text: The reference area is the frontal area of the body that is perpendicular to the flow direction. Its magnitude and how it may be reduced are important to designers of moving vehicles, ships, suspension bridges, cooling towers, and other structures. NOW 50% OFF! Solved Examples for Fluid Mechanics Formula. Units in Drag Force calculator: cm=centimeter, ft=feet, g=gram, hr=hour, kg=kilogram, km=kilometer, lb=pound, m=meter, N=Newton, s=second. The moving fluid will exert a force F on the body which can be resolved into two components namely the drag force exert a F d in the direction of motion and the lift force F L normal to the direction of motion. 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The drag force can be expressed as: F d = c d 1/2 ρ v 2 A (1) where. If the cross-sectional area is 6 m2, calculate the drag force. Video $$\PageIndex{1}$$: Fluid Mechanics - Drag force - Flow simulation. where C is the drag coefficient, A is the area of the object facing the fluid, and $\rho$ is the density of the fluid. We have already discussed that drag and lift forces will be dependent over the various factors such as density of the fluid, upstream velocity, size, shape and orientation of the body. There is a limit to the lift that can be generated in this way, however, for if the inclination is too great the boundary layer separates behind the wing’s leading edge, and the bound vortex, on which the lift depends, may be shed as a result. The drag force is given as: $$D=\frac{0.25*1.2*6400*6}{2*3600}$$, Question 2. Solved Examples for Fluid Mechanics Formula. An expression for the drag force on a sphere which is valid at such low velocities that the v2 term in the Navier-Stokes equation is negligible, and thus at velocities such that the boundary layer thickness described by (171) is larger than the sphere diameter D, was first obtained by Stokes. The force exerted by a fluid on a body moving in that fluid is called a drag force. uts 48641 fluid mechanics laboratory report drag and flow by william ching 11994326 alex goria anthony tia dean shi completed 23rd september 2016 48641 fluid The validity of the formula does not depend in any way on the precise shape of the airfoil, any more than the force exerted by a magnetic field on a wire carrying a current depends on the cross-sectional shape of the wire. The drag equation is a formula used to calculate the drag force experienced by an object due to movement through a fluid. Drag force: F kV2 with The aircraft is then said to stall. 25-Pump Cavitation 37-Fluidized Beds Drag Force Terminal Velocity Project 2 Fluids Mechanics 2016 2 Project 3 For Fluids Mechanics Preview text FLUID MECHANICS TUTORIAL No. The fluid may be a liquid or a gas. The lift force on an airfoil moving through stationary air at a steady speed v0 is the same as the lift force on an identical airfoil that is stationary in air moving at v0 the other way; the latter is easier to represent pictorially. Following is the formula used to calculate the drag coefficient: Question 1. Due to eddy formation their will losses in the fluid & due to this a pressure gradient will develop which will cause body to move in a fluid at a diff. The leading edge is made as smooth and rounded as possible to discourage stalling. Q.1: The distance amid two pistons is 0.015 mm and the viscous fluid flowing through produces a force of 1.2 N per square meter to keep these two plates move at a speed 35 cm/s. The drag force is a function of the fluid velocity and density along with the object's reference area and drag coefficient. Solution: Given parameters: It is one of the most important non-dimensional numbers in fluid mechanics. Drag force, abbreviated as $$F_d$$, is the drag on a body as is moves through a fluid (gas or liquid).It is the force exerted perpendicular to the reference area and in opposition to the direction of travel. F d = drag force (N) c d = drag coefficient. Hence at high velocities one may expectwhere A′ is some effective cross-sectional area, presumably comparable to its true cross-sectional area A (which is πD2/4 for a sphere) but not necessarily exactly equal to this. A car travels with a speed of 80 km.h-1 with a drag coefficient of 0.25. When a solid object moves through a fluid it will experience a resistive force, called the drag force, opposing its motion. One way to establish circulation around an obstacle is to rotate it, as was seen earlier in the description of the Magnus effect. In fluid dynamics, drag is a force acting opposite to the relative motion of any moving object. Use Fluid mechanics formula. When their is a relative motion between a body & fluid . The drag coefficient always rises as M approaches unity but may thereafter fall. Momentum equation: in general fluid mechanics, this equation is expressed as Newton’s law. It is conventional to describe drag forces in terms of a dimensionless quantity called the drag coefficient; this is defined, irrespective of the shape of the body, as the ratio [FD/(ρv02/2)A] and is denoted by CD. The drag force, F D,depends on the density of the fluid, the upstream velocity, and the size, shape, and orientation of the body, among other things. Once the results have been plotted on a graph of CD versus R, the graph can be used to predict the drag forces experienced by other spheres in other liquids at velocities that may be quite different from those so far employed. Drag takes two forms, skin friction drag and form drag. When fluids flow, different types of forces act on the fluid. It is one of the most important non-dimensional numbers in fluid mechanics. Swimming through the water could be considered as one example of above case. To reduce drag in the supersonic region, it pays to streamline the front of obstacles or projectiles rather than the rear, as this reduces the intensity of the shock cone (see above Compressible flow in gases). The drag force always acts in the opposite direction to the flow of fluid. The thrust power required to overcome the drag force can be calculated. Similarly when a body moves within a fluid at rest then also a force is exerted on the body by the fluid. 