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# Similarity solutions

(Difference between revisions)
 Revision as of 02:20, 13 July 2010 (view source)← Older edit Revision as of 02:25, 13 July 2010 (view source)Newer edit → Line 31: Line 31: |} |} - where ''$\eta$'' is a non-dimensional independent variable (function of both ''x'' and ''y'') that fulfills the similarity requirement. + where $\eta$ is a non-dimensional independent variable (function of both ''x'' and ''y'') that fulfills the similarity requirement. Mass and momentum boundary layer equations for flow over a flat plate (refer to [[Boundary layer approximations]]) can be converted to a new coordinate system $\eta$, ''f'', and ''g'' by using the notation ${f}'=\frac{df}{d\eta }$, ${g}'=\frac{dg}{dx}$ and${f}''=\frac{d^{2}f}{d\eta ^{2}}$. Mass and momentum boundary layer equations for flow over a flat plate (refer to [[Boundary layer approximations]]) can be converted to a new coordinate system $\eta$, ''f'', and ''g'' by using the notation ${f}'=\frac{df}{d\eta }$, ${g}'=\frac{dg}{dx}$ and${f}''=\frac{d^{2}f}{d\eta ^{2}}$. {| class="wikitable" border="0" {| class="wikitable" border="0" Line 163: Line 163: then, using eqs. (15) and (4.17): then, using eqs. (15) and (4.17): - + {| class="wikitable" border="0" - $c_{3}=0$ + |- + | width="100%" |
+ $\begin{matrix}{}&{}\\\end{matrix}c_{3}=0$ +
+ | + |} therefore, therefore,

## Revision as of 02:25, 13 July 2010

Boundary layer concept over a flat plate.

The form of the velocity, temperature, and concentration profiles for flow over a flat plate are presented in the figure to the right, and are based on qualitative measurements. These presentations indicate the possibility that these profiles are geometrically similar to each other along the flow direction for dependent variables such as velocity, temperature, and concentration, if the coordinates are stretched properly. For example, the velocity profiles are geometrically similar along the flow in the x-direction, differing only by a stretching factor (similarity factor) if the coordinates are properly sketched.

As noted before, not all boundary layer flow configurations have similar geometric profiles (similarity solution), but some do [1], especially for more conventional geometries and boundary conditions. The steady, two-dimensional, laminar, momentum boundary layer equation is a second order, nonlinear, partial differential equation. The two-dimensional, steady, laminar flow, energy, and species boundary layer equations are linear, second order, partial differential equations for most constant properties, and/or for decoupled heat and mass transfer problems.

If a similarity solution exists for a given situation, a mathematical transformation of coordinate systems can be performed to reflect this fact. A similarity technique converts conservation partial differential equations into ordinary differential equations and therefore, makes the solution much simpler. However, the analytical solutions may still require numerical integrations.

The conservation equations are uncoupled when each equation and its boundary condition can be solved independently of one another, except for continuity and momentum, which must be solved simultaneously. Coupled transport phenomena can occur in some applications because of coupled conservation governing equations and/or boundary conditions for mass diffusion, momentum, or heat transfer.

## Uncoupled Mass, Momentum, and Heat Transfer Problems

### Similarity Solution for Flow over a Flat Plate

We first apply the similarity solution to the classic problem of forced convective flow over a flat plate with constant free stream velocity, where mass, momentum, and heat transfer equations are uncoupled. Next, similarity solutions of control equations with appropriate boundary conditions of above figure are presented below. The viscous dissipation term is neglected for simplicity even though it is possible to obtain a similarity solution with viscous dissipation for this simple configuration. Assuming geometrically similar velocity profiles are possible along the flow over a flat plate using the following non-dimensional variables [2][3][1]:

 $\begin{matrix}{}\\\end{matrix}u=f(\eta)$ (1)
 $\eta =y\cdot g(x)$ (2)

where η is a non-dimensional independent variable (function of both x and y) that fulfills the similarity requirement. Mass and momentum boundary layer equations for flow over a flat plate (refer to Boundary layer approximations) can be converted to a new coordinate system η, f, and g by using the notation ${f}'=\frac{df}{d\eta }$, ${g}'=\frac{dg}{dx}$ and${f}''=\frac{d^{2}f}{d\eta ^{2}}$.

 $\frac{\partial u}{\partial y}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial \eta }\frac{\partial \eta }{\partial y}={f}'g$ (3)
 $\frac{\partial u}{\partial x}=\frac{\partial f}{\partial x}=\frac{\partial f}{\partial \eta }\frac{\partial \eta }{\partial x}={f}'y{g}'$ (4)
 $\frac{\partial ^{2}u}{\partial y^{2}}=\frac{\partial ^{2}f}{\partial y^{2}}=\frac{\partial }{\partial y}\left( \frac{\partial f}{\partial y} \right)=\frac{\partial }{\partial \eta }\left( \frac{\partial f}{\partial y} \right)\left( \frac{\partial \eta }{\partial y} \right)={f}''g^{2}$ (5)

Substituting eqs. (3)-(5) into mass and momentum boundary layer equations for flow over a flat plate yields:

 $\begin{matrix}{}\\\end{matrix}\nu {f}''g^{2}=f{f}'y{g}'+v{f}'g$ (6)
 ${f}'y{g}'+\frac{\partial v}{\partial y}=0$ (7)

Combining eqs. (6) and (7) to eliminate v and separating variables gives:

 $\frac{1}{f}\frac{d}{d\eta }\left( \frac{{{f}''}}{{{f}'}} \right)=\frac{1}{\nu }\frac{{{g}'}}{g^{3}}$ (8)

Considering that both η and x are independent variables, the left side of eq. (8) is a function of η, and the right side is a function of x, then each side must be a constant.

 $\frac{1}{\nu }\frac{{{g}'}}{g^{3}}=-c_{1}$ (9)

Integrating eq.(9) gives:

 $-\frac{1}{2g^{2}}=-c_{1}\nu x+c_{2}$ (10)

At the leading edge of the flat plate (x = 0), the boundary condition is $u=u_{\infty }$ which can be satisfied only at $\eta \to \infty$. In other words, η must be infinite when x = 0. It follows from eq. (2) that g(0) must be infinite and consequently, c2 in eq. (10) must be zero. Therefore

 $\eta =\frac{y}{\sqrt{2c_{1}\nu x}}$ (11)
 $g=\frac{1}{\sqrt{2c_{1}\nu x}}$ (12)

Since c1 is constant, the axial velocity is of the following form:

 $u=f\left( \frac{y}{\sqrt{x}} \right)$ (13)

From eq. (8)

 $\frac{1}{f}\frac{d}{d\eta }\left( \frac{{{f}''}}{{{f}'}} \right)=-c_{1}$ (14)

Integrating eq. (14) gives:

 $\frac{{{f}''}}{{{f}'}}=-c_{1}\int{fd\eta +c_{3}}$ (15)

The following boundary conditions are used to evaluate constants c1 and c3:

 $\begin{matrix}{}\\\end{matrix}\text{at }y=0,\text{ }u=v=0\text{ or at }\eta =0,\text{ }f=0$ (16)

Using eqs. (15) and (16):

 $\begin{matrix}{}\\\end{matrix}{f}''=0\text{ at }\eta =0$ (17)

then, using eqs. (15) and (4.17):

 $\begin{matrix}{}&{}\\\end{matrix}c_{3}=0$

therefore,

 $\frac{{{f}''}}{{{f}'}}=-c_{1}\int_{0}^{\eta }{fd\eta }$ (18)

Let the function ${\varsigma }'=d\varsigma /d\eta$ represent the non-dimensional axial velocity, $u/U_{\infty }$:

 ${\varsigma }'=\frac{u}{U_{\infty }}=\frac{f}{U_{\infty }}$ (19)

Then, using eq. (19)

 $\frac{{{f}''}}{{{f}'}}=\frac{{{\varsigma }'''}}{{{\varsigma }''}}$ (20)

Substituting (4.20) in (4.18) gives:

 $\frac{{{\varsigma }'''}}{{{\varsigma }''}}=-c_{1}\int_{0}^{\eta }{U_{\infty }\frac{d\varsigma }{d\eta }}d\eta =-c_{1}U_{\infty }\varsigma$ (21)

or

 ${\varsigma }'''+c_{1}U_{\infty }\varsigma {\varsigma }''=0$ (22)

From (4.22) we conclude that c1U is a constant, non-dimensional variable. Let $c_{1}U_{\infty }=1/2$, eq. (22) becomes

 ${\varsigma }'''+\frac{1}{2}\varsigma {\varsigma }''=0$ (23)

where

 $\eta =\frac{y}{\sqrt{\nu x/U_{\infty }}}\text{ and }{\varsigma }'(\eta )=\frac{u}{U_{\infty }}$ (24)

Equation (23) is known as the Blasius equation and is a nonlinear, third order, ordinary differential equation, which requires three boundary conditions.

 ${\varsigma }'(0)=0,\text{ }{\varsigma }'(\infty )=1,\text{ }\varsigma (0)=0$ (25)

Equation (23) can be numerically integrated using boundary conditions given by eq. (25). Blasius clearly integrated eq. (23) by hand without using a computer (Blasius, 1908). The numerical results using the shooting method are presented in Fig. 2, which shows that both u and vas functions of the independent variable η. A number of conclusions can be made from the numerical results presented in Fig. 2.

Figure 2: Dimensionless velocity distribution in a laminar boundary layer.

1. The edge of the momentum boundary layer is approximately at η=5.

2. The vertical velocity component v is non-zero at the edge of the boundary layer at η=5, even though the free stream flow is parallel to the plate.

