 2020
In this paper we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erdős-Rényi random graph Gn,p, in the regime of p where the edge of
The Large Deviation Principle for Interacting Dynamical Systems on Random Graphs
• Mathematics, Computer Science
Communications in Mathematical Physics
• 2022
The LDP for random graphs is translated to a class of interacting dynamical systems on such graphs and it is demonstrated that the solutions of the dynamical models depend continuously on the underlying graphs with respect to the cut-norm and apply the contraction principle.
Large deviation for the empirical degree distribution of an Erdos-Renyi graph
With $(d_1,\cdots,d_n)$ denoting the labeled degrees of an Erdos Renyi graph with parameter $\beta/n$, the large deviation principle for $\frac{1}{n}\sum\limits_{j=1}^n\delta_{d_j}$ (the empirical

## References

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• S. Chatterjee
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Random Struct. Algorithms
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This paper solves the problem of sharp large deviation estimates for the upper tail of the number of triangles in an Erdős‐Rényi random graph, by establishing a logarithmic factor in the exponent
Graph limits and exchangeable random graphs
• Computer Science
• 2007
A clear connection between deFinetti's theorem for exchangeable arrays and the emerging area of graph limits is developed and the graph theory is translated into more classical prob- ability.
Metrics for sparse graphs
• Mathematics, Computer Science
• 2007
This paper deals mainly with graphs with $o(n^2)$ but $\omega(n)$ edges: a companion paper [arXiv:0812.2656] will discuss the (more problematic still) case of {\em extremely sparse} graphs, with O( n) edges.
Applications of Stein's method for concentration inequalities
• Mathematics
• 2010
Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this
Random graphs with a given degree sequence
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Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is
Testing properties of graphs and functions
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We define an analytic version of the graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the
Generalized quasirandom graphs
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J. Comb. Theory, Ser. B
• 2008
Szemerédi’s Lemma for the Analyst
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Abstract.Szemerédi’s regularity lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of
Limits of dense graph sequences
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• 2006
Combinatorics, complexity, and chance : a tribute to Dominic Welsh
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• 2007
Preface 1. Orbit counting and the Tutte polynomial 2. Eulerian and bipartite orientable matroids 3. Tutte-Whitney polynomials: some history and generalisations 4. A survey on the use of Markov chains


text:
# What test can I use to compare slopes from two or more regression models?

I would like to test the difference in response of two variables to one predictor. Here is a minimal reproducible example.

library(nlme)
## gls is used in the application; lm would suffice for this example
m.set <- gls(Sepal.Length ~ Petal.Width, data = iris,
subset = Species == "setosa")
m.vir <- gls(Sepal.Length ~ Petal.Width, data = iris,
subset = Species == "virginica")
m.ver <- gls(Sepal.Length ~ Petal.Width, data = iris,
subset = Species == "versicolor")

I can see that the slope coefficients are different:

m.set$coefficients (Intercept) Petal.Width 4.7771775 0.9301727 m.vir$coefficients
(Intercept) Petal.Width
5.2694172   0.6508306
m.ver$coefficients (Intercept) Petal.Width 4.044640 1.426365  I have three questions: 1. How can I test the difference between slopes? 2. How can I test the difference between residual variances? 3. What is a simple, effective way to present these comparisons? A related question, Method to compare variable coefficient in two regression models, suggests re-running the model with a dummy variable to differentiate the slopes, are there options that would allow the use of independent data sets? - add comment ## 2 Answers How can I test the difference between slopes? Include a dummy for species, let it interact with$P_i$, and see if this dummy is significant. Let$L_i$be the sepal length and$P_i$be the pedal width and$S_1, S_2, S_3$be the dummy variables for the three species. The compare the model $$E(L_i) = \beta_0 + \beta_1 P_i$$ with the model that allows the effect of$P_i$to be different for each species: $$E(L_i) = \alpha_0 + \alpha_1 S_2 + \alpha_2 S_3 + \alpha_4 P_i + \alpha_5 P_iS_2 + \alpha_6 P_i S_3$$ The GLS estimators are MLEs and the first model is a submodel on the second, so you can use the likelihood ratio test here. The likelihoods can be extracted using the logLik function and the degrees of freedom for the test will be$4$since you've deleted$4$parameters to arrive at the submodel. What is a simple, effective way to present the comparison? I think the most appealing way would be to plot the regression lines for each species all on the same axes, maybe with error bars based on the standard errors. This would make the difference (or non-difference) between the species and their relationship to$P_i$very apparent. Edit: I noticed another question has been added to the body. So, I'm adding an answer to that: How can I test the difference between residual variances? For this, you'll need to stratify the data set and fit separate models since, the interaction-based model I suggested will constraint the residual variance to be the same in every group. If you fit separate models, this constraint goes away. In that case, you can still use the likelihood ratio test (the likelihood for the larger model is now calculated by summing the likelihoods from the three separate models). The "null" model depends on what you want to compare it with • if you only want to test the variance, while leaving the main effects in, then the "null" model should be the model with the interactions I've written above. The degrees of freedom for the test are then$2$. • If you want to test the variance jointly with the coefficients, then the null model should be the first model I've written above. The degrees of freedom for the test is then$6$. - Why is there no$S_1$in the second model? Is gls(Sepal.Length ~ species:Petal.Width, data = iris) the correct implementation of the model in R? – Abe Jul 25 '12 at 20:27 Hi @Abe.$S_1$is the "reference" species - the regression line for that species is given by$\alpha_0 + \alpha_4 P_i\$. If species is a categorical variable then I think gls(Sepal.Length ~ species*Petal.Width, data=iris) would be the syntax. –  Macro Jul 25 '12 at 21:31
@Macro Nice answer (+1)! I wonder whether you could fit the gls model but allowing for different residual variances for each Species with the option weights=varIdent(form=~1|Species) (regarding the second question)? –  COOLSerdash May 19 '13 at 14:06


text:
# General Solution to the Wave Equation (Inhomogeneous)

I am having trouble finding the general solution to the wave equation for a plucked string subject to gravity. The question is:

Find the general solution to: $$u_{tt}-c^2u_{xx}=-g$$ Where $g$ is the magnitude of the acceleration due to gravity.

Now I have tried to factor it as usual, i.e.: $$\left(\frac{\partial}{\partial t}-c\frac{\partial}{\partial x}\right)\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial x}\right)\cdot u+g=0$$ But then what are the PDE's in this case? Are the Transport Equations as the following?: $$u_t+cu_x=-g, \>\>\>\> u_t-cu_x=-g$$ I know it's just one constant added to the equation but it's really confusing me. If anyone could help itd be appreciated!

Edit: I have come to the realization that $u$ can be: $$u(x,t) = f(x+ct)+g(x-ct)+\frac{gx^2}{2c^2}+bx+c, \>\>\>\> b,c\in\mathbb{R}$$ But however, we can also have the solution: $$u(x,t) = f(x+ct)+g(x-ct)-\frac{gt^2}{2}+bt+c, \>\>\>\>\>\> b, c\in\mathbb R$$ Or even a combination of the two: $$u(x,t) = f(x+ct)+g(x-ct)+\frac{gx^2}{4c^2}+bx+c-\frac{gt^2}{4}+\beta t+\gamma, \>\>\>\> b,c,\beta,\gamma\in\mathbb R$$ But my question only asks for $\boldsymbol{one}$ solution. Which is the correct one??


text:
## Projection theorems using effective dimension

In this post I will describe how to use effective dimension to investigate how the fractal dimension of a set changes when projected onto a linear subspace. The first theorem investigating this behavior is the celebrated Marstrand projection theorem.