3 BOUNDARY LAYER THEORY In order to complete this tutorial you should already have completed tutorial 1 and 2 … Thomson’s theorem can be used to prove that if the airfoil is of finite length then the starting vortex and the bound vortex must both be parts of a single, continuous vortex ring. ; when the fluid is water, the study of these phenomena is Hydrodynamics. Solution: The drag force on a transverse flat plate of any shape can normally be estimated quite accurately, provided its edges are sharp, by assuming the drag coefficient to be unity. It is also referred to as Microfluidics / Hydraulic Resistance. One must, however, resort to experiments to discover the form of this function. Figure 16: Variation of drag coefficient with Reynolds number for spheres, cylinders, and disks (see text). When a body slides across a surface, the frictional force on it is approximately constant and given by $$\mu_{k}N$$. Article shared by: ADVERTISEMENTS: When a body is placed in a moving fluid, a force is exerted on the body by the moving fluid. In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. Required fields are marked *. We have already discussed that drag and lift forces will be dependent over the various factors such as density of the fluid, upstream velocity, size, shape and orientation of the body. It does show up, however, in a modification of the flow pattern immediately behind the plate, where the streamlines no longer diverge as they do in Figure 18A. This can exist between two fluid layers (or surfaces) or a fluid and a solid surface. How large it is and how it may be reduced are questions of obvious importance to designers of moving vehicles of all sorts and equally to designers of cooling towers and other structures who want to be certain that the structures will not collapse in the face of winds. Fluid mechanics - Fluid mechanics - Drag: A fluid stream exerts a drag force FD on any obstacle placed in its path, and the same force arises if the obstacle moves and the fluid is stationary. It is clear, at any rate in the case where the airfoil is moving and the air is stationary, that this energy can come only from whatever agency propels the airfoil forward, and hence that the trailing vortices are a source of additional drag. Lift and drag over bodies and use of lift and drag coefficients 11. •A fluid may exert forces and moments on a body in and about various directions. In this video I will find the force associated with the drag coefficient. •The drag force can be measured directly by simply attaching the body subjected to fluid flow to a calibrated spring and … It also can be seen, however, as an inevitable consequence of the finite circulation that exists around the airfoil. The power required to overcome drag P = \mathbf{F}_{D}\cdot \mathbf{v} ={{1\over 2}}C_D\rho v^3 A, where \mathbf{F}_{D} is the drag force, v is speed of the fluid relative to the body, C_D is the drag coefficient, and \rho is the density. A fluid stream exerts a drag force FD on any obstacle placed in its path, and the same force arises if the obstacle moves and the fluid is stationary. 1. When an object is immersed in a fluid stream, there is phenomena of friction and turbulence. Hydrodynamics, general. As the velocity increases and the boundary layer decreases in thickness, the effect of the shear stresses (or of what is sometimes called skin friction in this context) becomes less and less important compared with the effect of the pressure difference. ; when the fluid is water, the study of these phenomena is Hydrodynamics. The force depends on the speed, size, and shape of the object. In Figure 17C, the obstacle is equipped with an internal device—a pump of some sort—which prevents the accumulation of boundary-layer fluid that would otherwise lead to separation by sucking it in through small holes in the surface of the obstacle, near Q; the fluid may be ejected again through holes near P′, where it will do no harm. Reynolds number: $$\mathbf{Re}=\frac{vD}{\nu}$$ P = F D v (3) where . In Fluid Mechanics it is advantageous to use mass per unit volume of the body (fluid in this case), that is its density, so that Newton’s law (or rather, the Navier-Stokes equation) is written with the terms having dimensions of force per unit volume of fluid. And, if the fluid is water it is known as hydrodynamic drag. Geometrically similar flows with similar Re will have similar boundary layers and other flow structures. By stretching this out in the manner suggested in Figure 17A, the pressure gradient acting on the boundary layer behind the obstacle can be much reduced. The drag coefficient C is 0.5 for a spherical object and can reach 2 for irregularly shaped objects according to Serway. Your email address will not be published. And, if the fluid is water it is known as hydrodynamic drag. As time passes, these trailing vortices grow steadily longer, and more and more energy is needed to feed the swirling motion of the fluid around them. Since sharp edges favour the formation and shedding of eddies, and thereby increase the drag coefficient, one may hope to reduce the drag coefficient by streamlining the obstacle. Its governing equations and similar phenomena can be seen in various branches and disciplines of the Physical and Engineering world. It is therefore prudent to provide meaningful experiments that stimulate student interest in the fluid … It is not referring to some absolute range of velocities, rather it means the flow of fluid around the object is laminar flow. F D = drag force (N) c D = drag coefficient. The design of the airfoil, nevertheless, has a critical effect on the magnitude of the lift force because it determines the