3. The momentum boundary layer thickness is approximately

 $\frac{\delta }{x}=\frac{5}{\sqrt{\operatorname{Re}_{x}}}$ (26)

where $\operatorname{Re}_{x}=\frac{U_{\infty }x}{\nu }$

Using the definition of shear stress at the wall, τw, and the skin friction coefficient, cf, we obtain:

 $\tau _{w}=\mu \left. \frac{\partial u}{\partial y} \right|_{y=0}=c_{f}\frac{\rho U_{\infty }^{2}}{2}$ (27)

Rearranging the above equation in terms of $\varsigma$ gives:

 $\frac{c_{f}}{2}=\sqrt{\frac{\nu }{U_{\infty }x}}\left. {{\varsigma }''} \right|_{\eta =0}=\frac{\left. {{\varsigma }''} \right|_{y=0}}{\sqrt{\operatorname{Re}_{x}}}$ (28)
 $\frac{c_{f}}{2}=\frac{0.332}{\sqrt{\operatorname{Re}_{x}}}$ (29)

For the purposes of design and comparison with experimental measurements, it is more practical to obtain the average value of the skin friction, cf, which is usually determined by

 $\text{total drag}=\int_{A}{\tau _{w}dA=W\int_{0}^{x}{c_{f}\frac{\rho U_{\infty }^{2}}{2}dx=\frac{\rho U_{\infty }^{2}}{2}\bar{c}_{f}xW}}$ (30)

where W is the width of the plate. From eq. (4.29),

 $\frac{\bar{c}_{f}}{2}=\frac{1}{x}\int_{0}^{x}{\frac{c_{f}}{2}dx=0.664\operatorname{Re}_{x}^{-1/2}}$ (31)

It should be emphasized that the results presented above are for laminar flow parallel to a flat plate with constant free stream velocity. The approach to obtain the velocity distribution presented above can be extended to obtain the temperature distribution for laminar boundary layer flow over a flat plate with constant properties, constant free stream velocity, and constant temperature. The constant property assumption is important to decouple the momentum and energy equations in this configuration in order to obtain velocity first and temperature second. The non-dimensional temperature, θ, is defined as:

 $\theta =\frac{T-T_{w}}{T_{\infty }-T_{w}}$ (32)

The non-dimensional form of the energy equation (neglecting viscous dissipation) and boundary conditions and using the variables η and $\varsigma$ noted in eqs. (24) and (19) are as shown below.

 ${\theta }''+\frac{\Pr }{2}\varsigma {\theta }'=0$ (33)
 θ(0) = 0 (34)
 $\theta (\infty )=1$ (35)

with the notation that ${\theta }'=\frac{d\theta }{d\eta }$ and ${\theta }''=\frac{d^{2}\theta }{d\eta ^{2}}$. Equation (4.33) can be directly integrated:

 $\frac{d{\theta }'}{d\eta }+\frac{\Pr }{2}\varsigma {\theta }'=0$ (36)
 $\frac{d{\theta }'}{{{\theta }'}}+\frac{\Pr }{2}\varsigma d\eta =0$ (37)
 $\theta =c_{1}\int_{0}^{\eta }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta +c_{2}}$ (38)

Applying boundary conditions (34) and (35) gives

 $c_{2}=0\text{ and }c_{1}=\frac{1}{\int_{0}^{\infty }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}$ (39)

Therefore, we obtain the following equation for the dimensionless temperature, θ:

 $\theta (\eta )=\frac{\int_{0}^{\eta }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}{\int_{0}^{\infty }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}$ (40)
Figure 3: Dimensionless temperature distribution in a laminar boundary layer over a flat plate (Oosthuizen and Naylor, 1999; courtesy of Professors Oosthuizen and Naylor).

The energy equation (4.33) is a second order, linear, ordinary differential equation. The dimensionless temperature, θ, is a function of η and Pr. Equation (4.40) can be integrated numerically for a given Pr. The numerical results of θ as a function of η for different Pr values are presented in Fig.3. The following conclusions can be made upon reviewing the results of the momentum and thermal boundary layer equations, as presented in Figs. 2 and 3.

1. Temperature profile is identical to velocity for Pr = 1.

2. Temperature profile has strong dependence on the Prandtl number.

3. For Pr<1, the thermal boundary layer thickness is greater than the momentum boundary layer thickness, δT > δ.

4. For Pr>1 the thermal boundary layer thickness is less than the momentum boundary layer thickness, δT < δ.

The local heat transfer coefficient can be easily obtained from the numerical results of eq. (4.40) for a given Prandtl number by using Fourier’s law of heat conduction.

 $h_{x}=\frac{{q}''_{w}}{T_{w}-T_{\infty }}=\frac{-k\left. \frac{\partial T}{\partial y} \right|_{y=0}}{T_{w}-T_{\infty }}=\frac{-k\frac{d\theta }{d\eta }\left. \frac{\partial \eta }{\partial y} \right|_{w}}{T_{w}-T_{\infty }}$ (41)

The local Nusselt number is presented in terms of dimensionless variables as follows:

 $Nu_{x}=\frac{h_{x}x}{k}=\frac{x\left. {{\theta }'} \right|_{\eta =0}}{\sqrt{\nu x/U_{\infty }}}=\operatorname{Re}_{x}^{1/2}{\theta }'\left( 0 \right)$ (42)

or by using eq. (4.40):

 $Nu_{x}=\frac{\operatorname{Re}_{x}^{1/2}}{\int_{0}^{\infty }{\left[ \exp \left( -\frac{\Pr }{2}\int_{0}^{\eta }{\varsigma d\eta } \right) \right]d\eta }}$ (43)

The numerical results can be approximated for the range of Prandtl numbers from 0.5 to 15 by the following equation:

 $Nu_{x}=0.332\Pr ^{1/3}\operatorname{Re}_{x}^{1/2}$ (44)

The ratio of thermal boundary layer thickness, δT, to momentum boundary layer thickness, δ, for flow over a flat plate can also be approximated over this range of Prandtl numbers according to the relationship below:

 $\frac{\delta _{T}}{\delta }=\Pr ^{-1/3}\text{, }0.5\le \Pr <10$ (45)

For both design purposes and comparison with experimental measurement it is more convenient to calculate the average or mean heat transfer coefficient, $\bar{h}$.

 $q=\int_{A}^{{}}{h\left( T_{w}-T_{\infty } \right)dA}=W\int_{0}^{x}{h\left( T_{w}-T_{\infty } \right)}dx=\overline{h}\text{ }W\left( T_{w}-T_{\infty } \right)x$ (46)

From above, the mean heat transfer coefficient and mean Nusselt number become:

 $\overline{h}=\frac{1}{x}\int_{0}^{x}{h_{x}dx=0.664\frac{k}{x}\operatorname{Re}_{x}^{1/2}\Pr ^{1/3}}$ (47)
 $\overline{Nu}=\frac{\overline{h}x}{k}=2Nu_{x}=0.664\operatorname{Re}_{x}^{1/2}\Pr ^{1/3}$ (48)

This shows that the average heat transfer coefficient is twice the local heat transfer coefficient, just as the mean skin friction coefficient is twice the local skin friction coefficient for flow over a flat plate. It should be noted that these conclusions concerning the relation between the local and average transfer coefficients can not be extended to other configurations.

### Similarity Solution for Flow over a Wedge

Similarity solutions also exist for some other conventional geometries with simple boundary conditions. Some of these cases are discussed below. For flow over a wedge, as shown in Fig. 4.10, the free stream velocity varies according to potential (inviscid) flow theory:

 $\begin{matrix}{}\\\end{matrix}U=cx^{m}$ (49)

where U is the free stream velocity at the outer wedge surface and m is related to the wedge angle β by:

 $m=\frac{\beta /\pi }{2-(\beta /\pi )}=\frac{x}{U}\frac{dU}{dx}$ (50)

• When β (measured in radians) is positive, the free stream velocity increases along the wedge surface.
• When β is negative, the free stream velocity decreases along the wedge surface.
• β = 0 corresponds to flow parallel to a flat plate.
• β = π corresponds to flow perpendicular to the walls.

U∞ is the oncoming free stream velocity, which is constant, as shown in Fig. 4.10

Figure 3: Wedge flow configuration.

Equation (4.97) can be also applied near the leading edge of a blunt object. For example, the free stream velocity on the surface of a cylinder and a sphere near a stagnation point can be predicted by potential flow theory according to the equations below (Schlichting and Gersteu, 2000):

 (cylinder)$U=2U_{\infty }\sin \left( \frac{x}{R} \right)\approx 2U_{\infty }\left( \frac{x}{R} \right)$ (51)

 (sphere)$U=\frac{3}{2}U_{\infty }\sin \left( \frac{x}{R} \right)\approx \frac{3}{2}U_{\infty }\frac{x}{R}$ (52)

where R is the radius of the cylinder or sphere, U∞ is the oncoming velocity and U is the free stream velocity just outside the boundary layer. The pressure gradient term is related to free stream velocity by potential flow theory. This can be done using the inviscid momentum equation (μ = 0)

 $\frac{1}{\rho _{\infty }}\frac{dp}{dx}=-U\frac{dU}{dx}=-cx^{m}cmx^{m-1}=-\frac{U^{2}m}{x}$ (53)

The two dimensional, steady, constant property boundary layer equations for mass, momentum, and energy of flow over a wedge neglecting viscous dissipation are:

Continuity equation:

 $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (54)

Momentum (x-direction):

 $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}-\frac{1}{\rho }\frac{dP}{dx}$ (55)

Energy equation:

 $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}$ (56)

The above equations can be easily obtained by beginning with the general equations from Chapter 2 and making appropriate boundary layer approximations using scaling and order of magnitude analysis. Using eq. (4.101) for the pressure and comparable dimensionless similarity parameters to flow over a flat plate, we get the following ordinary differential equation for momentum and energy for constant wall temperature with no blowing or suction at the wall.

 ${\varsigma }'''+\frac{1}{2}\left( m+1 \right)\varsigma {\varsigma }''+m\left( 1-{\xi }'^{2} \right)=0$ (57)
 ${\theta }''+\frac{\Pr \left( m+1 \right)}{2}\varsigma {\theta }'=0$ (58)

The boundary conditions for velocity and temperature are the same as in the case of flow over a flat plate. Assuming that heating starts at the leading edge, the ordinary differential equations (4.105) and (4.106) for dimensionless velocity and temperature can be solved using the given boundary condition. The friction coefficient and Nusselt number can be calculated from the numerical results by the following equations:

 $c_{f,x}=\frac{2{\varsigma }''(0)}{\operatorname{Re}_{x}^{1/2}}$ (59)
 $Nu_{x}=A\operatorname{Re}_{x}^{1/2}$ (60)

where the constant A depends on m and Pr. Solutions of eqs. (4.105) and (4.106) for m > 0 are unique. For m < 0 two groups of solutions exist for a limited range of m values corresponding to negative β. These two groups correspond to a decelerating mainstream to the point of incipient separation or laminar boundary layer flows after separation. Falkner and Skan (1931) originally developed the similarity solution for flow over a wedge with U∞ = cxm and an impermeable wall. The numerical results for the friction coefficient for selected values of m and β are presented in Table 4.1. m = β = 0 corresponds to the case of flow parallel to a flat plate with constant free stream values (Blasius Solution). Eckert (1942) was the first to obtain the numerical results for eqs. (4.105) and (4.106) for constant free stream and wall temperature as presented in Table 4.2. In similarity solutions for both flow over a flat plate and over a wedge, it is assumed that there is no blowing or suction at the wall. It can be shown (see Problem 4.7) that similarity solutions with blowing or suction exist only if the vertical velocity at the wall changes along the flow according to the following equation:

 $v_{w}\propto x^{(m-1)/2}$ (61)

where vw is the blowing or suction velocity. Clearly this requirement restricts the use of the similarity solution to special cases that satisfy eq. (4.109).