Theorem (Marstrand’s projection theorem): Let $E \subseteq \mathbb{R}^2$ be an analytic set. Then for almost every angle $\theta$,

$\text{dim}_H(p_\theta E) = \min\{1, \text{dim}_H(E)\}$,

where

$p_\theta(x, y) = x\cos \theta + y\sin\theta.$

Note that, since the projection map $p_\theta$ is Lipschitz,

$\text{dim}_H(p_\theta E) \leq \text{dim}_H(E).$

Also, since $p_\theta$ maps into $\mathbb{R}$, we have

$\text{dim}_H(p_\theta E) \leq 1.$

Thus, Marstrand’s theorem shows that, for almost every angle $\theta$, the Hausdorff dimension of $p_\theta E$ is the maximum possible.

Remark: We cannot, in general, drop the almost every assumption. It is possible to construct a Borel set E of dimension, e.g. 1, such that

$\text{dim}_H(p_\theta E) < 1,$

for uncountably many angles $\theta$. There is great interest in proving bounds on the “exceptional” directions not satisfying the conclusion of MPT. Kauffman gave the first bounds in this direction, but sharp bounds are still an important open problem.

Remark: Marstrand also showed that, when the dimension of E is strictly greater than 1, the projection of E has positive measure for almost every angle. We do not yet have a proof of this theorem using algorithmic techniques (and I think this is an important problem that needs a solution).

Remark: Marstrand’s theorem can be generalized to sets in $\mathbb{R}^n$, for arbitrary n and to subspaces of arbitrary dimension m < n. In this post, I will restrict to subsets of $\mathbb{R}^2$ and projections onto lines.

In this post, I will describe the techniques Neil Lutz and I used to give a proof of Marstand’s theorem using effective dimension.

It turns out that these techniques can be also be used to prove two new theorems on the dimension of projected sets. The first result proves that the Marstrand’s theorem holds for sets whose Hausdorff and packing dimensions agree (not necessarily analytic). The second result gives lower bounds on the packing dimension of projected sets. Tuomas Orponen recently gave a new proof of these two theorems using classical techniques. I will explore these new theorems, and related work, in a future post.

### Effective dimension of projected points

The typical approach to proving (classical) theorems about the fractal dimension of sets using effective dimension is to prove the analogous bound for the effective dimension of points and then apply the point-to-set principle (discussed in the next sections). We follow this pattern, and begin with effective dimension.

Remark: It turns out that it is not so simple to convert the results of this section into a result on the dimension of sets. Still, it makes sense to start with bounding the effective dimension of projected points. In the next section, we will discuss how to convert this into a bound on the Hausdorff dimension of projected sets.

In the effective setting, bounds on the dimension of projected points follows almost verbatim from the techniques Neil and I used for points on lines (as described in this previous post). I will briefly recall the main ideas.

We first state the point-wise analog of Marstrand’s theorem. Let $z \in \mathbb{R}^2$ be a point. . Let $\theta \in [0, 2\pi)$ be an angle which is random relative to z. Finally, let

$d = \min\{\text{dim}^\theta(z), 1\}$.

Our goal is to show

$\text{dim}^\theta(p_\theta z) = d \ \ \ \ (1)$.

Remark: Note that we bounding the dimension of $p_\theta z$ relative to $\theta$. It is also important to note that this bound is the best possible, since

$\text{dim}^\theta(p_\theta z) \leq \text{dim}^\theta(z)$,

and

$\text{dim}^\theta(p_\theta z) \leq 1$.

The proof of (1) essentially follows from the techniques Neil and I developed for points on lines. For full details, see this previous post, but recall the main ideas: For every sufficiently large precision r we will do the following.

1. Use conditional complexity to construct an oracle A which reduces the complexity of z. Thus, $K^{A}_r(z) = \eta r$, where $\eta < d$. Note that this is not relative to $\theta$.
2. Using the linearity of $p_\theta$, we show that the complexity of any point w such that $p_\theta(z) = p_\theta(w)$ is lower bounded by the complexity of $\theta$ (this corresponds to the “intersection bound” for lines). Thus, since $\theta$ was chosen to be random, any other point w must have high complexity.
3. Since $\theta$ was chosen to be random relative to z we use (2) to show that z is essentially the only point of complexity less than $\eta r$ which projects to $p_\theta(z)$. Thus, simple enumeration allows us to compute z given $\theta$ and $p_\theta(z)$.

The formal statement corresponding to step 2 is encapsulated by the following lemma.

Lemma 1: Let $z \in \mathbb{R}^2$ and $\theta \in [0, 2\pi)$. Let $w \in \mathbb{R}^2$ such that $p_\theta(z) = p_\theta(w)$. Let $r \in \mathbb{N}$ and $A$ be any oracle. Then

$K^{A, z}_{r-t}(\theta) \leq K^{A,z}_r(w) + o(r)$,

where $t = -\log\|w - z\|$.

As with points on lines, the purpose of this lemma is to give lower bounds on the complexity of any point w which projects to $p_\theta(z)$.

For the first step, recall that we can construct an oracle A such that

$K^{A}_r(z) = \eta r + O(\log r)$.

An important property of this construction is that A does not contain any information not already contained in z, and therefore, if $\theta$ is random relative to z,

$K^{A, z}_{r-t}(\theta) = K^{z}_{r-t}(\theta) = r - t - O(\log r)$,

for all t < r. As in the case of lines, this combined with Lemma 1 shows that any w such that

1. $p_\theta(z) = p_\theta(w)$, and
2. $K^{A}_r(w) \leq \eta r$

must be very close to z.

Therefore we can proceed to step 3, allowing us to conclude that

$K^{A, \theta}_r(z) \leq K^{A, \theta}_r(p_\theta z) + o(r)$.

It is important to note that this does not immediately imply what we want (equation (1)). We still need to show that

$K^{A, \theta}_r(z) = K^A_r(z) - o(r) = \eta r - o(r)$.

Fortunately, this follows from a standard symmetry of information argument. Since $\theta$ is random relative to z, $\theta$ gives no information about z. Since A contains only information contained in z, there is no mutual information between oracles A and $\theta$, and the conclusion follows.

### The point-to-set principle

In this section, I describe how to convert the lower bound on the effective dimension of projected points to lower bounds on the fractal dimensions of projected sets. The bridge between the classical and effective worlds, proved by Jack Lutz and Neil Lutz, is called the point-to-set principle.

Theorem 2 (Point-to-set principle): Let $E \subseteq \mathbb{R}^n$. Then

$\text{dim}_H(E) = \min\limits_{A\subseteq\mathbb{N}} \sup\limits_{x\in E} \text{dim}^A(x)$,

and

$\text{dim}_P(E) = \min\limits_{A\subseteq\mathbb{N}} \sup\limits_{x\in E} \text{Dim}^A(x)$.