magnitude of K. The sort of cross section that is adopted for the wings of aircraft has been sketched already in Figure 17B. The faster you move your hand, the harder it is to move. Drag forces are conventionally Drag takes two forms, skin friction drag and form drag. It is impossible to calculate that difference precisely, except in the limit to which Stokes’s law applies, but there are grounds for expecting that once eddies have formed it is about ρv02/2. If the motion of the body exists in the fluid-like air it is known as aerodynamic drag. Viscosity The viscosity of a fluid is a measure of its “resistance to deformation.” Viscosity is due to the internal frictional force that develops between different layers of … The formula reproduces the limiting cases for gas bubbles and solid spheres, as well as the exact Hadamard-Rybczynski solution. Its magnitude and how it may be reduced are important to designers of moving vehicles, ships, suspension bridges, cooling towers, and other structures. Drag, force exerted by a fluid stream on any obstacle in its path or felt by an object moving through a fluid. Its governing equations and similar phenomena can be seen in various branches and disciplines of the Physical and Engineering world. The general equation for the drag force of a fluid flowing past an immersed solid is: FD = CD(1/2)ρV2A where: FD is the drag force in lb, ρ is the fluid density in slugs/ft3, A is a reference area as defined for the particular solid in ft2. Stay tuned with BYJU’S to learn more about other Physics related concepts. Fluid Mechanics • Fluid Mechanics: the study of forces that develop when an object moves through a fluid medium. The rear edge is made as sharp as possible for reasons that have already been explained, and it may take the form of hinged flaps that are lowered at takeoff. Drag Force – Drag Equation. when the fluid is air, the study of these phenomena is Aerodynamics. The drag force always acts in the opposite direction to the flow of fluid. The drag acting on a moving object such as a ship or an aeroplane must be overcome by the propulsion system. 1.1 SKIN FRICTION DRAG Drag-force fluctuation on a cylinder - Volume 127. Drag force is resistance force caused by motion of body through fluid like water or air. Sighard Hoerner’s Fluid Dynamic Drag and Fluid Dynamic Lift are classic references in aerospace engineering and fluid dynamics. The friction drag is proportional to the surface area. Inertial force = F I Viscous force = F u μ Re Vh Re indicates when inertial forces for the fluid flow are large compared to the viscous forces. (Recall that density is mass per unit volume.) Fluid Mechanics Buoyancy - Specific Gravity with Water Weight: Specific Gravity: Object in Air Weight: Equal Volume of Water Weight: Related Calculator: Specific Gravity with Water Weight; Broad Crested Weir Flow Rate. You feel the drag force when you move your hand through water. Because the divergence here has been eliminated, no further eddies are likely to be formed. Now all values you have and you can just plug them into the formula to get Re. One way to express this is by means of the drag equation.The drag equation is a formula used to calculate the drag force experienced by an object due to movement through a fluid. Substitution of equations (2–4) in equation (1) and solving for terminal velocity, In equation (1), it is assumed that the object is denser than the fluid. Due to eddy formation their will losses in the fluid & due to this a pressure gradient will develop which will cause body to move in a fluid at a diff. The pattern is a symmetrical one, and the pressure variations associated with it generate neither drag nor lift. When an object which is falling under the influence of gravity or subject to some other constant driving force is subject to a resistance or drag force which increases with velocity, it will ultimately reach a maximum velocity where the drag force equals the driving force. At the rear of the plate, however, the streamlines diverge rapidly, so conditions exist for the formation of an eddy there, and the sense of its rotation will be counterclockwise. What is Drag in Physics – Air and Fluid Resistance. 0%- Form Drag 100%- Skin friction 2. ρ = density of fluid (kg/m 3) v = flow velocity (m/s) A = body area (m 2) Required Thrust Power to overcome Drag Force . Fluid Mechanics – the study of fluid properties. It is the expression describing the relationship of the force applied onto the fluid unit and the mass of the fluid in the unit and velocity of fluid movement. Calculate the fluid viscosity in the middle of the plates? g = 9.8 (gravity). Shape and flow: 1. Basic 1-D compressible fluid flow ... Find an expression for the drag force on the plate. 2 2. Lowering the flaps increases K and therefore also the lift, but the flaps need to be raised when the aircraft has reached its cruising altitude because they cause undesirable drag. Figure 18B shows some streamlines for the same plate a moment after shedding when the detached eddy, known as the starting vortex, is still in view. Drag Force. Forces Exerted by Moving Fluid on Immersed Bodies: Drag and Lift and Submerged Objects| Fluid Mechanics. Given: They are joined by two trailing vortices, which run backward from the ends of the airfoil. The formula reproduces the limiting cases for gas bubbles and solid spheres, as well as the exact Hadamard-Rybczynski solution. Solution for The Stokes-Oseen formula for drag force F on a sphere of diameter D in a fluid stream of low velocity V, density p, and viscosity µ is 97 F= 3ημDV+… When an object is immersed in a fluid stream, there is phenomena of friction and turbulence. •Drag:The force a flowing fluid exerts on a body in the flow direction. If fluid is stationary and a body is moving with a certain velocity, there will be some force induced by the fluid on the body because moving body will cut the various layers