Table 1 The local friction coefficient for laminar boundary layer flow over a wedge, with U = cxm and an impermeable wall

 β m ${\varsigma }''(0)=\frac{1}{2}C_{f}\operatorname{Re}_{x}^{1/2}$ 2πππ/2π/50-0.14-0.18-0.199 ∞1.00.3330.1110-0.0654-0.0826-0.0904 ∞1.233 Stagnation0.7570.5120.332 Flat plate0.1640.0820 Separation

Table 2 Local values of NuxRex-1/2 for laminar boundary layer flow over a wedge with constant wall and free stream temperature and an impermeable wall with U∞ = cxm

 A B Pr 1 2 3 4 5 -0.5120π/5π/2π8π/5 -0.075300.1110.3331.04 0.2420.2920.3310.3840.4960.813 0.2530.3070.3480.4030.5230.858 0.2720.3320.3780.4400.5700.938 0.450.5850.6690.7921.0431.736 0.5700.7300.8511.0131.3442.236

## Coupled Mass, Momentum, and Heat Transfer Problems

Coupled transport phenomena occur frequently in problems associated with evaporation, sublimation, absorption, or combustion. Sublimation over a flat plate can find its application in an analogy between heat and mass transfer and will be used here as an example to show the methodology used in solving these coupled problems.

Figure 4: Phase diagram for solid-liquid and solid-vapor phase change.

When the pressure and temperature of ice are above the triple point and the ice is then heated, melting occurs. However, when the ice is exposed to moist air with a partial pressure of water below its triple point pressure, heating of the ice will result in a phase change from ice directly to vapor without going through the liquid phase. Spacecrafts and space suits can reject heat by sublimating ice into the vacuum of space. Another application for sublimation of ice is the preparation of specimens using freeze-drying for a scanning electronic microscope (SEM) or a transmission electronic microscope (TEM). This type of phase change is referred to as sublimation. The opposite process is deposition, which describes the process of vapor changing directly to solid without going through condensation. The phase-change processes related to solids can be illustrated by a phase diagram in Fig. 4.11. Sublimation and deposition will be the subjects of this sub-section. When a subcooled solid is exposed to its superheated vapor, as shown in Fig. 4.12(a), the vapor phase temperature is above the temperature of the solid-vapor interface and the temperature of the solid is below the interfacial temperature. The boundary condition at the solid-vapor interface is

 $k_{s}\frac{\partial T_{s}}{\partial x}-h_{\delta }(T_{\infty }-T_{\delta })=\rho _{s}h_{sv}\frac{d\delta }{dt}$ (62)

where hδ is the convective heat transfer coefficient at the solid-vapor interface, hsv is the latent heat of sublimation, and δ is the thickness of the sublimated or deposited material. The interfacial velocity d?/dt in eq. (4.110) can be either positive or negative, depending on the direction of the overall heat flux at the interface. While a negative interfacial velocity signifies sublimation, a positive interfacial velocity signifies deposition. When the vapor phase is superheated, as shown in Fig. 4.12(a), the solid-vapor interface is usually smooth and stable.

Figure 5: Temperature distribution in sublimation and deposition.

In another possible scenario, as shown in Fig. 4.12 (b), the solid temperature is above the interfacial temperature and the vapor phase is supercooled. The interfacial energy balance for this case can still be described by eq. (4.110). Depending on the degrees of superheating in the solid phase and supercooling in the vapor phase (the relative magnitude of the first and second terms in eq. (4.110)) both sublimation and deposition are possible. During sublimation, a smooth and stable interface can be obtained. During deposition, on the other hand, the interface is dendritic and unstable, because supercooled vapor is not stable. The solid formed by deposition of supercooled vapor has a porous structure. During sublimation or deposition, the latent heat of sublimation can be supplied from or absorbed by either the solid phase or the vapor phase, depending on the temperature distributions in both phases. Naphthalene sublimation is also a case whereby a heat transfer coefficient can be obtained through the measurement of a mass transfer coefficient and the analogy between heat and mass transfer (Eckert and Goldstein, 1976). The significant advantages of this method include its high accuracy and the simplicity of the experimental apparatus. In addition, the local heat transfer coefficient can be obtained by measuring the local sublimed depth of the specimen. Figure 4.13 shows the physical model of a sublimation problem, where a flat plate is coated with a layer of sublimable material and is subject to constant heat flux underneath (Zhang et al., 1996). A gas with the ambient temperature and mass fraction of sublimable material flows over the flat plate at a velocity of U∞. The heat flux applied from the bottom of the flat plate will be divided into two parts: one part is used to supply the latent heat of sublimation, and the other is transferred to the gas through convection. The sublimated vapor is injected into the boundary layer and is removed by the gas flow.

Figure 6: Sublimation on a flat plate with constant heat flux.

The following assumptions are made in order to solve the problem: 1. The flat plate is very thin, so the thermal resistance of the flat plate can be neglected. 2. The gas is incompressible, with no internal heat source in the gas. 3. The sublimation problem is two-dimensional steady state.

The governing equations for mass, momentum, energy, and species of the problem are

 $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (63)
 $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{\partial ^{2}u}{\partial y^{2}}$ (64)
 $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{\partial ^{2}T}{\partial y^{2}}$ (65)
 $u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}=D\frac{\partial ^{2}\omega }{\partial y^{2}}$ (66)

A no slip condition at the surface of the flat plat requires that u = 0 at y = 0 . For a binary mixture that contains the vapor, sublimable substance, and gas, the molar flux of the sublimable substance at the surface of the flat plate is [see eq. (1.109)]

 ${\dot{m}}''=-\frac{\rho D}{1-\omega }\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix}$ (67)

Since the mass fraction of the sublimable substance in the mixture is very low, i.e., ω$\ll$1, the mass flux at the wall can be simplified to

 ${\dot{m}}''=-\rho D\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix}$ (68)

Sublimation at the surface causes a normal blowing velocity,$v_{w}=\dot{{m}''}/\rho$. The normal velocity at the surface of the flat plate is therefore

 $v=v_{w}=-D\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix}$ (69)

The energy balance at the surface of the flat plate is

 $-k\frac{\partial T}{\partial y}-\rho h_{sv}D\frac{\partial \omega }{\partial y}={q}''_{w}\begin{matrix} , & y=0 \\ \end{matrix}$ (70)

Another reasonable, practical, representable boundary condition at the surface of the flat plate emerges by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature. The mass fraction and the temperature at the surface of the flat plate have the following relationship (Kurosaki, 1974):

 $\omega =aT+b\begin{matrix} , & y=0 \\ \end{matrix}$ (71)

where a and b are constants that depend on the sublimable material and its temperature. As $y\to \infty ,$ the boundary conditions are

 $u\to U_{\infty }$, $T\to T_{\infty }$,$\omega \to \omega _{\infty }$ (72)

Introducing the stream function ψ,

 $u=\frac{\partial \psi }{\partial y}\begin{matrix} {} & v=-\frac{\partial \psi }{\partial x} \\ \end{matrix}$ (73)

the continuity equation (4.111) is automatically satisfied, and the momentum equation in terms of the stream function becomes

 $\frac{\partial \psi }{\partial y}\frac{\partial ^{2}\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{\partial ^{2}\psi }{\partial y^{2}}=\nu \frac{\partial ^{3}\psi }{\partial y^{3}}$ (74)

Similarity solutions for eq. (4.122) do not exist unless the injection velocity vw is proportional to x-1/2, and the incoming mass fraction of the sublimable substance, ω∞, is equal to the saturation mass fraction corresponding to the incoming temperature, T∞ (Kurosaki, 1974; Zhang et al., 1996). The governing equations cannot be reduced to ordinary differential equations. The local nonsimilarity solution proposed by Zhang et al. (1996) will be presented here. Defining the following similarity variables:

 \begin{align} & \xi =\frac{x}{L},\text{ }\eta =y\sqrt{\frac{U_{\infty }}{2\nu L\xi }}\begin{matrix} , & f=\frac{\psi }{\sqrt{2\nu U_{\infty }L\xi }} \\ \end{matrix} \\ & \theta =\frac{k(T-T_{\infty })}{{q}''_{w}\sqrt{2\nu L\xi /U_{\infty }}}\begin{matrix} , & \phi =\frac{\rho h_{sv}D(\omega -\omega _{\infty })}{{q}''_{w}\sqrt{2\nu L\xi /U_{\infty }}} \\ \end{matrix} \\ \end{align} (75)

eqs. (4.122) and (4.113) – (4.114) become

 ${f}'''+f{f}''=2\xi \left( {f}'{F}'-{f}''F \right)$ (76)
 ${\theta }''+\Pr (f{\theta }'-{f}'\theta )=2\Pr \xi \left( {f}'\Theta -{\theta }'F \right)$ (77)
 ${\phi }''+\operatorname{Sc}(f{\phi }'-{f}'\phi )=2\operatorname{Sc}\xi \left( {f}'\Phi -{\phi }'F \right)$ (78)

where prime ' represents partial derivative with respect to η, and all upper case variables represent partial derivatives of primary similarity variables with respect to ξ.