Oracles testifying to the above characterizations are called Hausdorff or packing oracles.

In many cases, proving a classical bound on the Hausdorff or packing dimension of a set via effective dimension reduces to proving the analogous bound for the effective dimension of points, and relativizing with respect to a Hausdorff (or packing) oracle. However, in the case of projections, it is not so simple.

The key issue is Step 1 from the previous section, where we lowered the complexity of our point. The problem with trying to naively apply the point-to-set principle is that a Hausdorff oracle B for $p_\theta E$ could give a lot of information about $\theta$.

If we followed the argument of the previous section, relative to B, we run into a problem in Step 2. Lemma 1 does relativize, and so we can conclude that $K^{B, A, z}_{r-t}(\theta) \leq K^{B, A, z}_{r-t}(w) + o(r)$. However, because B can give a lot of information about $\theta$, this will not give a good enough bound on the complexity of w.

It is important to note that this is not a limitation of our tools, but a fundamental obstacle that we can’t avoid, in general. It is known (due to a construction of Davies) that Marstrand’s theorem does not hold for all sets, assuming CH. Therefore, since there is no restriction of the set in the point-to-set principle, we are doomed to fail.

Remark: This is a very interesting phenomenon, and in a later post I will describe it in more detail. It turns out that it is possible to extract sufficient conditions for our techniques to work. In particular, we can use algorithmic information theory to understand why assumptions of the set E are needed for MPT.

To overcome this obstacle, Neil and I turned to a restricted version of the point-to-set principle proved by John Hitchcock.

Theorem 3: Let $E \subseteq \mathbb{R}^n$ be $\Sigma^0_1$ relative to oracle B. Then

$\text{dim}_H(E) = \sup\limits_{x\in E} \text{dim}^B(x).$

Remark: The key feature of this restricted point-to-set principle is that we can eliminate the oracle (for $\Sigma^0_1$ sets). Informally, this removes the main obstacle preventing us from applying our pointwise projection bound.

We can combine this with a classical result of geometric measure theory.

Theorem 4: Let $E \subseteq \mathbb{R}^n$ be analytic with Hausdorff dimension s. Then there is a $\mathbf{\Sigma}^0_1$ subset F of E such that the Hausdorff dimension of F is s.

### Effective proof of Marstrand’s theorem

We can now prove Marstrand’s theorem. Let E be an analytic subset of the plane with dimension s. Let F be a $\mathbf{\Sigma}^0_1$ subset of E. Let B be an oracle such that F is $\Sigma^0_1$ relative to B. We note that, for any angle $\theta$, the projection $p_\theta F$ is $\Sigma^0_1$ relative to $(B, \theta)$.

For each n, let $z_n \in F$ be an element such that

$\text{dim}^B(z_n) > s - 1/n$.

Let $\theta$ be random relative to the join of $B, z_1, z_2,\ldots$. By Hitchcock’s point-to-set principle, it suffices to show that, for every n,

$\text{dim}^{B, \theta}(p_\theta z_n) > \text{dim}^{B,\theta}(z_n) = s - 1/n$.

The point here is that we chose $\theta$ so that it is random relative to $(B, z_n)$ for every n. Therefore, we can simply relativize (with respect to B) the pointwise bound of the previous section, and the conclusion follows.

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# isotope of a groupoid

Let $G,H$ be groupoids (http://planetmath.org/Groupoid). An isotopy $\phi$ from $G$ to $H$ is an ordered triple: $\phi=(f,g,h)$, of bijections from $G$ to $H$, such that

 $f(a)g(b)=h(ab)\qquad\mbox{for all }a,b\in G.$

$H$ is called an isotope of $G$ (or $H$ is isotopic to $G$) if there is an isotopy $\phi:G\to H$.

Some easy examples of isotopies:

1. 1.

If $f:G\to H$ is an isomorphism, $(f,f,f):G\to H$ is an isotopy. By abuse of language, we write $f=(f,f,f)$. In particular, $(1_{G},1_{G},1_{G}):G\to G$ is an isotopy.

2. 2.

If $\phi=(f,g,h):G\to H$ is an isotopy, then so is

 $\phi^{-1}:=(f^{-1},g^{-1},h^{-1}):H\to G,$

for if $f^{-1}(a)=c$ and $g^{-1}(b)=d$, then $ab=f(c)g(d)=h(cd)$, so that $f^{-1}(a)g^{-1}(b)=cd=h^{-1}(ab)$

3. 3.

If $\phi=(f,g,h):G\to H$ and $\gamma=(r,s,t):H\to K$ are isotopies, then so is

 $\gamma\circ\phi:=(r\circ f,s\circ g,t\circ h):G\to K,$

for $(r\circ f)(a)(s\circ g)(b)=r(f(a))s(g(b))=t(f(a)g(b))=t(h(ab))=(t\circ h)(ab)$.

From the examples above, it is easy to see that “groupoids being isotopic” on the class of groupoids is an equivalence relation, and that an isomorphism class is contained in an isotopic class. In fact, the containment is strict. For an example of non-isomorphic isotopic groupoids, see the reference below. However, if $G$ is a groupoid with unity and $G$ is isotopic to a semigroup $S$, then it is isomorphic to $S$. Other conditions making isotopic groupoids isomorphic can be found in the reference below.

An isotopy of the form $(f,g,1_{H}):G\to H$ is called a principal isotopy, where $1_{H}$ is the identity function on $H$. $H$ is called a principal isotope of $G$. If $H$ is isotopic to $G$, then $H$ is isomorphic to a principal isotope $K$ of $G$.

###### Proof.

Suppose $(f,g,h):G\to H$ is an isotopy. To construct $K$, start with elements of $G$, which will form the underlying set of $K$. The binary operation on $K$ is defined by

 $a\cdot b:=(f^{-1}\circ h)(a)(g^{-1}\circ h)(b).$

Then $\cdot$ is well-defined, since $f,g$ are bijective, for all pairs of elements of $G$. Hence $K$ is a groupoid. Furthermore, $(f^{-1}\circ h,g^{-1}\circ h,1_{K}):G\to K$ is an isotopy by definition, so that $K$ is a principal isotope of $G$. Finally, $h(a\cdot b)=h(f^{-1}(h(a))g^{-1}(h(b)))=f(f^{-1}(h(a)))g(g^{-1}(h(b)))=h(a)h(b)$, showing that $h:K\to H$ is a bijective homomorphism, and hence an isomorphism. ∎

Remark. In the literature, the definition of an isotope is sometimes limited to quasigroups. However, this is not necessary, as the follow proposition suggests:

###### Proposition 1.

Any isotope of a quasigroup is a quasigroup.

###### Proof.

Suppose $(f,g,h):G\to H$ is an isotopy, and $G$ a quasigroup. Pick $x,z\in H$. Let $a,c\in G$ be such that $f(a)=x$ and $h(c)=z$. Let $b\in G$ be such that $ab=c$. Set $y=g(b)\in H$. Then $xy=f(a)g(b)=h(ab)=h(c)=z$. Similarly, there is $t\in H$ such that $tx=z$. Hence $H$ is a quasigroup. ∎

On the other hand, an isotope of a loop may not be a loop. Nevertheless, we sometimes say that an isotope of a loop $L$ as a loop isotopic to $L$.