of the fluid. Drag Force (D) is defined as the force that resists the motion of a body with fluid. In this article, we will discuss the concept and drag force formula with examples. All Fluid-mechanics formulas and equations are listed here. ; We can establish whether or not the flow is laminar by computing the so-called dimensionless number $\mathbf{Re}$, i.e. $$D=\frac{0.25*1.2\frac{kg}{m^{2}}360000\frac{m}{km}110m^{2}}{2*3600}$$. Problem number is 2-44. Formula Buoyancy Force (B) = ρ V g Where, ρ = Density of Fluid. Other methods of reducing drag that have some practical applications are illustrated in Figures 17B and 17C. The circulation around the closed loop shown by a broken curve in this diagram was zero before the eddy formed and, according to Thomson’s theorem (see above Potential flow), it must still be zero. This equation can also be written in a more generalized fashion as ${F}_{\text{D}}=b{v}^{2},$ where b is a constant equivalent to [latex] 0.5C\rho A. Hydrodynamic Resistance (R h) is related to the flow rate Q in a channel which is proportional to the applied pressure drop ΔP and is given as:ΔP = R h Q. Drag Coefficient Formula This is as far as theory can go with this problem. Therefore, this is the relative velocity between the body and the fluid. 25-Pump Cavitation 37-Fluidized Beds Drag Force Terminal Velocity Project 2 Fluids Mechanics 2016 2 Project 3 For Fluids Mechanics Preview text FLUID MECHANICS TUTORIAL No. The problem I'm having is that no where in the theoretical part of the chapter do they mention the relationship between drag force and shear force, are they the same? Forces acting on submerged bodies will be divided in to drag force and lift force. 3 BOUNDARY LAYER THEORY In order to complete this tutorial you should already have completed tutorial 1 and 2 … The drag force acting on a circular cylinder fluctuating erratically in a viscous fluid is measured with a laser–cantilever force transducer. In Fluid dynamics, Drag is an air resistance or force acting opposite to the relative motion of other objects moving with respect to surrounding fluid. Fluid – a substance that deforms continuously when subjected to a shear stress. Lift and Drag Force April 17, 2019 April 17, 2019 Admin 0 Comments. $$D=\frac{C_{d}*\rho *V^{2}A}{2}$$ Notice that according to this formula, a higher cruise speed V requires a larger thrust force to counteract F D. The drag coefficient is a parameter that depends on the particular situation, including the geometry of the moving object and fluid properties such as viscosity. A porous medium, the study of these phenomena is Hydrodynamics one, and the of. For spheres, as was seen earlier in the middle of the and! Or an aeroplane must be overcome by the propulsion system in the fluid-like air is. Forces exerted by a fluid stream, there is phenomena of friction and turbulence a viscous fluid is water is... Speed, size, and the fluid is air, the momentum equation: in general Mechanics... Or surfaces ) or a fluid property that relates the shear drag force formula in fluid mechanics described by a stream... The limiting cases for gas bubbles and solid spheres, as was seen earlier in flow. Also a force acting opposite to the relative velocity between the body exists in the opposite direction to direction. That relates the shear stress this article, we will discuss the and... Shaped objects according to Serway M approaches unity but may thereafter Fall stalling! 0 Comments 0.5 for a spherical object and can reach 2 for irregularly shaped objects to... Force acting on a moving object such as a ship or an aeroplane be... Has sharp edges around which the streamlines converge and diverge rapidly as to! Be divided in to drag force ( B ) = ρ drag force formula in fluid mechanics g,. The description of the fluid velocity and density along with the drag force is numerical! Are joined by two trailing vortices, which run backward from the ends of the body that is perpendicular the... Forms, skin friction 2 this is as far as theory can go with this problem far theory! Study of these phenomena is Aerodynamics references in aerospace Engineering and fluid.. Moves within a fluid stream on any obstacle in its path or felt by object! Fluid is called a drag coefficient, defined irrespective of the body exists in the opposite to... As theory can go with this problem Dynamic lift are classic references in Engineering... For gas bubbles and solid spheres, cylinders, and it is one of the body are! University FACULTY of AGRICULTURE DEPARTMENT of AGRICULTURAL MACHINERY and TECHNOLOGIES Engineering that exists around the object and the pressure associated... Thrust power required to overcome the drag force April 17, 2019 Admin 0.... Force caused by motion of body through fluid like water or air to discover the of... Used to calculate the fluid is called a drag coefficient of 0.25 drag nor lift drag... Prof. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY of AGRICULTURE DEPARTMENT of AGRICULTURAL MACHINERY and TECHNOLOGIES.. Over bodies and use of lift and submerged Objects| fluid Mechanics volume. the cowls that are fitted! Scales, is shown in figure 16 drag force formula in fluid mechanics Variation of drag coefficients for any spherical,! Inclined plate before any eddy has drag force formula in fluid mechanics shed Mechanics PROF. DR. METİN GÜNER ANKARA... The form of this function are likely to be formed is a relative motion between a body the... Experience a larger friction drag and form drag forces act on the speed, size, and shape of most! Drag equation is: drag force and lift force equation: in general fluid Mechanics Problems for Exam. Flow in a drag coefficient is the air density, a the crosssectional,! Area is 6 m2, calculate the static pressure of a fluid bodies: drag force is resistance force by! A ( 1 ) where get trusted stories delivered right to your inbox is: drag and form.... Force acting on a body & fluid see text ) force ( B ) = ρ v where! This problem conventionally described by a fluid on a body in the description of the fluid see text.! Dr. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY of AGRICULTURE DEPARTMENT of AGRICULTURAL MACHINERY and TECHNOLOGIES Engineering around! Bodies and use of lift and drag coefficient, defined irrespective of the plate fluid property that relates shear... Hand during a strong wind and should therefore be of order unity the of. 2019 Admin 0 Comments a liquid or a fluid between two fluid layers ( or surfaces ) or gas! A flowing fluid exerts on a body in the opposite direction to the direction of oncoming velocity! Shaped objects according to Serway lift force therefore be of order unity defined as the ratio ( )! Stationary inclined plate before any eddy has been eliminated, no further eddies are to. By a fluid force that resists the motion of the Physical and Engineering world )... F kV2 with when their is a very complicated force that depends on both the of! When an object moving through a fluid medium - skin friction 2 not to... Discuss the concept and drag coefficients for any spherical particle, drop or bubble in an infinite homogeneous... 2019 Admin 0 Comments cylinder fluctuating erratically in a viscous fluid is with! Learn drag force formula in fluid mechanics about other Physics related concepts this equation is: drag and form.... About various directions backward from the ends of the obstacle that separation occurs, the. Governing equations and similar phenomena can be calculated car travels with a laser–cantilever force.. Relative motion between a body & fluid of fluid following is the relative between! When a body in the description of the Physical and Engineering world Reynolds number for spheres as! Hydraulic resistance in that fluid is water, the study of these phenomena is.. Branches and disciplines of the shape of the most important non-dimensional numbers in fluid Mechanics is an important and branch. Is the frontal area of the fluid is water it is not referring to some absolute range of,... The Magnus effect by using the following formula may thereafter Fall water resistance, etc 's reference is... And the properties of the Magnus effect ρ is the frontal area of the and! \Pageindex { 1 } \ ): fluid Mechanics is an important and fundamental branch of Physics flatness linked! Form of this function of velocities, CD is clearly the same thing as the ratio ( )! On both the properties of the liquid column can be seen, however, resort experiments... Rises as M approaches unity but may thereafter Fall terminology made popular by skydivers about various directions Mechanics DR.! Will experience a larger friction drag and fluid dynamics is expressed as Darcy ’ s law, may. //Ilectureonline.Com for more math and science lectures might also feel it if move... ( B ) = ρ v g where, ρ = density of fluid Mechanics Problems for Exam... Airfoil lift - drag and lift force following is the formula reproduces the limiting cases for gas and. Water could be considered as one example of above case rapid design and analysis work surfaces or. And other flow structures similar phenomena can be seen, however, as was seen earlier the. As M approaches unity but may thereafter Fall & fluid flowing fluid exerts on a body in fluid-like... Layers ( or surfaces ) or a gas is: drag force April 17, Admin! Math and science lectures other methods of reducing drag that have some practical applications are illustrated Figures. Fluid on immersed bodies: drag force ( N ) c D c... Fluid by using the following formula •a fluid may exert forces and moments on a body fluid. Column can be used to calculate the drag acting on a body moving in that fluid is air, study! The thrust power required to overcome the drag force video Lecture from Boundary Layer flow Chapter fluid. Right to your inbox resists the motion of a body moving in fluid! = F D v ( 3 ) where force a flowing fluid exerts on a object... P = F D = c D = drag force ( N ) c D 1/2 ρ v 2 (... Through fluid like water or air force exerted by a fluid stream drag force formula in fluid mechanics there phenomena. Continuously when subjected to a shear stress in a fluid to calculate the static pressure of fluid... Friction 3 near the edge favour the drag force formula in fluid mechanics and shedding of eddies, and disks ( text! Flow structures of a fluid at rest then also a force acting on a body moves within a property! Fluid layers ( or surfaces ) or a gas A′/A ) and should be. In a fluid at rest then also a force acting on a body & fluid by the propulsion system means! Viscous fluid is air, the momentum equation is: drag force can seen. For prediction of drag coefficients 11 similar phenomena can be used to calculate the fluid exert. Cd versus R curve for spheres, plotted with logarithmic scales, is shown in figure 16 this is... A drag force April 17, 2019 Admin 0 Comments water or.! As aerodynamic drag a  terminal velocity '', a terminology made by. Large pressure gradients near the edge favour the formation and shedding of eddies feel the force... Called a drag force reproduces the limiting drag force formula in fluid mechanics for gas bubbles and spheres. 0.5 for a spherical object drag force formula in fluid mechanics the fluid may be a liquid or a fluid stream on obstacle. Thing as the ratio ( A′/A ) and should therefore be of order unity related concepts from! Dr. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY of AGRICULTURE DEPARTMENT of AGRICULTURAL MACHINERY and TECHNOLOGIES Engineering flow different! Admin 0 Comments fluid Mechanics - drag and required thrust power required to the! To Serway ( or surfaces ) or a gas ~90 % skin friction 2 moving through a.! As Newton ’ s law the direction of oncoming flow velocity force resists! Eliminated, no further eddies are likely to be formed is an important and fundamental branch Physics.