 $F=\frac{\partial f}{\partial \xi }\begin{matrix} , & \Theta =\frac{\partial \theta }{\partial \xi }\begin{matrix} , & \Phi =\frac{\partial \phi }{\partial \xi } \\ \end{matrix} \\ \end{matrix}$ (79)

It can be seen from eqs. (4.124) – (4.126) that the similarity solution exists only if F = Θ = Φ = 0. In order to use eqs. (4.124) – (4.126) to obtain a solution for the sublimation problem, supplemental equations for F, Θ, and Φ must be obtained. Taking partial derivatives of eqs. (4.124) – (4.126) with respect to ξ and neglecting the higher order term, one obtains

 ${F}'''+F{f}''+{F}''f=2\left( {f}'{F}'-{f}''F \right)$ (80)
 ${\Theta }''+\Pr (F{\theta }'+f{\Theta }'-{F}'\theta -{f}'\Theta )=2\Pr \left( {f}'\Theta -{\theta }'F \right)$ (81)
 ${\Phi }''+\operatorname{Sc}(F{\phi }'+f{\Phi }'-{F}'\phi -{f}'\Phi )=2\operatorname{Sc}\left( {f}'\Phi -{\phi }'F \right)$ (82)

The boundary conditions of eqs. (4.124) – (4.126) and eqs. (4.128) – (4.130) are

 ${f}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix}$ (83)
 $f(\xi ,0)=-\frac{2}{3}B\left[ \xi ^{1/2}{\phi }'(\xi ,0)-\xi ^{3/2}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ \end{matrix}$ (84)
 ${f}'(\xi ,\infty )=1\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (85)
 ${F}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix}$ (86)
 $F(\xi ,0)=-\frac{1}{3}B\left[ \frac{1}{2}\xi ^{-1/2}{\phi }'(\xi ,0)-\xi ^{1/2}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ \end{matrix}$ (87)
 ${F}'(\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (88)
 ${\theta }'(\xi ,0)+{\phi }'(\xi ,0)=-1\begin{matrix} , & \eta =0 \\ \end{matrix}$ (89)
 $\theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (90)
 ${\Theta }'(\xi ,0)+{\Phi }'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix}$ (91)
 $\Theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (92)
 $\phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\theta (\xi ,0)+\phi _{s}\xi ^{-1/2}\begin{matrix} , & \eta =0 \\ \end{matrix}$ (93)
 $\phi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (94)
 $\Phi (\xi ,0)=\frac{ah_{sv}}{c_{p}}\frac{1}{Le}\Theta (\xi ,0)-\frac{\phi _{s}}{2\xi ^{3/2}}\begin{matrix} , & \eta =0 \\ \end{matrix}$ (95)
 $\Phi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix}$ (96)

where

 $B=\frac{{q}''_{w}}{\rho h_{sv}\nu }\sqrt{\frac{2\nu L}{U_{\infty }}}$ (97)

reflects the effect of injection velocity at the surface due to sublimation, and

 $\phi _{s}=\frac{\rho h_{sv}D(\omega _{sat,\infty }-\omega _{\infty })}{{q}''_{w}\sqrt{2\nu L/U_{\infty }}}$ (98)

represents the effect of the mass fraction of the sublimable substance in the incoming flow. $\omega _{sat,\infty }$

is saturation mass fraction corresponding to the incoming temperature:

 $\omega _{sat,\infty }=aT_{\infty }+b$ (99)

Figure 7: Temperature and mass fraction distributions (Zhang et al. 1996).

The set of ordinary differential equations (4.124) – (4.126) and (4.128) – (4.130) with boundary conditions specified by eqs. (4.131) – (4.144) are boundary value problems that can be solved using a shooting method (Zhang et al., 1996). Figure 4.14 shows typical dimensionless temperature and mass fraction profiles obtained by numerical solution. It can be seen that the dimensionless temperature and mass fraction at different ξ are also different, which is further evidence that a similarity solution does not exist. Once the converged solution is obtained, the local Nusselt number based on the total heat flux at the bottom of the flat plate is

 $Nu_{x}=\frac{h_{w}x}{k}=\frac{[{q}''_{w}/(T_{w}-T_{\infty })]x}{k}=\frac{\operatorname{Re}_{x}^{1/2}}{\sqrt{2}\theta (\xi ,0)}$ (100)

and the Nusselt number based on convective heat transfer is

 $Nu_{x}^{*}=\frac{h_{x}x}{k}=-\frac{x}{T_{w}-T_{\infty }}\left( \frac{\partial T}{\partial y} \right)_{y=0}=-\frac{{\theta }'(\xi ,0)}{\sqrt{2}\theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ (101)

The Sherwood number is

 $Sh_{x}=\frac{h_{m}x}{D}=-\frac{x}{\omega _{w}-\omega _{\infty }}\left. \frac{\partial \omega }{\partial y} \right|_{y=0}=-\frac{{\phi }'(\xi ,0)}{\sqrt{2}\phi \theta (\xi ,0)}\operatorname{Re}_{x}^{1/2}$ (102)
Figure 8: Nusselt number based on convection and Sherwood number (Zhang et al. 1996).

Figure 4.15 shows the effect of blowing velocity on the Nusselt number based on convective heat transfer and the Sherwood number for $\phi _{sat,\infty }=0$, i.e., the mass fraction of sublimable substance is equal to the saturation mass fraction corresponding to the incoming temperature. It can be seen that the effect of blowing velocity on mass transfer is stronger than that on heat transfer.

## References

1. 1.0 1.1 Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ.
2. Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY
3. Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, Hoboken, NJ.


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# Biot Number Calculator

Created by Miłosz Panfil, PhD
Reviewed by Dominik Czernia, PhD and Steven Wooding
Last updated: Oct 31, 2022

This Biot number calculator helps you compute the Biot number. What is the Biot number? The Biot number determines how quickly heat transfers from the surface of the body to its interior. The text below explains the physics behind the heat transfer and gives the Biot number formula.

## What is the Biot number?

The Biot number helps us answer the following question: How much will the temperature inside a body vary if we heat up a part of its surface?

If the Biot number is small (much less than 1), then the temperature on the surface and interior will be very similar.

On the other hand, if the Biot number is large (much larger than 1), there will be a large temperature gradient inside the body.

💡 See our temperature conversion tool for all your temperature conversion needs.

## Heat transfer

If we warm the surface of a material, two things happen. First, we warm the surface. The efficiency of this process depends on the heat transfer coefficient. Secondly, the heat from the surface starts to flow through the rest of the material, heating the interior. How quickly this happens depends on the thermal conductivity of a material. Check the thermal conductivity calculator to learn more about this phenomenon.

The Biot number compares the efficiency of these two processes.

1. If the heat transfer is more efficient than the thermal conductivity, the surface will warm up quicker than the rest of the body – the Biot number is larger than 1.

2. On the other hand, if the material conducts heat well, then besides warming it up only in one place, its temperature will be pretty uniform – the Biot number is smaller than 1.

🙋 You might also be interested in our specific heat calculator.

## Biot number formula

The Biot number formula is given by:

${\rm Bi} = \frac{h}{k}L_{\rm c},$

where:

• ${\rm Bi}$ – Biot number;
• $L_{\rm c}\ \rm [m]$ – Characteristic length of the material;
• $h\ \rm [W/m^2\! \cdot\! K]$ – Heat transfer coefficient at the material's surface; and
• $k\ \rm [W/m\! \cdot\! K]$ – Thermal conductivity of the material.

We can compute the characteristic length $L_{\rm c}\ \rm [m]$ knowing the volume $V\ \rm [m^3]$ and the area $A\ \rm [m^2]$ of a surface through which the material is heated up (or cooled down):

$L_{\rm c} = \frac{V}{A}$

For example, for a copper pan of water with characteristic length $\small L_{\rm c} = 15\ \rm cm$, the Biot number is $\small \rm Bi = 2.807$. This means that when we heat the pan, there will be a significant difference in temperatures between the water at the bottom and at the top of the pot. If we decrease the amount of water (decrease the characteristic length), the Biot number drops, signaling a more uniform temperature inside the pot.

In the Biot number calculator computations, we took the water thermal conductivity as $\small k = 0.7\ \rm W/m\! \cdot\! K$ and the heat transfer coefficient between copper and water as $\small h = 13.1\ \rm W/m\! \cdot\! K$.

Miłosz Panfil, PhD
Characteristic length
Surface area
Volume
Biot number
Characteristic length
m
Heat transfer coefficient
W/(m²·K)
Thermal conductivity
W/(m·K)
Biot number
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# Practice Questions on Permutation-CombinationAptitude Questions and Answers

## Moderate Permutation-Combination Question - 26

 Q26. $a,b,c,d$ and $e$ are five natural numbers. Find the number of ordered sets $(a,b,c,d,e)$ possible such that $a+b+c+d+e =64$.
 A. ${^{64}C_5}$ B. ${^{63}C_4}$ C. ${^{65}C_4}$ D. ${^{63}C_5}$

## Moderate Permutation-Combination Question - 27

 Q27. There are five cards lying on the table in one row. Five numbers from among 1 to 100 have to be written on them, one number per card, such that the difference between the numbers on any two adjacent cards is not divisible by 4. The remainder when each of the 5 numbers is divided by 4 is written down on another card (the 6th card) in order. How many sequences can be written down on the 6th card?
 A. $2^10$ B. $2^{10} × 3^3$ C. $4 × 3^4$ D. ${4^2}{3^3}$

## Moderate Permutation-Combination Question - 28

 Q28. From a total of six men and four ladies a committee of three is to be formed. If Mrs. $X$ is not willing to join the committee in which Mr. $Y$ is a member, whereas Mr.$Y$ is willing to join the committee only if Mrs $Z$ is included.  How many such committee are possible?
 A. 138 B. 128 C. 112 D. 91

## Moderate Permutation-Combination Question - 29

 Q29. Goldenrod and No Hope are in a horse race with 6 contestants. How many different arrangements of finishes are there if No Hope always finishes before Goldenrod and if all of the horses finish the race?
 A. $700$ B. $360$ C. $120$ D. $24$ E. $21$

## Moderate Permutation-Combination Question - 30

 Q30. Jay wants to buy a total of 100 plants using exactly a sum of Rs 1000. He can buy Rose plants at Rs 20 per plant or marigold or Sun flower plants at Rs 5 and Re 1 per plant respectively. If he has to buy at least one of each plant and cannot buy any other type of plants, then in how many distinct ways can Jay make his purchase?
 A. 2 B. 3 C. 4 D. 5
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# Exploring The Universe With LHC

in #physics2 years ago

How can LHC test astrophysics phenomena? How is relativity used to calculate collisions at LHC? Are you better than a computer at finding the Higgs boson? We answer all of this, and much more in this interactive experience.

For the optimal experience, and for all interactive elements to work, read the series on kasperfred.com

# Introduction

This project is an interactive experience consisting of a series of interlinked webpages which contain text, pictures, equations, videos, and other interactive elements to explore the topic: Exploring the universe with LHC.