## References

• 1 R. H. Bruck: A Survey of Binary Systems.  Springer-Verlag. New York (1966).
 Title isotope of a groupoid Canonical name IsotopeOfAGroupoid Date of creation 2013-03-22 18:35:54 Last modified on 2013-03-22 18:35:54 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 8 Author CWoo (3771) Entry type Definition Classification msc 20N02 Classification msc 20N05 Synonym isotopism Synonym homotopism Defines isotopy Defines isotope Defines homotopy Defines homotope Defines isotopic Defines homotopic Defines principal isotopy Defines principal isotope


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 Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/24473

 Title: Past Cone Dynamics and Backward Group Preserving Schemes for Backward Heat Conduction Problems Authors: C.-S. Liu;C.-W. Chang;J.-R. Chang Contributors: NTOU:Department of Systems Engineering and Naval Architecture國立臺灣海洋大學:系統工程暨造船學系 Keywords: Past cone dynamics;Backward group preserving scheme;Backward heat conduction problem;Strongly ill-posed problem. Date: 2006 Issue Date: 2011-10-20T08:12:26Z Publisher: Computer Modeling in Engineering & Science Abstract: Abstract:In this paper we are concerned with the backward problems governed by differential equations. It is a first time that we can construct a backward time dynamics on the past cone, such that an augmented dynamical system of the Lie type$\mathaccentV {dot}05F{\bf X}={\bf B}({\bf X},t){\bf X}$ with$t \in {\@mathbb R}^-$,${\bf X} \in {\@mathbb M}^{n+1}$ lying on the past cone and${\bf B} \in so(n,1)$, was derived for the backward differential equations system$\mathaccentV {dot}05F{\bf x} = {\bf f}({\bf x},t)$,$t \in {\@mathbb R}^{-}, {\bf x} \in {\@mathbb R}^n$. These two differential equations systems are mathematically equivalent. Then we apply the backward group preserving scheme (BGPS), which is an explicit single-step algorithm formulated by an exponential mapping to preserve the group preperties of$SO_o(n,1)$, on the backward heat conduction problem (BHCP). It can retrieve all the initial data with high order accuracy. Several numerical examples of the BHCP were work out, and we show that the BGPS is applicable to the BHCP, even those of strongly ill-posed ones. Under the noisy final data the BGPS is also robust to against the disturbance. The one-step BGPS effectively reconstructs the initial data from a given final data, with a suitable grid length resulting into a high accuracy never seen before. The results are very significant in the computations of BHCP. Relation: 12(1), pp.67-82 URI: http://ntour.ntou.edu.tw/handle/987654321/24473 Appears in Collections: [系統工程暨造船學系] 期刊論文

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# Sum of Pairs of Integers less than N

What is the sum of all (ordered) pairs of integers less than N? That is, what is the value of this:

$\sum_{i \le n} \sum_{j \le n} ij$

Now this is very easy to determine. One way is to notice that it is the same as:

$(1+2+\ldots+n)^2 = \frac{1}{4}[n(n+1)]^2$

You can see this by imagining the expansion of square on the left hand side. Alternatively you can get the same result by just evaluating the sum in the obvious way. (Move the i outside the inner sum, evaluate the inner sum, and move the result out of the remaining sum.)

However, I have remarked before about how the square of the sum of the first N numbers is equal to the sum of the first N cubes. Therefore we have the equality:

$\sum_{i\le n}\sum_{j \le n} ij = \sum_{i \le n} i^3$

So this is easy to prove it is true by recognising what the sum on the left really is, but is there a more direct way of seeing it? It just isn’t obvious that the sum of the pairs of integers should be the sum of cubes less than that number!

In fact it is very easy to see. The sum can just be rewritten as:

$\sum_{i\le n}\sum_{j \le n} ij = \sum_{i\le n}\sum_{j \le i} ij + i(i-j)$

To convince yourself that no terms have been lost, note that on the left hand side there are $n^2$ terms, whilst on the right there are $2i-1$ for each i(There are not 2i as when i=j there is nothing for it to pair with) and so, using the formula for the sum of the first N numbers, there are $n^2$ terms on the right. Obviously no term has been repeated and so they are equal.

Then we just have:

$\sum_{i\le n}\sum_{j \le i} ij + i(i-j) = \sum_{i\le n}\sum_{j \le i} i^2 = \sum_{i\le n} i^3$

And so it is shown. A very simple result, but I like seeing these relationships directly.


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Back to Mover page.

Documentation added 25 Mar 2021 by Vikram K. Mulligan, Flatiron Institute (vmulligan@flatironinstitute.org).

## trRosettaConstraintGenerator

The trRosettaConstraintGenerator takes as input a multiple sequence alignment, converts this to a one-hot 3D input tensor, and runs this through the trRosetta neural network to generate predictions of the probability distributions of inter-residue distances and orientations. It then converts this information to a list of Rosetta constraints.

These constraints can be useful for de novo structure prediction. In this case, however, it is not necessary to script this constraint generator directly. Instead, the trRosetta application and the trRosettaProtocol mover wrap this constraint generator, and provide a full structure prediction pipeline that takes an MSA as input and produces a pose as output. If one wishes to do more exotic things, such as using trRosetta distance and orientation constraints in design, docking, or loop modelling, it can be useful to have direct access to the trRosettaConstraintGenerator, however.

### A note on nomenclature

Although "omega" and "phi" are commonly used to refer to the third and first mainchain backbone dihedrals of an alpha amino acid, and "theta" is used to refer to the second mainchain backbone dihedral of a beta-amino acid, in the context of trRosetta-related protocols, these Greek letters are assigned new meanings. Here, "omega" refers to the inter-residue dihedral angle between the CA and CB atoms of a first residue and the CB and CA atoms of a second residue. "Theta" refers to the inter-residue dihedral angle between the N, CA, and CB atoms of a first residue and the CB atom of a second residue. And "phi" refers to the inter-residue angle between the CA and CB atoms of a first residue and the CB atom of a second residue.

### Compilation requirements

The trRosettaConstraintGenerator requires that Rosetta be compiled with Tensorflow support. See the autogenerated description below for details on how to compile Rosetta and link Tensorflow.

## All options

Autogenerated Tag Syntax Documentation:

The trRosettaConstraintGenerator takes as input a file containing a multiple sequence alignment, feeds this to the trRosetta neural network, and uses the output to generate distance and angle constraints between pairs of residues as described in Yang et al. (2020) Improved protein structure prediction using predicted interresidue orientations. Proc. Natl. Acad. Sci. USA 117(3):1496-503. https://doi.org/10.1073/pnas.1914677117.