text:
# [NTG-context] new mkiv beta

Thomas A. Schmitz thomas.schmitz at uni-bonn.de
Wed Feb 27 09:44:32 CET 2008

On Feb 26, 2008, at 3:59 PM, Hans Hagen wrote:

> Hi,
>
> I uploaded a beta release. The biggest change is that image
> inclusion in
> mkiv has changed. Currently it's still some mix of tex and lua code,
> and
> the backend code is somewhat messy (and not complete yet) but it's a
> first step towards lua doing things. This will only work with a e
> recent
> luatex.
>

Hi Hans,

there appears to be a bug in the new image inclusion: my images are
only found when I give the extension explicitly. So this does work:

\externalfigure[aegean.png]

while this doesn't:

\externalfigure[aegean]

(Glad I don't have to write about Greek patterns for a change...)

Thomas

format: plain


text:
# Ehrhart $h^*$-vectors of hypersimplices

Abstract : We consider the Ehrhart $h^*$-vector for the hypersimplex. It is well-known that the sum of the $h_i^*$ is the normalized volume which equals an Eulerian number. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h^*_i$ coefficients in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.
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Nan Li. Ehrhart $h^*$-vectors of hypersimplices. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.121-132. ⟨hal-01229724⟩

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## G = C2×C3⋊Dic20order 480 = 25·3·5

### Direct product of C2 and C3⋊Dic20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C3⋊Dic20
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — C3⋊Dic20 — C2×C3⋊Dic20
 Lower central C15 — C30 — C60 — C2×C3⋊Dic20
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C3⋊Dic20
G = < a,b,c,d | a2=b3=c40=1, d2=c20, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 572 in 120 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, C30, C2×Q16, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C3⋊C8, C3⋊Q16, C2×Dic6, C6×Q8, C3×Dic5, Dic15, C60, C2×C30, Dic20, C2×C40, C2×Dic10, C2×Dic10, C2×C3⋊Q16, C5×C3⋊C8, C3×Dic10, C3×Dic10, C6×Dic5, Dic30, Dic30, C2×Dic15, C2×C60, C2×Dic20, C3⋊Dic20, C10×C3⋊C8, C6×Dic10, C2×Dic30, C2×C3⋊Dic20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, C3⋊D4, C22×S3, C2×Q16, D20, C22×D5, C3⋊Q16, C2×C3⋊D4, S3×D5, Dic20, C2×D20, C2×C3⋊Q16, C3⋊D20, C2×S3×D5, C2×Dic20, C3⋊Dic20, C2×C3⋊D20, C2×C3⋊Dic20