For the best experience, we recommend using the latest version of Chrome desktop with Javascript and cookies enabled.

Along the way, you can refer to the interactive glossary for terms and theorems.

To each page, there's an associated public discussion thread which meant to provide an outlet for any questions or comments the reader might have while exploring the interactive experience.

You can access this thread by clicking on the blue "discuss" button at the top of any article.

# Glossary

To access the glossary, read the series on kasperfred.com

# What is LHC:ATLAS?

Accelerating particles close to the speed of light and colliding them hoping something interesting will happen.

## LHC

The Large Hadron Collider (LHC) is the result of a multinational collaboration with over 10,000 scientists and 100 countries.

By colliding charged particles, mostly protons, at speeds close to that of the speed of light, the LHC has made discoveries ranging from the confirmation of the existence and properties of the Higgs boson, to the likely non-existence of supersymmetrical particles and, recently, the discovery of two baryons composed of one bottom quark and two up quarks, and one bottom quark and two down quarks.

## ATLAS

A picture of the caliometers of the ATLAS detector.

ATLAS is one of the four main detectors at LHC, and measures the decay products from the collisions. When protons hit head on in inelastic relativistic collisions, they create new particles which spread around in the detector. The atlas detector is constructed as a cylinder with multiple layers. The different layers can measure different properties of the decay products.

### ATLAS Structure

The ATLAS detector is constructed as a cylinder through which the particle beams shoot. Upon a head-on collision, the protons decay to other particles which scatter in all directions. This is possible while conserving momentum precisely because the protons collided head on.

Since we cannot predict where the decay go, the detectors are constructed as approximate onion-like shells around the beam direction as illustrated below.

Schematic of ATLAS from [email protected]

In the innermost shell, we have tracking devices such as the Xenon gas tubes, which can track as charged particles go through them, and the PIXEL/SCT trackers. Which is explained well in the following video:

To see the video, read the article on kasperfred.com

Knowing the trajectory of the particles along with strong magnets which curve the trajectory allows us, with some difficulty when dealing with multiple particles, to calculate the relativistic momentum of the charged particles.

In the next shell, the particles pass through calorimeters which tries to measure the energy of the particles by absorbing them passively through high density materials such as lead, and actively through liquid argon for example.

Electromagnetic and Hadronic calorimeters can stop most particles, but neutrinos and muons.

The next shells detects Cherenkov and transition radiation, and at the outermost shell, we have the the muon spectrometers.

ATLAS detectors

On the picture above, we see the inner shells where the trackers can see the charged particles such as the proton, electron and muon. Notice, how the magnets bend the paths of the particles.

In the calorimeters, we see most particles being absorbed and decay to other particles. By observing the decay particles, and considering quantum symmetries, we can figure out the types of decay particles as has been colored in on the image.

Notice further that we cannot directly detect neutrinos in the detector. Instead, we have to rely on inference from looking at the missing total energy in the system because we know there's conservation of energy even if some of it goes to creating new particles.

### Listen to collisions detected by ATLAS

You can listen to the audio of detected collisions from ATLAS courtesy of Ewan Hill below.

To listen to the collisions, read the article on kasperfred.com

The audio is generated by associating the values from different detectors of a subset of the collisions with MIDI tones and normalizing the frequencies to be within the audible range. 30 seconds is approximately equivalent to one event.

# What is Special Relativity?

Time and space are the same thing, and it's all relative?

When we read What is LHC:ATLAS?, I mentioned that since the speed of the particles approaches the speed of light, we have to account for the relativistic effects.

"However, we do have Galilean relativity which has worked perfectly fine for me" you might inject, and you'd be right. Galilean relativity serves us just well in most of our lives. In fact, for speeds far below the speed of light Einstein relativity and Galilean relativity are nearly indistinguishable from each other.

However, in certain circumstances such as when you're using a particle accelerator, or even something as mundane as using GPS, Galilean relativity is not enough, and you have to use Einstein's relativity.

"Okay, so I need this Einstein's relativity. What is it?" you ask begrudgingly.

Einstein's special relativity (refered to just as special relativity from here) can be derived entirely from two fundamental postulates.

## The Fundamental Postulates

1. All intertial reference frames are equivalent.

This version of the postulate, however, follows a somewhat circular argument as an object which does not accelerate does not have a net force, however, the only way we know that it doesn't have a net force is by checking if the object accelerates.

This inconsistency lead Einstein to develop his general theory of relativity which solves the problem by working for any reference system, and not just inertial systems. However, if you ignore this detail, you'll be fine.

1. The speed of light in vacuum, $c$, is a universal constant in all reference frames.

While today, we know this to be true as we have measured it empirically, Einstein couldn't possibly have known this at his time, you rebut. And you'd be right. In fact, it's possible to derive that given that light is a electromagnetic wave that light must travel at $c$ from Maxwell's equations. However, Einstein describes in his notes from 1949 himself finding it as a solution to an apparent paradox he stumbled upon when he was 16. This approach leads to the stronger claim that $c$ is constant in all reference frames.

## Notation

### Standard Configuration

The standard configuration is a configuration of coordinate systems often used in special relativity to ease, and somewhat normalize the notation.

The standard configuration consists of an rest system $S$ with coordinates $(x,y)$, and another system $S'$ with coordinates $(x',y')$ moving with $v$ along the $x$-axis with respect to $S$. Furthermore, at $t=0$, we set $x=x'=0$.

This setup is illustrated below.

### Four vectors

Since handling the individual dimensions separately can quickly become a mess, it's an often used convention to combine the $3+1$ dimensions into a four vector with the unit vectors:

$$\textbf{e}_t,\textbf{e}_x,\textbf{e}_y,\textbf{e}_z$$

The index of the time dimension can wary between authors. One convention is to use the zeroth dimension as the time dimension, so the unit vectors become

$$\textbf{e}_0,\textbf{e}_1,\textbf{e}_2,\textbf{e}_3$$

which gives us the four vector

$$\textbf{a}=\sum_{a=0}^3 a^\alpha \textbf{e}_\alpha.$$

By using Einstein notation, we get

$$\textbf{a}=a^\alpha \textbf{e}_\alpha$$

where it's assumed we sum over repeated greek letters.

We have adopted the convention where we denote four vectors with bold face such as, $\textbf{a}$, while we continue to denote three vectors with an arrow such as, $\vec{a}$.

For some more syntatic sugar, let

$$\eta_{\alpha \beta} = \textbf{e}\alpha \cdot \textbf{e}\beta = \begin{bmatrix} -1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 0&0&0&1 \end{bmatrix}.$$

In which case

$$\textbf{a}\cdot \textbf{b}=\eta_{\alpha \beta} a^\alpha b^\beta.$$

The reason we have a $-1$, and not $1$ in the time dimension is to avoid messing up causality which the philosophers tell me is a good thing.

One neat thing about using four vectors is that the dot product between any two four vectors (or a chain of them) is invariant. So

$$\textbf{a} \cdot \textbf{b} = \textbf{a}' \cdot \textbf{b}' .$$

### Lorentz Transformation

In the standard configuration, the componentwise Lorentz transformation is given by

\begin{aligned} &a'^{0} = \gamma(a^{0} - \beta a^1 ), \ &a'^{1} = \gamma(a^{1} - \beta a^0 ), \ &a'^{2} = a^2, \ &a'^{3} = a^3. \ \end{aligned}

## Time Contraction

While deriving every formula of special relativity is beyond the scope of this project, deriving time contraction seems worthwhile due to it's broad applicability.

To see the video read the article on kasperfred.com

Let's consider a light-clock consisting of two perfect mirrors put parallel with a distance $L$ to each other, and let a foton travelling at speed $c$ bounce between the mirrors.

Obviously, the time it takes for the foton to bounce from the bottom mirror to the top mirror and back again is going to be

$$t = \frac{2L}{c}$$

because time equals distance over speed.

This can be rewritten as

$$L = \frac{ct}{2}$$

Now, let's consider the same system, but this time the mirrors move along the $x$-axis with velocity $v$ while the foton continous to bounce between the mirrors.

Let the time it takes for the foton to complete one full bounce be $t'$, and let the distance the mirrors have moved before the foton comes back be $2s$. Then we know that

$$2s = v\cdot t'.$$

So we can write $s=\frac{v\cdot t'}{2}$.

We know that the foton travels along $h$ twice for a total distance of $2h$, so we find that

$$2h=c\cdot t.$$

We now have all the sides in a right-angled triangled, and per pythagoras, we know that

$$\big(\frac{vt'}{2}\big)^2 + \big(\frac{ct}{2}\big)^2 = \big(\frac{ct'}{2}\big)^2.$$

Which gives us

$$c^2t^2 = c^2 t'^2 - v^2t'^2.$$

By dividing by $c^2$, we find

$$t^2 = t'^2 - \frac{v^2t'^2}{c^2}=t'^2 \big( 1 - \frac{v^2}{c^2} \big).$$

Isolating $t'$, we find

$$t' = t \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.$$

Or when $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$,

$$t' = t \gamma.$$

## Relativistic Mechanics

Just like classcal mechanics, we still have conservation of momemtum and total energy in relativistic collisions.

We consider a system consisting of a photon and an electron. In the initial state, the photon has a four momentum vector $\textbf{p}_\gamma$, and the electron is at rest, $\textbf{p}_e = m_e c(1,\vec{0})$.

In the final state, the photon has the four momentum vector $\bar{\textbf{p}}_\gamma$, and the electron has the four momentum vector $\bar{\textbf{p}}_e$.

The setup is illustrated below.

This is an example of Compton spread. The angle of spreading $\theta$ depends on the wavelength of the incoming photon with the relation

$$\bar{\lambda}=\lambda+\lambda_c(1-\cos(\theta))$$

where $\lambda_c = \frac{h}{m_e c}$.

# The Higgs

Catchy subtitle is not needed

The confirmation of the existence of the Higgs boson is probably one of the most important findings made by the LHC. Not only did it complete the Standard model, but also gave insight into the early cosmological universe by having a better understanding of how the processes could function.

So not only does the LHC play an important role in furthering particle physics, but its findings are of high significance for astrophysicists who seek to understand the early universe some of whose properties are best emulated in a particle accelerator.

## Finding the Higgs

Finding the Higgs boson is not easy. Not only does it decay about an order of magnitude faster than other elementary particles making it impossible to detect directly in ATLAS, or other detectors, while it does decay into a lot of things, all of those are very common making isolating the Higgs boson very difficult.