The trRosettaConstraintGenerator requires compilation with Tensorflow support. To compile with Tensorflow support:

2. Unzip/untar the archive into a suitable directory (~/mydir/ is used here as an example), and add the following environment variables: Linux, Windows: LIBRARY_PATH=$LIBRARY_PATH:~/mydir/lib LD_LIBRARY_PATH=$LD_LIBRARY_PATH:~/mydir/lib MacOS: LIBRARY_PATH=$LIBRARY_PATH:~/mydir/lib DYLD_LIBRARY_PATH=$DYLD_LIBRARY_PATH:~/mydir/lib

3. Edit your user.settings file (Rosetta/main/source/tools/build/user.settings), and uncomment (i.e. remove the octothorp from the start of) the following lines: import os 'program_path' : os.environ['PATH'].split(':'), 'ENV' : os.environ,

4. Compile Rosetta, appending extras=tensorflow (for CPU-only) or extras=tensorflow_gpu (for GPU) to your scons command. For example: ./scons.py -j 8 mode=release extras=tensorflow bin

<trRosettaConstraintGenerator name="(&string;)" msa_file="(&string;)"
generate_distance_constraints="(true &bool;)"
generate_omega_constraints="(true &bool;)"
generate_theta_constraints="(true &bool;)"
generate_phi_constraints="(true &bool;)"
distance_constraint_prob_cutoff="(0.05 ℜ)"
omega_constraint_prob_cutoff="(0.55 ℜ)"
theta_constraint_prob_cutoff="(0.55 ℜ)"
phi_constraint_prob_cutoff="(0.65 ℜ)"
distance_constraint_weight="(1.0 ℜ)"
omega_constraint_weight="(1.0 ℜ)"
theta_constraint_weight="(1.0 ℜ)"
phi_constraint_weight="(1.0 ℜ)" />
• msa_file: Filename for a multiple sequence alignment file, in a3m format. Dashes indicate gap sequences, and lowercase characters will be removed (and flanking regions ligated). If not provided, the commandline option -trRosetta:msa_file will be used. One or the other is required.
• generate_distance_constraints: Set whether this will generate distance constraints. Distance constraints constrain the distance between pairs of amino acids. These are symmetric, and are only generated once per amino acid pair (since dist(a,b) == dist(b,a)). Defaults to commandline setting -trRosetta:use_distance_constraints.
• generate_omega_constraints: Set whether this will generate omega dihedral constraints. Omega constraints constrain the dihedral between CA1-CB1-CB2-CA2 in pairs of amino acids. These are symmetric, and are only generated once per amino acid pair (since omega(a,b) == omega(b,a)). Note that this is NOT omega the backbone dihedral torsion! Defaults to commandline setting -trRosetta:use_omega_constraints.
• generate_theta_constraints: Set whether this will generate theta dihedral constraints. Theta constraints constrain the dihedral between N1-CA1-CB1-CB2 in pairs of amino acids. These are asymmetric (i.e. theta(a,b)!=theta(b,a)), so there are two per amino acid pair (unless a == b, which is skipped). Defaults to commandline setting -trRosetta:use_theta_constraints.
• generate_phi_constraints: Set whether this will generate phi angle constraints. Phi constraints constrain the angle between CA1-CB1-CB2 in pairs of amino acids. These are asymmetric (i.e. phi(a,b)!=phi(b,a)), so there are two per amino acid pair (unless a == b, which is skipped). Note that this is NOT phi the backbone dihedral torsion! Defaults to commandline setting -trRosetta:use_phi_constraints.
• distance_constraint_prob_cutoff: Set the probability cutoff below which we omit a distance constraint. Default 0.05, or whatever is set on the commandline with the -trRosetta:distance_constraint_prob_cutoff commandline option.
• omega_constraint_prob_cutoff: Set the probability cutoff below which we omit a omega dihedral constraint. Default 0.55, or whatever is set on the commandline with the -trRosetta:omega_constraint_prob_cutoff commandline option.
• theta_constraint_prob_cutoff: Set the probability cutoff below which we omit a theta dihedral constraint. Default 0.55, or whatever is set on the commandline with the -trRosetta:theta_constraint_prob_cutoff commandline option.
• phi_constraint_prob_cutoff: Set the probability cutoff below which we omit a phi angle constraint. Default 0.65, or whatever is set on the commandline with the -trRosetta:phi_constraint_prob_cutoff commandline option.
• distance_constraint_weight: Set the weight for trRosetta-generated distance constraints. Defaults to 1.0, or whatever was set on the commandline with the -trRosetta:distance_constraint_weight commandline option.
• omega_constraint_weight: Set the weight for trRosetta-generated omega dihedral constraints. Defaults to 1.0, or whatever was set on the commandline with the -trRosetta:omega_constraint_weight commandline option.
• theta_constraint_weight: Set the weight for trRosetta-generated theta dihedral constraints. Defaults to 1.0, or whatever was set on the commandline with the -trRosetta:theta_constraint_weight commandline option.
• phi_constraint_weight: Set the weight for trRosetta-generated phi angle constraints. Defaults to 1.0, or whatever was set on the commandline with the -trRosetta:phi_constraint_weight commandline option.

### Best practices

At the time of this writing, it is recommended to leave all options set to defaults unless one has reason to customize the settings.

## Example

The following example roughly reproduces the protocol used by the trRosettaProtocol mover and trRosetta application. Note that (a) this is somewhat simplified, limiting its accuracy as a structure prediction protocol, and (b) it is not necessary to manually script the full structure prediction protocol, since the mover or the application can be used instead. This is only for demonstration purposes to show how the trRosettaConstraintGenerator can be scripted.

<ROSETTASCRIPTS>
<SCOREFXNS>
<ScoreFunction name="cen" weights="score0.wts" >
<Reweight scoretype="atom_pair_constraint" weight="5.0" />
<Reweight scoretype="angle_constraint" weight="1.0" />
<Reweight scoretype="dihedral_constraint" weight="1.0" />
</ScoreFunction>
<ScoreFunction name="r15" weights="ref2015.wts" />
<ScoreFunction name="r15_cst" weights="ref2015_cst.wts" />
</SCOREFXNS>
<SIMPLE_METRICS>
<RMSDMetric name="measure_rmsd"
use_native="true"
super="true"
custom_type="RMSD_after_centroid_phase_"
rmsd_type="rmsd_protein_bb_heavy"
/>
<RMSDMetric name="measure_rmsd2"
use_native="true"
super="true"
custom_type="RMSD_after_fullatom_phase_"
rmsd_type="rmsd_protein_bb_heavy"
/>

</SIMPLE_METRICS>
<CONSTRAINT_GENERATORS>
<trRosettaConstraintGenerator name="gen_csts"
msa_file="inputs/1r6j_msa.a3m"
/>
</CONSTRAINT_GENERATORS>
<MOVERS>
<InitializeByBins name="randomize_bb"
bin_params_file="ABBA"
/>
constraint_generators="gen_csts"
/>
<MinMover name="minimize"
scorefxn="cen"
tolerance="0.0000001"
bb="true" chi="false" jump="0"
/>
<ClearConstraintsMover name="remove_csts" />
<SwitchResidueTypeSetMover name="make_fullatom" set="fa_standard"/>
<FastRelax name="frlx" repeats="3" scorefxn="r15_cst" />
</MOVERS>
<PROTOCOLS>
</PROTOCOLS>
<OUTPUT scorefxn="r15" />

The input multiple sequence alignment file can be generated with HHBlits or other software. See the trRosettaProtocol mover documentation for more details and an example of the .a3m file format.