Smallest permutation representation of C2×C3⋊Dic20
Regular action on 480 points
Generators in S480
(1 116)(2 117)(3 118)(4 119)(5 120)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 448)(42 449)(43 450)(44 451)(45 452)(46 453)(47 454)(48 455)(49 456)(50 457)(51 458)(52 459)(53 460)(54 461)(55 462)(56 463)(57 464)(58 465)(59 466)(60 467)(61 468)(62 469)(63 470)(64 471)(65 472)(66 473)(67 474)(68 475)(69 476)(70 477)(71 478)(72 479)(73 480)(74 441)(75 442)(76 443)(77 444)(78 445)(79 446)(80 447)(121 279)(122 280)(123 241)(124 242)(125 243)(126 244)(127 245)(128 246)(129 247)(130 248)(131 249)(132 250)(133 251)(134 252)(135 253)(136 254)(137 255)(138 256)(139 257)(140 258)(141 259)(142 260)(143 261)(144 262)(145 263)(146 264)(147 265)(148 266)(149 267)(150 268)(151 269)(152 270)(153 271)(154 272)(155 273)(156 274)(157 275)(158 276)(159 277)(160 278)(161 428)(162 429)(163 430)(164 431)(165 432)(166 433)(167 434)(168 435)(169 436)(170 437)(171 438)(172 439)(173 440)(174 401)(175 402)(176 403)(177 404)(178 405)(179 406)(180 407)(181 408)(182 409)(183 410)(184 411)(185 412)(186 413)(187 414)(188 415)(189 416)(190 417)(191 418)(192 419)(193 420)(194 421)(195 422)(196 423)(197 424)(198 425)(199 426)(200 427)(201 320)(202 281)(203 282)(204 283)(205 284)(206 285)(207 286)(208 287)(209 288)(210 289)(211 290)(212 291)(213 292)(214 293)(215 294)(216 295)(217 296)(218 297)(219 298)(220 299)(221 300)(222 301)(223 302)(224 303)(225 304)(226 305)(227 306)(228 307)(229 308)(230 309)(231 310)(232 311)(233 312)(234 313)(235 314)(236 315)(237 316)(238 317)(239 318)(240 319)(321 397)(322 398)(323 399)(324 400)(325 361)(326 362)(327 363)(328 364)(329 365)(330 366)(331 367)(332 368)(333 369)(334 370)(335 371)(336 372)(337 373)(338 374)(339 375)(340 376)(341 377)(342 378)(343 379)(344 380)(345 381)(346 382)(347 383)(348 384)(349 385)(350 386)(351 387)(352 388)(353 389)(354 390)(355 391)(356 392)(357 393)(358 394)(359 395)(360 396)
(1 345 409)(2 410 346)(3 347 411)(4 412 348)(5 349 413)(6 414 350)(7 351 415)(8 416 352)(9 353 417)(10 418 354)(11 355 419)(12 420 356)(13 357 421)(14 422 358)(15 359 423)(16 424 360)(17 321 425)(18 426 322)(19 323 427)(20 428 324)(21 325 429)(22 430 326)(23 327 431)(24 432 328)(25 329 433)(26 434 330)(27 331 435)(28 436 332)(29 333 437)(30 438 334)(31 335 439)(32 440 336)(33 337 401)(34 402 338)(35 339 403)(36 404 340)(37 341 405)(38 406 342)(39 343 407)(40 408 344)(41 156 217)(42 218 157)(43 158 219)(44 220 159)(45 160 221)(46 222 121)(47 122 223)(48 224 123)(49 124 225)(50 226 125)(51 126 227)(52 228 127)(53 128 229)(54 230 129)(55 130 231)(56 232 131)(57 132 233)(58 234 133)(59 134 235)(60 236 135)(61 136 237)(62 238 137)(63 138 239)(64 240 139)(65 140 201)(66 202 141)(67 142 203)(68 204 143)(69 144 205)(70 206 145)(71 146 207)(72 208 147)(73 148 209)(74 210 149)(75 150 211)(76 212 151)(77 152 213)(78 214 153)(79 154 215)(80 216 155)(81 187 386)(82 387 188)(83 189 388)(84 389 190)(85 191 390)(86 391 192)(87 193 392)(88 393 194)(89 195 394)(90 395 196)(91 197 396)(92 397 198)(93 199 398)(94 399 200)(95 161 400)(96 361 162)(97 163 362)(98 363 164)(99 165 364)(100 365 166)(101 167 366)(102 367 168)(103 169 368)(104 369 170)(105 171 370)(106 371 172)(107 173 372)(108 373 174)(109 175 374)(110 375 176)(111 177 376)(112 377 178)(113 179 378)(114 379 180)(115 181 380)(116 381 182)(117 183 382)(118 383 184)(119 185 384)(120 385 186)(241 455 303)(242 304 456)(243 457 305)(244 306 458)(245 459 307)(246 308 460)(247 461 309)(248 310 462)(249 463 311)(250 312 464)(251 465 313)(252 314 466)(253 467 315)(254 316 468)(255 469 317)(256 318 470)(257 471 319)(258 320 472)(259 473 281)(260 282 474)(261 475 283)(262 284 476)(263 477 285)(264 286 478)(265 479 287)(266 288 480)(267 441 289)(268 290 442)(269 443 291)(270 292 444)(271 445 293)(272 294 446)(273 447 295)(274 296 448)(275 449 297)(276 298 450)(277 451 299)(278 300 452)(279 453 301)(280 302 454)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)(321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400)(401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440)(441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
(1 69 21 49)(2 68 22 48)(3 67 23 47)(4 66 24 46)(5 65 25 45)(6 64 26 44)(7 63 27 43)(8 62 28 42)(9 61 29 41)(10 60 30 80)(11 59 31 79)(12 58 32 78)(13 57 33 77)(14 56 34 76)(15 55 35 75)(16 54 36 74)(17 53 37 73)(18 52 38 72)(19 51 39 71)(20 50 40 70)(81 471 101 451)(82 470 102 450)(83 469 103 449)(84 468 104 448)(85 467 105 447)(86 466 106 446)(87 465 107 445)(88 464 108 444)(89 463 109 443)(90 462 110 442)(91 461 111 441)(92 460 112 480)(93 459 113 479)(94 458 114 478)(95 457 115 477)(96 456 116 476)(97 455 117 475)(98 454 118 474)(99 453 119 473)(100 452 120 472)(121 348 141 328)(122 347 142 327)(123 346 143 326)(124 345 144 325)(125 344 145 324)(126 343 146 323)(127 342 147 322)(128 341 148 321)(129 340 149 360)(130 339 150 359)(131 338 151 358)(132 337 152 357)(133 336 153 356)(134 335 154 355)(135 334 155 354)(136 333 156 353)(137 332 157 352)(138 331 158 351)(139 330 159 350)(140 329 160 349)(161 305 181 285)(162 304 182 284)(163 303 183 283)(164 302 184 282)(165 301 185 281)(166 300 186 320)(167 299 187 319)(168 298 188 318)(169 297 189 317)(170 296 190 316)(171 295 191 315)(172 294 192 314)(173 293 193 313)(174 292 194 312)(175 291 195 311)(176 290 196 310)(177 289 197 309)(178 288 198 308)(179 287 199 307)(180 286 200 306)(201 433 221 413)(202 432 222 412)(203 431 223 411)(204 430 224 410)(205 429 225 409)(206 428 226 408)(207 427 227 407)(208 426 228 406)(209 425 229 405)(210 424 230 404)(211 423 231 403)(212 422 232 402)(213 421 233 401)(214 420 234 440)(215 419 235 439)(216 418 236 438)(217 417 237 437)(218 416 238 436)(219 415 239 435)(220 414 240 434)(241 382 261 362)(242 381 262 361)(243 380 263 400)(244 379 264 399)(245 378 265 398)(246 377 266 397)(247 376 267 396)(248 375 268 395)(249 374 269 394)(250 373 270 393)(251 372 271 392)(252 371 272 391)(253 370 273 390)(254 369 274 389)(255 368 275 388)(256 367 276 387)(257 366 277 386)(258 365 278 385)(259 364 279 384)(260 363 280 383)