One common decay pattern is for the Higgs boson to decay to two photons ($H\rightarrow \gamma_1 \gamma_2$) with the 4 momntum vectors $\textbf{p}^\alpha_{\gamma_1}= \frac{E_1}{c} \cdot (1,\hat{n}1)$ and $\textbf{p}^\alpha{\gamma_2}= \frac{E_2}{c} \cdot (2,\hat{n}_2)$. Furthermore, we know the angle between the two photon beams is given by $\theta$, and $\cos(\theta)=\hat{n}_1\cdot\hat{n}_2$.

Since $\textbf{p}^2=-m_0^2c^2$, and photons don't have a invariant mass, we know that $\textbf{p}_1^2 = \textbf{p}_2^2 = 0$.

We therefore find that $(\textbf{p}^\alpha_{\gamma_1}+\textbf{p}^\alpha_{\gamma_2})^2=2\textbf{p}^\alpha_{\gamma_1}\cdot\textbf{p}^\alpha_{\gamma_2}$ which gives us

$$(\textbf{p}^\alpha_{\gamma_1}+\textbf{p}^\alpha_{\gamma_2})^2=-2 \frac{E_1E_2}{c^2} (1-cos(\theta)).$$

Remembering that $\textbf{p}^2=-m_0^2c^2$, we can find the invariant mass of the parent particle (the Higgs boson)

$$m_H = \frac{1}{c^2} \sqrt{2E_1E_2(1-\cos(\theta))}.$$

Since $E_1,E_2,\theta$ are all measureable by the detector, we can therefore get a value of the mass of the Higgs boson. Exactly this was done on data from 2011 and 2012 where it was found that the mass of the Higgs boson was around $m_H=126.8 \text{GeV}$.

While the above algebra may seem easy, it's not quite as simple to isolate $H\rightarrow \gamma \gamma$ events. In order to find such events a likelihood function must be developed based on statistical knowledge as well as the physical understanding of the problem. It took most of the first run of the LHC to tune to likelihood function, due to the many tunable parameters, and it's still limiting how many collisions are possible in each batch due to the added noise of more collisions.

These are some of the reasons why experiments in using machine learning algoirthms such as neural networks to let computer automatically find an optimal likelihood function have begun.

Below has a simple neural network been implemented to classify Higgs decay based on the Higgs ML dataset from 2014.

You can use the two features to compete against the neural network in detecting Higgs decays. (The feature values have been normalised, and the $y$-axis is a kernel density estimation of the value. The features have been crafted and chosen from the detector data as those with the highest predictive power)

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1. ## Rational calculation performance

I have this method which calculates a new Rational (this is simply a class that holds a numerator and denominator) given an input Rational.

public static Rational RightBound(Rational left)
{
long lNum = (long)left.Numerator;
long lDen = (long)left.Denominator;
for (long i = 1; i <= lDen; i++)
{
if ((lNum * i + 1) % lDen == 0)
{
return new Rational((int)((lNum * i + 1) / lDen), (int)i);
}
}
}

It calculates the value of i based on the equation (lNum * i + 1) % lDen
(% is a modulus operator).
This is fine and works (although to be honest I dont really understand what it is doing). The problem is that when the input denominator becomes large the performance is not good - it can sit in the loop several million times and we call this method millions of times!

Does anyone recognize what this is doing and can suggest an alternative way to calculate i?

Or alternatively a condition to reduce the loop size eg i is always less than the input numerator (I know it is not but....)

2. Originally Posted by Geetarman
I have this method which calculates a new Rational (this is simply a class that holds a numerator and denominator) given an input Rational.

Code:
public static Rational RightBound(Rational left)
{
long lNum = (long)left.Numerator;
long lDen = (long)left.Denominator;
for (long i = 1; i <= lDen; i++)
{
if ((lNum * i + 1) % lDen == 0)
{
return new Rational((int)((lNum * i + 1) / lDen), (int)i);
}
}
}
It calculates the value of i based on the equation (lNum * i + 1) % lDen
(% is a modulus operator).
This is fine and works (although to be honest I dont really understand what it is doing). The problem is that when the input denominator becomes large the performance is not good - it can sit in the loop several million times and we call this method millions of times!

Does anyone recognize what this is doing and can suggest an alternative way to calculate i?

Or alternatively a condition to reduce the loop size eg i is always less than the input numerator (I know it is not but....)
At a guess this is trying to rewrite the rational in lowest terms (but that is a guess, it is not easy to see what this is trying to do). In effect you want to find the greatest common divisor (gcd) of the numerator and denominator then divide both numerator and denominator by this. We usually use the Euclidean Algorithm for this.

Having animated this in another package I can say it is not reducing the rational to lowest terms - sorry

It appears to be finding the smallest positive i <= left.Denominator such that left.Numerator*i+1 is divisible by left.Denominator and returning a rational with numerator (left.Numerator*i+1)/left.Denominator and denominator i.

CB


text:
# Using the inversion formula of a Fourier Transform to calculate a limit

Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by

$$f(x)=\begin{cases} 1-|x|/2 \quad \text{if} \ |x|\leq 2, \\ 0 \qquad \quad\quad\ \text{otherwise}. \end{cases}$$ Calculate the fourier transform of $f$ and hence, using inversion formula, show that
$$\int_{-\infty}^\infty \frac{\sin^2 (t)}{t^2}dt =2\pi.$$ I computed the fourier transform to be
$$\hat{f}(t) = \frac{1-\cos(2t)}{t^2} = \frac{2\sin^2(t)}{t^2}.$$
I'm not sure how to use it now. I thought that the inversion would give us our old function $f$ back... let alone a constant value.

The definition of the inversion formula is
$$f(x) = \frac{1}{2\pi}\int_{\mathbb{R}}\hat{f}(t)e^{ixt}dt.$$

• Just take $x=0$. – uniquesolution Oct 17 '17 at 12:00

The inversion formula give you : $$\forall x\in \mathbb{R},\quad f(x) =\frac{1}{2\pi}\int_{-\infty}^{+\infty}\hat{f}(t)e^{ixt}\mathrm{d}t =\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{\sin^2(t)}{t^2}e^{ixt}\mathrm{d}t$$ Hence, $$f(0) =1= \frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{\sin^2(t)}{t^2}\mathrm{d}t$$ So $$\int_{-\infty}^{+\infty}\frac{\sin^2(t)}{t^2}\mathrm{d}t = \pi .$$
It appears that there is not $2$ if your Fourier transform is correct.
• Thank you. Could you please double check my Fourier Transform? I used wolframalpha.com/input/?i=integral+from+-2+to+0+of+(1%2Bx%2F2)*e%5E(-ixa)dx+%2B+integral+from+0+to+2+of+(1-x%2F2)*e%5E(-ixa)dx to compute my integral and it seems to get what I got... The definition the fourier transform given is $$\hat{f}(t) = \int_{\mathbb{R}} f(x)e^{-ixt}dx$$ – Twenty-six colours Oct 18 '17 at 0:48
• I've checked and your Fourier Transform is correct. The answer of $\int_\mathbb{R}\sin^2(t)/t^2 dt$ is $\pi$ (I've checked it with maple). – Zanzi Oct 18 '17 at 9:32


text:
# Should I include punctuation marks in \emph, or should I place those marks after the command?

I wonder if I should place punctuation marks directly after the \emph{} command, or should I include those marks within the curly brackets? Thus:

The languages only differ in their \emph{grammar}, their pronunciation and their most common words.

vs.

The languages only differ in their \emph{grammar,} their pronunciation and their most common words.

-
Vaguely related question (namely, there's a reference to a style manual which comments on things like this): tex.stackexchange.com/q/13048/86 –  Loop Space Apr 5 '11 at 8:52
Closely related question of my own: english.stackexchange.com/questions/9878/… –  Jon Purdy Apr 5 '11 at 16:06

I would say: it depends, namely on the style guide you have to use, or on the typographer you are relying on. Bringhurst, in his Elements of Typographic Style, writes (on page 60 in version 3.2):

When boldface is used to emphasize words, it is usually best to leave the punctuation in the background, which is to say, in the basic text font. It is the words, not the punctuation, that merit emphasis in a sequence such as the following:

… on the islands of Lombok, Bali, Flores, Timor and Sulawesi, the same textiles …

But if the same names are emphasized by setting them in italic rather than bold, there is no advantage in leaving the punctuation in roman. With italic text, italic punctuation normally gives better letterfit and thus looks less obtrusive:

… on the islands of Lombok, Bali, Flores, Timor and Sulawesi, the same textiles …

Regarding parentheses, Bringhurst states (on page 85 in version 3.2):

### Use upright (i.e., “roman”) rather than sloped parentheses, brackets and braces, even if the context is italic.

Parentheses and brackets are not letters, and it makes little sense to speak of them as roman or italic. There are vertical parentheses and sloped ones, and the parentheses on italic fonts are almost always sloped, but vertical parentheses are generally to be preferred. That means they must come from the roman font, and may need extra spacing when used with italic letterforms.

(efg) (efg)

The sloped square brackets usually found on italic fonts are, if anything, even less useful than sloped parentheses. If, perish the thought, there were a book or film entitled The view from My [sic] Bed, sloped brackets might be useful as a way of indicating that the brackets and their contents are actually part of the title. Otherwise, vertical brackets should be used, no matter whether the text is roman or italic: “The View from My [sic] Bed” and “the view from my [sic] bed.

Note that the spacing looks ugly here, but not in Bringhurst’s book …

I quoted from Bringhurst’s work in such length because I wanted to show the rationale behind his statements. Of course, other typographers might find other explanations and solutions, but I like Bringhurst’s.

-

This is more a style question than one about TeX per se. Traditional typographic rules were to have the punctuation match the preceding text, so

\emph{Some text,} or \textbf{some text,} or ...

This is still seen in some places, for example the journal Nature. Most texts nowadays have moved away from this, and view the punctuation as following the style of the overall context. So

\emph{Some text}, or \textbf{some text}, or ...

would I think be more usual today. As I say, this is a matter of style, and so if you are sending material for publication you should seek advice from the publisher.

-

In Lamport's LaTeX manual there is an emphasis example on page 16 not including the final period: Here is some silly \emph{emphasized text}. So I guess it is standard to not include the punctuation mark, unless you emphasize a complete sentence with interspersed commas and final period.