## Code organization

Please see the trRosetta application documentation for information about the trRosetta code organization.

## References

• The trRosetta neural network is described in Yang et al. (2020) Proc Natl Acad Sci USA 117(3):1496-1503 (doi 10.1073/pnas.1914677117).
• The trRosettaProtocol mover, trRosettaConstriantGenerator, trRosetta application, and other C++ infrastructure were written by Vikram K. Mulligan (vmulligan@flatironinstitute.org), and are currently unpublished.


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# Question #a33e0

##### 1 Answer
Apr 8, 2014

It is easier to understand by looking at pictures:

$I C {l}_{2}^{-}$

$N {O}_{2}$

$I C {l}_{4}^{-}$


text:
## Wavelength

1. The problem statement, all variables and given/known data
A tube of air is open at only one end and has a length of 1.7 m. This tube sustains a standing wave at its third harmonic. What is the distance between one node and the adjacent antinode?

2. Relevant equations
v = f (lambda)

3. The attempt at a solution
If it's in the third harmonic, that means the length of the tube is equal to 1.25 wavelengths. So, 1.7 / 1.25 = 1.36. The distance between an antinode and the adjacent node is .25 lambda, so 1.36 * .25 = .34m. So .34m is the answer.
 PhysOrg.com science news on PhysOrg.com >> Leading 3-D printer firms to merge in $403M deal (Update)>> LA to give every student an iPad;$30M order>> CIA faulted for choosing Amazon over IBM on cloud contract
 Recognitions: Homework Help If it's in the third harmonic, who many nodes and anti nodes are formed in the tube?
 Doesn't it look like this? http://img101.imageshack.us/my.php?i...armonicoq0.png

## Wavelength

Can someone tell me what I did wrong?
 Recognitions: Homework Help In the tube closed at one end, third harmonic is the first overtone. So L = 3*lambda/4

 Similar discussions for: Wavelength Thread Forum Replies Introductory Physics Homework 11 Quantum Physics 1 Introductory Physics Homework 2 Introductory Physics Homework 1 Introductory Physics Homework 2


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BACKGROUND:

Although pharyngitis is common, group A Streptococcus is an uncommon etiology, and sequelae are rare in patients <3 years old. Inappropriate testing leads to increased cost of health care and unnecessary exposure to antibiotics. Rapid streptococcal tests (RSTs) for group A Streptococcus pharyngitis are not routinely indicated in this age group. At our urban, tertiary pediatric emergency department (ED), on average, 20 RSTs were performed each month for patients <3 years of age. Our objective was to reduce RSTs in the ED in patients aged <3 years by 50% in 18 months.

METHODS:

We initiated this project in October 2016 at an urban, tertiary pediatric ED. We surveyed pertinent multidisciplinary stakeholders to identify factors leading to RSTs in children <3 years of age. We conducted multiple interventions and collected weekly data on the number of RSTs in children aged <3 years (outcome measure) and the number of family complaints and return visits for complications of pharyngitis (balancing measure). We used statistical process control for analysis.

RESULTS:

The mean number of RSTs ordered per month in patients aged <3 years declined by 52% in 10 months. The majority of tests during the study phase were ordered by nurse practitioners (62%) for patients aged 25 to 36 months (66%). There has been 1 family grievance and no patient complications attributable to the project.

CONCLUSIONS:

Our interventions led to a successful and sustained reduction of RSTs in patients aged <3 years. A local clinical practice guideline was developed, and the project was expanded to other acute care settings.

There are 12 million outpatient pediatric visits per year for sore throat, accounting for $224 million to$539 million per year in health care–related costs.1,2  Streptococcal pharyngitis is common in the pediatric population; however, in patients aged <3 years, group A Streptococcus (GAS) is a rare cause of sore throat, and sequelae (such as acute rheumatic fever) are rare.3  Inappropriate testing leads to increased cost of health care and unnecessary exposure to antibiotics. At our urban, tertiary pediatric emergency department (ED), 242 rapid streptococcal tests (RSTs) were ordered for patients aged <3 years from October 1, 2015, to September 30, 2016, for an average of 20.1 tests per month.

According to the national guidelines published by the Infectious Diseases Society of America (IDSA) in 2012, RSTs for GAS pharyngitis are not routinely indicated in this age group unless the patient both meets clinical criteria and has a household contact with documented streptococcal pharyngitis.4  Despite strong recommendations from national committees, including the IDSA, against routinely testing patients aged <3 years, there is a gap in clinical practice for ordering RSTs at our local institution.5

On the basis of information gathered with our baseline data, including surveys and key stakeholder opinions, we identified gaps (namely lack of knowledge and retention and family expectation) and designed interventions to address them. Our multidisciplinary quality improvement (QI) team used a variety of methods to understand factors leading to ordering RSTs. This allowed us to develop multiple sequential interventions to reduce testing in this population. Another identified gap was the need for reminders, and we developed a mechanism for in-person reminders for the short-term and focused reminders in the ordering process as a more sustainable intervention. Lastly, interventions were designed to address family questions and expectations from ED management. We believed this multipronged and targeted approach would be best suited to optimize the ordering of testing.

Our aim with this multidisciplinary project was to reduce the mean number of RSTs ordered per month in the ED in patients aged <3 years by 50% over 18 months between October 1, 2016, and March 31, 2018.

The Children’s Mercy Kansas City ED at the Adele Hall Campus is an urban, tertiary pediatric ED with 75 000 visits annually. Medical providers in the ED include faculty physicians trained in pediatric emergency medicine (PEM), pediatrics and/or emergency medicine physicians, resident physicians (pediatrics, internal medicine–pediatrics, emergency medicine, and family medicine), fellows (PEM, hospital medicine, and child abuse and neglect), medical students, and nurse practitioners (NPs). The patients are triaged according to the Emergency Severity Index criteria and roomed accordingly in 1 of 3 different zones of the ED. NPs see a large number of low-acuity patients, including those who present with sore throat.

We assembled an interdisciplinary team of personnel, including 3 ED nurses (Jacque Brown, Angela Wiedner, and Amy Scott), 2 ED NPs (J.M. and J.E.-S.), 2 PEM attending physicians (S.J. and Theodore Barnett), 1 PEM fellow (T.A.), and 2 pediatric infectious diseases (ID) physicians (L.N. and A.M.). One PEM attending with formal QI training also served as quality consultant.

Our team surveyed representatives from pertinent disciplines (NPs, nurses, PEM physicians, pediatric ID physicians, and PEM leadership) to identify factors leading to RSTs in this age group. With this information, we built a process map and key driver diagram to identify gaps and build interventions. Figure 1 shows our process map for pharyngitis diagnosis at baseline, with highlighted areas representing opportunities for educational intervention. We found that lack of knowledge and/or retention, family expectations, and practices driven by adult literature were all factors in choosing to obtain an RST. We developed multiple sequential interventions on the basis of focused discussions with key stakeholders (PEM physicians, PEM fellows, NPs, and nurses) as well as expert opinion from PEM and ID faculty and PEM management. The order of interventions was based on the discussions with stakeholders as well as the interventions’ feasibility and importance.