-


text:
# Neural network

(Redirected from Neural nets)
Jump to: navigation, search

## Contents

See also Artificial neural network.

A neural network is a computing paradigm that is loosely modeled after cortical structures of the brain. It consists of interconnected processing elements called neurons that work together to produce an output function. The output of a neural network relies on the cooperation of the individual neurons within the network to operate. Processing of information by neural networks is often done in parallel rather than in series (or sequentially). Since it relies on its member neurons collectively to perform its function, a unique property of a neural network is that it can still perform its overall function even if some of the neurons are not functioning. That is, they are very robust to error or failure (i.e., fault tolerant).

Neural network is sometimes used to refer to a branch of computational science that uses neural networks as models to either simulate or analyze complex phenomena and/or study the principles of operation of neural networks analytically. It addresses problems similar to artificial intelligence (AI) except that AI uses traditional computational algorithms to solve problems whereas neural networks use 'networks of agents' (software or hardware entities linked together) as the computational architecture to solve problems. Well-designed neural networks are trainable systems that can often "learn" to solve complex problems from a set of exemplars and generalize the "acquired knowledge" to solve unforeseen problems, i.e., they are self-adaptive systems.

Traditionally, a neural network is used to refer to a network of biological neurons. In modern usage, the term is often used to refer to artificial neural networks, which are composed of artificial neurons. Thus the term 'Neural Network' has two distinct connotations:

1. Biological neural networks are made up of real biological neurons that are connected or functionally-related in the peripheral nervous system or the central nervous system. In the field of neuroscience, they are often identified as groups of neurons that perform a specific physiological function in laboratory analysis.
2. Artificial neural networks are made up of interconnecting artificial neurons (usually simplified neurons) designed to model (or mimic) some properties of biological neural networks. Artificial neural networks can be used to model the modes of operation of biological neural networks, whereas cognitive models are theoretical models that mimic cognitive brain functions without necessarily using neural networks while artificial intelligence are well-crafted algorithms that solve specific intelligent problems (such as chess playing, pattern recognition, etc.) without using neural network as the computational architecture.

Please see the corresponding articles for details on artificial neural networks or biological neural networks. This article focuses on the relationship between the two concepts.

Image:Neural network.svg
Simplified view of an artificial neural network

## Characterization

In general, a biological neural network is composed of a group or groups of physically connected or functionally associated neurons. A single neuron can be connected to many other neurons and the total number of neurons and connections in a network can be extremely large. Connections, called synapses, are usually formed from axons to dendrites, though dendrodentritic microcircuits [Arbib, p.666] and other connections are possible. Apart from the electrical signalling, there are other forms of signaling that arise from neurotransmitter diffusion, which have an effect on electrical signaling. As such, neural networks are extremely complex. While a detailed description of neural systems seems currently unattainable, progress is made towards a better understanding of basic mechanisms.

Artificial intelligence and cognitive modeling try to simulate some properties of neural networks. While similar in their techniques, the former has the aim of solving particular tasks, while the latter aims to build mathematical models of biological neural systems.

In the artificial intelligence field, artificial neural networks have been applied successfully to speech recognition, image analysis and adaptive control, in order to construct software agents (in computer and video games) or autonomous robots. Most of the currently employed artificial neural networks for artificial intelligence are based on statistical estimation, optimisation and control theory.

The cognitive modelling field is the physical or mathematical modelling of the behaviour of neural systems; ranging from the individual neural level (e.g. modelling the spike response curves of neurons to a stimulus), through the neural cluster level (e.g. modelling the release and effects of dopamine in the basal ganglia) to the complete organism (e.g. behavioural modelling of the organism's response to stimuli).

## The brain, neural networks and computers

While historically the brain has been viewed as a type of computer, and vice-versa, this is true only in the loosest sense. Computers do not provide us with accurate hardware for describing the brain (even though it is possible to describe a logical process as a computer program, or to simulate a brain using a computer) as they do not possess the parallel processing architectures that have been described in the brain. Even when speaking of multiprocessor computers, the functions are not nearly as distributed as in the brain.

Neural networks, as used in artificial intelligence, have traditionally been viewed as simplified models of neural processing in the brain, even though the relation between this model and brain biological architecture is very much debated. To answer this question, Marr has proposed various levels of analysis which provide us with a plausible answer for the role of neural networks in the understanding of human cognitive functioning.

The question of what is the degree of complexity and the properties that individual neural elements should have in order to reproduce something resembling animal intelligence is a subject of current research in theoretical neuroscience.

Historically computers evolved from Von Neumann architecture, based on sequential processing and execution of explicit instructions. On the other hand origins of neural networks are based on efforts to model information processing in biological systems, which are primarily based on parallel processing as well as implicit instructions based on recognition of patterns of 'sensory' input from external sources. In other words, rather than sequential processing and execution, at their very heart, neural networks are complex statistic processors.

## Neural networks and Artificial intelligence

Main article: Artificial neural network

An artificial neural network (ANN), also called a simulated neural network (SNN) or commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical or computational model for information processing based on a connectionist approach to computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network.

In more practical terms neural networks are non-linear statistical data modeling tools. They can be used to model complex relationships between inputs and outputs or to find patterns in data.

### Background

An artificial neural network involves a network of simple processing elements (neurons) which can exhibit complex global behaviour, determined by the connections between the processing elements and element parameters.

In a neural network model, simple nodes (called variously "neurons", "neurodes", "PEs" ("processing elements") or "units") are connected together to form a network of nodes — hence the term "neural network". While a neural network does not have to be adaptive per se, its practical use comes with algorithms designed to alter the strength (weights) of the connections in the network to produce a desired signal flow.

In modern software implementations of artificial neural networks the approach inspired by biology has more or less been abandoned for a more practical approach based on statistics and signal processing. In some of these systems neural networks, or parts of neural networks (such as artificial neurons) are used as components in larger systems that combine both adaptive and non-adaptive elements.

### Applications

The utility of artificial neural network models lies in the fact that they can be used to infer a function from observations. This is particularly useful in applications where the complexity of the data or task makes the design of such a function by hand impractical.

#### Real life applications

The tasks to which artificial neural networks are applied tend to fall within the following broad categories:

Application areas include system identification and control (vehicle control, process control), game-playing and decision making (backgammon, chess, racing), pattern recognition (radar systems, face identification, object recognition and more), sequence recognition (gesture, speech, handwritten text recognition), medical diagnosis, financial applications, data mining (or knowledge discovery in databases, "KDD"), visualisation and e-mail spam filtering.

###  Neural network software

Main article: Neural network software

Neural network software is used to simulate, research, develop and apply artificial neural networks, biological neural networks and in some cases a wider array of adaptive systems.

### Learning paradigms

There are three major learning paradigms, each corresponding to a particular abstract learning task. These are supervised learning, unsupervised learning and reinforcement learning. Usually any given type of network architecture can be employed in any of those tasks.

#### Supervised learning

In supervised learning, we are given a set of example pairs $(x, y), x \in X, y \in Y$ and the aim is to find a function f in the allowed class of functions that matches the examples. In other words, we wish to infer the mapping implied by the data and the cost function is related to the mismatch between our mapping and the data.

#### Unsupervised learning

In unsupervised learning we are given some data $x$, and the cost function to be minimised can be any function of the data $x$ and the network's output, $f$. The cost function is determined by the task formulation. Most applications fall within the domain of estimation problems such as statistical modelling, compression, filtering, blind source separation and clustering.

#### Reinforcement learning

In reinforcement learning, data $x$ is usually not given, but generated by an agent's interactions with the environment. At each point in time $t$, the agent performs an action $y_t$ and the environment generates an observation $x_t$ and an instantaneous cost $c_t$, according to some (usually unknown) dynamics. The aim is to discover a policy for selecting actions that minimises some measure of a long-term cost, i.e. the expected cumulative cost. The environment's dynamics and the long-term cost for each policy are usually unknown, but can be estimated. ANNs are frequently used in reinforcement learning as part of the overall algorithm. Tasks that fall within the paradigm of reinforcement learning are control problems, games and other sequential decision making tasks.

####  Learning algorithms

There are numerous algorithms available for training neural network models; most of them can be viewed as a straightforward application of optimization theory and statistical estimation. This is done by simply taking the derivative of the cost function with respect to the network parameters and then changing those parameters in a gradient-related direction.

Evolutionary computation methods, simulated annealing, expectation maximization and non-parametric methods are among other commonly used methods for training neural networks. See also machine learning.

Recent developments in this field also saw the use of Particle Swarm Optimizations and other Swarm Intelligence techniques used in the training of neural networks, in this case fitness would be defined as the accuracy of the neural network.

## Neural networks and neuroscience

Theoretical and computational neuroscience is the field concerned with the theoretical analysis and computational modeling of biological neural systems. Since neural systems are intimately related to cognitive processes and behaviour, the field is closely related to cognitive and behavioural modeling.

The aim of the field is to create models of biological neural systems in order to understand how biological systems work. To gain this understanding, neuroscientists strive to make a link between observed biological processes (data), biologically plausible mechanisms for neural processing and learning (biological neural network models) and theory (statistical learning theory and information theory).

###  Types of models

Many models are used in the field, each defined at a different level of abstraction and trying to model different aspects of neural systems. They range from models of the short-term behaviour of individual neurons, through models of how the dynamics of neural circuitry arise from interactions between individual neurons, to models of how behaviour can arise from abstract neural modules that represent complete subsystems. These include models of the long-term and short-term plasticity of neural systems and its relation to learning and memory, from the individual neuron to the system level.

### Current research

While initially research had been concerned mostly with the electrical characteristics of neurons, a particularly important part of the investigation in recent years has been the exploration of the role of neuromodulators such as dopamine, acetylcholine, and serotonin on behaviour and learning.

### References

• Peter Dayan, L.F. Abbott. Theoretical Neuroscience. MIT Press.
• Wulfram Gerstner, Werner Kistler. Spiking Neuron Models:Single Neurons, Populations, Plasticity. Cambridge University Press.

## History of the neural network analogy

Main article: Connectionism

The concept of neural networks started in the late-1800s as an effort to describe how the human mind performed. These ideas started being applied to computational models with the Perceptron.

In early 1950s Friedrich Hayek was one of the first to posit the idea of spontaneous order in the brain arising out of decentralized networks of simple units (neurons). In the late 1940s, Donald Hebb made one of the first hypotheses for a mechanism of neural plasticity (i.e. learning), Hebbian learning. Hebbian learning is considered be a 'typical' unsupervised learning rule and it (and variants of it) was an early model for long term potentiation.

The Perceptron is essentially a linear classifier for classifying data $x \in R^n$ specified by parameters $w \in R^n, b \in R$ and an output function $f = w'x + b$. Its parameters are adapted with an ad-hoc rule similar to stochastic steepest gradient descent. Because the inner product is linear operator in the input space, the Perceptron can only perfectly classify a set of data for which different classes are linearly separable in the input space, while it often fails completely for non-separable data. While the development of the algorithm initially generated some enthusiasm, partly because of its apparent relation to biological mechanisms, the later discovery of this inadequacy caused such models to be abandoned until the introduction of non-linear models into the field.

The Cognitron (1975) was an early multilayered neural network with a training algorithm. The actual structure of the network and the methods used to set the interconnection weights change from one neural strategy to another, each with its advantages and disadvantages. Networks can propagate information in one direction only, or they can bounce back and forth until self-activation at a node occurs and the network settles on a final state. The ability for bi-directional flow of inputs between neurons/nodes was produced with the Hopfield's network (1982), and specialization of these node layers for specific purposes was introduced through the first hybrid network.

The parallel distributed processing of the mid-1980s became popular under the name connectionism.

The backpropagation network was probably the main reason behind the repopularisation of neural networks after the publication of "Learning Internal Representations by Error Propagation" in 1986. The original network utilised multiple layers of weight-sum units of the type $f = g(w'x + b)$, where $g$ was a sigmoid function. Training was done by a form of stochastic steepest gradient descent. The employment of the chain rule of differentiation in deriving the appropriate parameter updates results in an algorithm that seems to 'backpropagate errors', hence the nomenclature. However it is essentially a form of gradient descent. Determining the optimal parameters in a model of this type is not trivial, and steepest gradient descent methods cannot be relied upon to give the solution without a good starting point. In recent times, networks with the same architecture as the backpropagation network are referred to as Multi-Layer Perceptrons. This name does not impose any limitations on the type of algorithm used for learning.

The backpropagation network generated much enthusiasm at the time and there was much controversy about whether such learning could be implemented in the brain or not, partly because a mechanism for reverse signalling was not obvious at the time, but most importantly because there was no plausible source for the 'teaching' or 'target' signal.

In more recent times, neuroscientists have successfully made some associations between reinforcement learning and the dopamine system of reward. However, the role of this and other neuromodulators is still under active investigation.

## Criticism

A. K. Dewdney, a former Scientific American columnist, wrote in 1997, “Although neural nets do solve a few toy problems, their powers of computation are so limited that I am surprised anyone takes them seriously as a general problem-solving tool.” (Dewdney, p.82)

Arguments against Dewdney's position are that neural nets have been successfully used to solve many complex and diverse tasks, ranging from autonomously flying aircraft [1] to detecting credit card fraud [2].

Technology writer Roger Bridgman commented on Dewdney's statements about neural nets: "Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be "an opaque, unreadable table...valueless as a scientific resource".

In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers. An unreadable table that a useful machine could read would still be well worth having."[3]

## References

• Template:Cite journal
• Wasserman, P.D. (1989). Neural computing theory and practice. Van Nostrand Reinhold.
• Jeffrey T. Spooner, Manfredi Maggiore, Raul Ord onez, and Kevin M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques, John Wiley and Sons, NY, 2002.

## Neural network

##### what is world wizzy?
• World Wizzy is a static snapshot taken of Wikipedia in early 2007. It cannot be edited and is online for historic & educational purposes only.


text:
# A study on 4 reactions forming the ${}^{46}Ti^{*}$.

Cicerchia M., Gramegna F. per la Collaborazione NUCLE-EX

Comunicazione
I - Fisica nucleare e subnucleare
Aula 32C-3 - Venerdì 21 h 15:30 - 18:30
» Download abstract

The study of pre-equilibrium emitted particles is a useful tool to examine nuclear clustering; to study how possible cluster structures affect nuclear reactions, the NUCL-EX collaboration (INFN, Italy) is carrying out an extensive research campaign on pre-equilibrium emission of light charged particles from hot nuclei. In this framework, the reactions ${}^{16}O + {}^{30}Si$, ${}^{18}O + {}^{28}Si$ and ${}^{19}F + {}^{27}Al$ at 7 MeV/u have been measured at the GARFIELD+RCo array in Legnaro National Laboratories. After a general introduction on the experimental campaign, this contribution will focus on the analysis results obtained so far; effects related to the entrance channel and to the colliding ions cluster nature are emphasized through differences between the theoretical predictions and the experimental data.

Società Italiana di Fisica - Via Saragozza 12 40123 Bologna P.IVA 00308310374 | credits


text:
This page's content is no longer actively maintained, but the material has been kept on-line for historical purposes.
The page may contain broken links or outdated information, and parts may not function in current web browsers.

## Hypertext Help with LaTeX

### \emph

This is LaTeX2e.

The \emph command is considered more suitable for short stretches of emphasized text. Like the LaTeX 2.09 \em command it toggles between italics and Roman typefaces. A typical use would be

    This is \emph{emphasized text}.


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# Are anti-unitary gates possible?

According to Wigner’s theorem, every symmetry operation must be represented in quantum mechanics by an unitary or an anti-unitary operator. To see this, we can see that given any two states $$|\psi\rangle$$ and $$|\psi'\rangle$$, you would like to preserve

$$|\langle\psi|\psi\rangle'|^2=|\langle O\psi|O\psi \rangle'|^2 \tag{1}$$

under some transformation $$O$$. If $$O$$ is unitary, that works. Yet, we can verify that an anti-unitary $$A$$ operator such that

$$\langle A\psi| A\psi'\rangle= \langle\psi|\psi'\rangle^* ,\tag{2}$$ works too; where $${}^*$$ is the complex conjugate. Note that I cannot write it as $$\color{red}{\langle \psi| A^\dagger A|\psi\rangle'}$$ as $$A$$ does not behave as usual unitary operators, it is only defined on kets $$|A\psi\rangle=A|\psi\rangle$$ not bras.

I was wondering if one could make an anti-unitary gate? Would that have any influence on what can be achieved with a quantum computer?

## Example

I was thinking on some kind of time reversal gate $$T$$. The time reversal operator is the standard example of anti-unitarity.

As a qubit is a spin-1/2 like system, for a single qubit we need to perform $$T=iYK\tag{3}$$ where $$K$$ is the complex conjugate. If you have an state $$|\psi\rangle=a|0\rangle+b|1\rangle\tag{4}$$ then $$K|\psi\rangle=a^*|0\rangle+b^*|1\rangle.\tag{5}$$ I do not think you can build the operator $$K$$ from the usual quantum gates without knowing the state of the qubit beforehand. The gate has to be tailored to the state of the qubit.

## Experiments

I know now of one preprint Arrow of Time and its Reversal on IBM Quantum Computer (2018) where the authors claim to have achieved this on a IBM backend. But I do not know if their time reversal algorithm is the same as making an anti-unitary gate.

• What is an anti-unitary gate? $U U^\dagger = -I$? Aug 17, 2021 at 16:15
• @Rammus I think it is more than that. You need some anti-linearity. An example is the complex conjugate operation $K$. If you have a statevector $[a,b]$ then $K[a,b]=[a^*,b^*]$ . It can be antilinear and still $KK^\dagger=I$ Aug 17, 2021 at 16:29
• Ahh interesting, maybe it is good to add the definition of antiunitary operators to the question. Aug 17, 2021 at 16:45
• Mauricio: ah, I think I understand now. An antilinear map still defines a proper action on the projective space, thus it's still meaningful to ask how it acts on quantum states. I never thought about it this way, it's really interesting. Also, an antilinear $f:V\to V$ is still linear when thought of as $f:\bar V\to V$ with $\bar V$ complex conjugate space, so it might make sense to describe it as a matrix maybe? Granted, care would likely be needed when performing things such as basis changes on such a "matrix representation"
– glS
Aug 18, 2021 at 12:03
• @glS indeed, the action on rays is equivalent to action on projectors ... And yes, the action on density matrices is well-defined since the mapping is positive. Finally, you're completely right, applying it to one half of an entangled state will bring you out of state space. Aug 18, 2021 at 12:09

No, you can't perform anti-unitary gates. Or rather: if they are possible, then faster than light communication is possible.

This works because conjugating sqrt(X) is equivalent to inverting it. The conjugation between the X^(1/2) and the X^(-1/2) makes them add up to an X instead of cancelling to an I. So they act as conjugation detectors.

(It might be a bit silly to associate a conjugation with a specific qubit, as it is a global effect. Perhaps it is coming along for the ride as part of another operation, such as the "Universal Not".)

Of course, this diagram assumes a particular notion of simultaneity (indicated by the dashed vertical line). But that means antiunitary operations would reveal a preferred rest frame to the universe! In fact it's even worse than that. Antiunitary operations also break rotational symmetry! The X rotations act as conjugation detectors, as do Z rotations, but Y rotations don't, so you could find the preferred Y axis of the universe. (Well... okay, technically the exact thing you expect to break, and exactly how you expect it to break, will depend on what your favorite interpretation of quantum mechanics is.)

• Sorry, could you explain a bit more the details of the circuit ? I am not sure I am getting what it’s supposed to do. Aug 17, 2021 at 20:04
• Also thanks for the paper! I would also argue that their universal NOT (a gate that finds an orthogonal state to any state) is exactly the time reversal $T$ I defined above (as long as we talk of single qubits). Aug 17, 2021 at 20:10
• @Mauricio There's not much to say about the circuit. The sqrt(X) gate has the property that its conjugation is equal to its inverse, so when you put a conjugation between it and its inverse you un-inverse the inverse so you get a full X gate instead of cancelling to an I gate. So the two sqrt X pieces act as a conjugation detector that works no matter where the conjugation occurs. Aug 17, 2021 at 21:02
• Fun fact: QM threw a monkey wrench into what was thought to be a proof that FTL communication was impossible, and it's still undecided. We believe FTL is simply not possible but can't prove it. Aug 18, 2021 at 3:53
• Maybe it would help to clean up the circuit. You could remove all the unneccesary registers (all but the first and second last). Also you can remove the "off" and "on" (what do they mean anyway?). Aug 19, 2021 at 6:16