FIGURE 1

Flow diagram for RSTs in patients <3 years old. This figure shows our process map for pharyngitis diagnosis at baseline. The bar on the left shows the patient, bedside nurse, and provider. The highlighted areas represent opportunities for educational intervention.

FIGURE 1

Flow diagram for RSTs in patients <3 years old. This figure shows our process map for pharyngitis diagnosis at baseline. The bar on the left shows the patient, bedside nurse, and provider. The highlighted areas represent opportunities for educational intervention.

Close modal

Elements of the interventions are as follows.

1. Provider education: PEM physicians, fellows, and NPs were included. We used several forums for this purpose, including division meetings, huddles, and emails, and created educational handouts. We chose provider education for the first step on the basis of feedback from the QI team, who wanted to start with education and culture change before instituting any changes to the ordering process. Education was important because it enabled us to identify more champions for the project and allowed for repeat educational sessions for newer staff. We also conducted 3 focused discussions with the champions to provide continued education and feedback.

2. Nursing education: Education for nurses was focused on reminders about RST guidelines at weekly huddles, informational posters about streptococcal pharyngitis and RST guidelines in the staff break room, and education about the existing standing order for RSTs. This step was particularly important because RSTs at our institution may be performed in triage by the nurses. It also served as an opportunity for the nurses to review other standing orders in place at our institution with the new staff.

3. Daily management system (DMS) and lean process: Our ED uses DMS, a structured management system that includes visual boards, huddles, leadership support, continuous improvement, and methodical problem-solving to ensure consistent and reliable service for patients. We used our DMS process for this project to provide weekly visual updates about the number of RSTs obtained in children <3 years in the previous week, which were attended daily by physician, nursing, and NP representatives. This visual step was important to increase the awareness of inappropriate RSTs, provide education, gather support for the project, and offer frequent reminders to the front-line staff about the RST guidelines.

4. Resident education: Education for residents included an annual case-based didactic lecture during the project time frame and a short educational session at a housestaff meeting. There was informal education provided by supervising physicians during clinical shifts if a resident considered ordering an RST in a patient <3 years. Residents were also provided with resources, including relevant journal articles.

5. Computerized order process alert: We used our electronic medical record (EMR) system (Cerner, North Kansas City, MO) to create a real-time alert and reminder. When an RST order was placed on a patient <3 years old, an alert would remind the user of the recommendations about when RSTs are appropriate. It would also ask the user to either cancel the test or select a rationale for ordering the test among the following options: recent exposure to a sick contact with documented streptococcal pharyngitis, parent and/or guardian requesting the test, or other (qualitative free-text response). The user had the opportunity to review Centers for Disease Control and Prevention guidelines for RSTs.

6. Family education: We modified an existing handout for families that was taken from a different QI project focused on improving guideline-based streptococcal pharyngitis testing at a private practice6  (Supplemental Fig 4). Family education also included sample dialogue for NPs to discuss streptococcal pharyngitis testing with families. This step was considered important to engage families in an educated decision-making process and in light of the responses gathered from the RST order process alert (parent and/or guardian requesting the test).

We collected weekly RST data in addition to data on family complaints, admissions, and return visits within 7 days for complications of pharyngitis. For our outcome measure, we chose the number of RSTs ordered per month in patients aged <3 years. Although having patients with pharyngitis as the denominator would have been ideal and was considered, identifying such visits from our data was not feasible. This decision was based on several confounders, including the variability in documentation of diagnoses. We were able to collect data on the proportion of appropriate testing with data from our computerized process alert for a portion of the study. On the basis of the IDSA guidelines, there is only a small proportion of testing in this age group that would be considered appropriate; hence, we chose to track the number of tests per month. We chose our process measure to be the total number of canceled tests in patients aged <3 years, which we determined from the order process alert. The order alerts including reasons for ordering and canceling RSTs provided useful process information. The data from the outcome and process measure were collected from our EMR system. To ensure that the decrease in RSTs did not negatively impact other ED metrics, the balancing measure was the number of grievances reported to the hospital’s patient advocacy department and the number of admissions or return visits for complications of pharyngitis within 7 days of the patient’s initial visit.

We established a baseline between October 1, 2015, and September 30, 2016, on which to test improvement. Once baseline data were established, we initiated our interventions, starting with education, in October 2016. We achieved our goal of reducing RSTs by 50% in 10 months and wanted to monitor sustainability in the busy winter and spring seasons. We used failure modes and effects analysis principles and closely monitored the data for an additional 8 months. We did not have stop rules a priori. We chose the lengthy duration of the entire project to negate the effect of seasonal variation in presentation and testing for pharyngitis. We obtained a monthly report of the number of RSTs ordered. All streptococcal testing was performed in the microbiology laboratory, and results were reported in the EMR. We validated the data with corroborative information from the microbiology laboratory and regularly checked the data for accuracy and completeness. Each month, the authors reviewed a random sample of patient charts with RSTs associated to check for data accuracy. This enabled us to refine the automated data collection and provided an opportunity to inform our interventions including timely education.

We used Microsoft Excel and QI Macros to develop statistical process control charts for analysis and document plan-do-study-act cycles for interventions. We agreed on using the statistical control QI chart rule of having 8 points on either side of the process mean for discerning special versus common cause.7,8  The project was initiated after receiving an institutional review board exemption. The board determined that it met criteria to be classified as a QI project.

Our baseline data from October 1, 2015, to September 30, 2016, showed that 242 RSTs were ordered on patients <3 years old. The baseline mean was 20.1 tests per month over 12 months. Most of the tests were ordered by NPs (69%), followed by residents (13%) and attending physicians (10%). Tests were ordered for patients aged 25 to 36 months (77%) and 13 to 24 months (23%).

The mean number of RSTs ordered per month in patients aged <3 years declined by 52% over 10 months, from 20.1 to 9.8 per month. The results for the outcome measure are shown in a control chart in Fig 2. Most of the tests during the study period were ordered by NPs (62%), followed by residents (21%), faculty (10%), nurses (3%), medical students (2%), and fellows (2%). Figure 3 shows the cumulative results for RSTs by ordering personnel for the entire data set analyzed. Tests were ordered for patients aged 25 to 36 months (66%), 13 to 24 months (31%), and ≤12 months (0.03%). The majority of patients were white (34%) and African American (29%), and 55% of patients were boys.

FIGURE 2

Control chart for RSTs in patients <3 years old. Shown is a c-chart for the number of RSTs ordered for patients <3 years old. CL, control limit; LCL, lower control limit; UCL, upper control limit.

FIGURE 2

Control chart for RSTs in patients <3 years old. Shown is a c-chart for the number of RSTs ordered for patients <3 years old. CL, control limit; LCL, lower control limit; UCL, upper control limit.

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FIGURE 3

RST by ordering provider. A Pareto chart shows the ordering personnel who ordered the RST. Sixty-six percent of tests were ordered by NPs.

FIGURE 3

RST by ordering provider. A Pareto chart shows the ordering personnel who ordered the RST. Sixty-six percent of tests were ordered by NPs.

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Results from the RST order process alert starting in April 2017 showed that symptoms consistent with streptococcal pharyngitis and a recent sick contact with GAS were the reported reasons for RSTs in 43% cases. Parental and/or guardian request was the reason for testing in 23% of patients. The remaining 33% of patients had RSTs ordered for a variety of other reasons. Qualitative data indicated that almost one-quarter of parents or guardians requested RSTs, leading us to use a family-centered educational handout on pharyngitis for families.

Of the patients with RSTs, 25% had a positive result. Using data on this group from the order process alert, we found that 40% had symptoms of pharyngitis and a recent exposure to GAS, and 33% had a parent and/or guardian requesting the test. The remaining 27% of patients had RSTs ordered for symptoms alone.

Ultimately, our process measure showed that 11 (7%) RSTs were canceled. Our balance measures revealed 1 grievance for a patient <3 years old presenting with sore throat and no RST in our ED who was later found to have a positive test result at his primary care physician’s office. Although there was a patient who presented initially with a peritonsillar abscess, there were no patients who developed a complication as a result of not having an RST performed at their initial ED visit.

In a busy academic pediatric ED, we used QI methodology to reduce RSTs in patients <3 years old by 52% in 10 months. We monitored our data for an additional 8 months, and the successful reduction has been sustained. Our interventions led to a significant decline in RSTs in this population without a significant increase in streptococcal pharyngitis–related admissions, return visits, or grievances. Although 1 patient had a peritonsillar abscess, robust evidence suggests that children present with an abscess at initial presentation rather than as a complication of a missed or delayed diagnosis of streptococcal pharyngitis.1,9,10

Key interventions involved education for health care providers and families about GAS, the use of DMS, and an order process alert. The success of this project was largely due to an interdisciplinary approach and an interest in practice improvement. The collaboration among physicians, NPs, and nurses, in addition to the partnership between PEM and pediatric ID subspecialists, was a valuable asset. Through educational support and teamwork, we were able to build trust to engage our colleagues in this project. We reviewed data collectively in terms of type of health care provider group without identifying any individual provider, and there were no punitive actions for those who ordered an RST. This project also enabled culture change before process change, specifically by offering education before making changes to our EMR system. We believe the order process alert was a critical and measurable process change. To activate the alert in our EMR, we worked closely with a microbiology information technology champion. We performed a pilot of our customized order prompt, and once successful, we launched it. Of our interventions, the EMR prompt provided the most data regarding rationale for ordering RSTs and helped achieve sustainability in reduced testing. The options allowed us to identify reasons for RSTs, modify the process, and provide feedback if necessary. This information indicated that providers may need additional strategies to better explain the need for testing to caregivers, prompting us to share suggested dialogue techniques with providers and use an educational handout for families.

In December 2017, there was an increase in RSTs, which we attribute to increased testing for influenza and streptococcal pharyngitis during a severe influenza outbreak.10,11  The increase in GAS testing prompted us to conduct further investigation and educational interventions for providers regarding GAS testing guidelines for patients <3 years old. Continuing targeted education for providers with high use may be of benefit.

We developed a structured process, from weekly data collection to educational interventions and the order process alert, that served as the foundation for wider implementation of QI projects focused on reducing RSTs at our institution. Members of our group conducted a similar study in the community setting, which successfully reduced RSTs in patients <3 years old, further suggesting that this project may be generalizable to a variety of settings.6  The scope of these projects has extended to include affiliated urgent-care centers and outpatient settings at the hospital. Additional systems changes were achieved by the development of a hospital-wide clinical practice guideline for diagnosis and treatment of pharyngitis.

Our study was limited by several factors. We did not account for interprovider variability; for example, RSTs may have been ordered more frequently by a single provider compared with their group, and we did not obtain data on individual providers. Second, although we captured return visits to our ED, we were unable to acquire data on return visits to outside hospitals or clinics. Consequently, we do not know if this population had any complications prompting a return medical visit at another facility. Additionally, families may have been unsatisfied with their medical care, specifically with their children not having RSTs, but did not submit a formal grievance. Although our educational interventions may be generalized to other settings, the order process alert may not be appropriate for institutions that lack an EMR. Replication of our interventions may also be hindered by lack of buy-in from key stakeholders at outside institutions.

Although it was not the established foci for this project, we believe this project was impactful for antibiotic and financial stewardship efforts. By reducing RSTs, we in turn may have decreased unnecessary antibiotic use in this population. Additionally, although we did not plan this as a cost-saving project, a reduction in the number of RSTs and streptococcal cultures led to a $2200-per-month reduction in health care costs, from$3781 per month at baseline to \$1580 per month since starting the project. Patient charges may differ, and these costs do not include the antibiotics or the charges the patient would incur with a health care visit due to adverse events from them.

We used QI methodology to reduce RSTs for patients aged <3 years. Our interventions were focused on education for health care providers and families and used lean processes. Collectively, our interventions led to a significant reduction of RSTs in patients in this age group. We believe that early collaboration with stakeholders led to provider buy-in, which was a key driver of success. The success of the project has remained sustained. Our findings suggest that other sites might also achieve reductions in RSTs in this age group by focusing on a similar interdisciplinary approach. We have implemented additional systems changes, including a clinical practice guideline for pharyngitis. Additionally, we have expanded this project to include affiliated urgent-care centers and all outpatient settings at the hospital.

We send a special thank you to our QI team: Jacque Brown, Angela Wiedner, Theodore Barnett, and Amy Scott.

Dr Ahluwalia made substantial contributions to the study, including the conceptualization and design, methodology, investigation, data curation, formal analysis, and resources, and participated in drafting the initial manuscript; Drs Jain and Myers made substantial contributions to the study, including the conceptualization and design, methodology, investigation, supervision and oversight, data curation, formal analysis, and resources, and participated in drafting the initial manuscript; Dr Norton made substantial contributions to the study, including the methodology, investigation, and resources; Ms Meade and Ms Etherton-Still made contributions to the study, including the methodology and investigation; and all authors reviewed or edited the manuscript, approved the final manuscript as submitted, and agree to be accountable for all aspects of the work.

This work was presented in part at the American Academy of Pediatrics National Conference and Exhibition; September 16–19, 2017; Chicago, IL; and at the Pediatric Academic Society Annual Meeting; May 5–8, 2018; Toronto, ON.

Dr Ahluwalia’s current affiliation is Children’s National Medical Center, Washington, District of Columbia.

FUNDING: No external funding.

DMS

daily management system

ED

emergency department

EMR

electronic medical record

GAS

group A Streptococcus

ID

infectious diseases

IDSA

Infectious Diseases Society of America

NP

nurse practitioner

PEM

pediatric emergency medicine

QI

quality improvement

RST

rapid streptococcal test

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## Competing Interests

POTENTIAL CONFLICT OF INTEREST: The authors have indicated they have no potential conflicts of interest to disclose.

FINANCIAL DISCLOSURE: The authors have indicated they have no financial relationships relevant to this article to disclose.