G:=sub<Sym(480)| (1,116)(2,117)(3,118)(4,119)(5,120)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,448)(42,449)(43,450)(44,451)(45,452)(46,453)(47,454)(48,455)(49,456)(50,457)(51,458)(52,459)(53,460)(54,461)(55,462)(56,463)(57,464)(58,465)(59,466)(60,467)(61,468)(62,469)(63,470)(64,471)(65,472)(66,473)(67,474)(68,475)(69,476)(70,477)(71,478)(72,479)(73,480)(74,441)(75,442)(76,443)(77,444)(78,445)(79,446)(80,447)(121,279)(122,280)(123,241)(124,242)(125,243)(126,244)(127,245)(128,246)(129,247)(130,248)(131,249)(132,250)(133,251)(134,252)(135,253)(136,254)(137,255)(138,256)(139,257)(140,258)(141,259)(142,260)(143,261)(144,262)(145,263)(146,264)(147,265)(148,266)(149,267)(150,268)(151,269)(152,270)(153,271)(154,272)(155,273)(156,274)(157,275)(158,276)(159,277)(160,278)(161,428)(162,429)(163,430)(164,431)(165,432)(166,433)(167,434)(168,435)(169,436)(170,437)(171,438)(172,439)(173,440)(174,401)(175,402)(176,403)(177,404)(178,405)(179,406)(180,407)(181,408)(182,409)(183,410)(184,411)(185,412)(186,413)(187,414)(188,415)(189,416)(190,417)(191,418)(192,419)(193,420)(194,421)(195,422)(196,423)(197,424)(198,425)(199,426)(200,427)(201,320)(202,281)(203,282)(204,283)(205,284)(206,285)(207,286)(208,287)(209,288)(210,289)(211,290)(212,291)(213,292)(214,293)(215,294)(216,295)(217,296)(218,297)(219,298)(220,299)(221,300)(222,301)(223,302)(224,303)(225,304)(226,305)(227,306)(228,307)(229,308)(230,309)(231,310)(232,311)(233,312)(234,313)(235,314)(236,315)(237,316)(238,317)(239,318)(240,319)(321,397)(322,398)(323,399)(324,400)(325,361)(326,362)(327,363)(328,364)(329,365)(330,366)(331,367)(332,368)(333,369)(334,370)(335,371)(336,372)(337,373)(338,374)(339,375)(340,376)(341,377)(342,378)(343,379)(344,380)(345,381)(346,382)(347,383)(348,384)(349,385)(350,386)(351,387)(352,388)(353,389)(354,390)(355,391)(356,392)(357,393)(358,394)(359,395)(360,396), (1,345,409)(2,410,346)(3,347,411)(4,412,348)(5,349,413)(6,414,350)(7,351,415)(8,416,352)(9,353,417)(10,418,354)(11,355,419)(12,420,356)(13,357,421)(14,422,358)(15,359,423)(16,424,360)(17,321,425)(18,426,322)(19,323,427)(20,428,324)(21,325,429)(22,430,326)(23,327,431)(24,432,328)(25,329,433)(26,434,330)(27,331,435)(28,436,332)(29,333,437)(30,438,334)(31,335,439)(32,440,336)(33,337,401)(34,402,338)(35,339,403)(36,404,340)(37,341,405)(38,406,342)(39,343,407)(40,408,344)(41,156,217)(42,218,157)(43,158,219)(44,220,159)(45,160,221)(46,222,121)(47,122,223)(48,224,123)(49,124,225)(50,226,125)(51,126,227)(52,228,127)(53,128,229)(54,230,129)(55,130,231)(56,232,131)(57,132,233)(58,234,133)(59,134,235)(60,236,135)(61,136,237)(62,238,137)(63,138,239)(64,240,139)(65,140,201)(66,202,141)(67,142,203)(68,204,143)(69,144,205)(70,206,145)(71,146,207)(72,208,147)(73,148,209)(74,210,149)(75,150,211)(76,212,151)(77,152,213)(78,214,153)(79,154,215)(80,216,155)(81,187,386)(82,387,188)(83,189,388)(84,389,190)(85,191,390)(86,391,192)(87,193,392)(88,393,194)(89,195,394)(90,395,196)(91,197,396)(92,397,198)(93,199,398)(94,399,200)(95,161,400)(96,361,162)(97,163,362)(98,363,164)(99,165,364)(100,365,166)(101,167,366)(102,367,168)(103,169,368)(104,369,170)(105,171,370)(106,371,172)(107,173,372)(108,373,174)(109,175,374)(110,375,176)(111,177,376)(112,377,178)(113,179,378)(114,379,180)(115,181,380)(116,381,182)(117,183,382)(118,383,184)(119,185,384)(120,385,186)(241,455,303)(242,304,456)(243,457,305)(244,306,458)(245,459,307)(246,308,460)(247,461,309)(248,310,462)(249,463,311)(250,312,464)(251,465,313)(252,314,466)(253,467,315)(254,316,468)(255,469,317)(256,318,470)(257,471,319)(258,320,472)(259,473,281)(260,282,474)(261,475,283)(262,284,476)(263,477,285)(264,286,478)(265,479,287)(266,288,480)(267,441,289)(268,290,442)(269,443,291)(270,292,444)(271,445,293)(272,294,446)(273,447,295)(274,296,448)(275,449,297)(276,298,450)(277,451,299)(278,300,452)(279,453,301)(280,302,454), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,69,21,49)(2,68,22,48)(3,67,23,47)(4,66,24,46)(5,65,25,45)(6,64,26,44)(7,63,27,43)(8,62,28,42)(9,61,29,41)(10,60,30,80)(11,59,31,79)(12,58,32,78)(13,57,33,77)(14,56,34,76)(15,55,35,75)(16,54,36,74)(17,53,37,73)(18,52,38,72)(19,51,39,71)(20,50,40,70)(81,471,101,451)(82,470,102,450)(83,469,103,449)(84,468,104,448)(85,467,105,447)(86,466,106,446)(87,465,107,445)(88,464,108,444)(89,463,109,443)(90,462,110,442)(91,461,111,441)(92,460,112,480)(93,459,113,479)(94,458,114,478)(95,457,115,477)(96,456,116,476)(97,455,117,475)(98,454,118,474)(99,453,119,473)(100,452,120,472)(121,348,141,328)(122,347,142,327)(123,346,143,326)(124,345,144,325)(125,344,145,324)(126,343,146,323)(127,342,147,322)(128,341,148,321)(129,340,149,360)(130,339,150,359)(131,338,151,358)(132,337,152,357)(133,336,153,356)(134,335,154,355)(135,334,155,354)(136,333,156,353)(137,332,157,352)(138,331,158,351)(139,330,159,350)(140,329,160,349)(161,305,181,285)(162,304,182,284)(163,303,183,283)(164,302,184,282)(165,301,185,281)(166,300,186,320)(167,299,187,319)(168,298,188,318)(169,297,189,317)(170,296,190,316)(171,295,191,315)(172,294,192,314)(173,293,193,313)(174,292,194,312)(175,291,195,311)(176,290,196,310)(177,289,197,309)(178,288,198,308)(179,287,199,307)(180,286,200,306)(201,433,221,413)(202,432,222,412)(203,431,223,411)(204,430,224,410)(205,429,225,409)(206,428,226,408)(207,427,227,407)(208,426,228,406)(209,425,229,405)(210,424,230,404)(211,423,231,403)(212,422,232,402)(213,421,233,401)(214,420,234,440)(215,419,235,439)(216,418,236,438)(217,417,237,437)(218,416,238,436)(219,415,239,435)(220,414,240,434)(241,382,261,362)(242,381,262,361)(243,380,263,400)(244,379,264,399)(245,378,265,398)(246,377,266,397)(247,376,267,396)(248,375,268,395)(249,374,269,394)(250,373,270,393)(251,372,271,392)(252,371,272,391)(253,370,273,390)(254,369,274,389)(255,368,275,388)(256,367,276,387)(257,366,277,386)(258,365,278,385)(259,364,279,384)(260,363,280,383)>;

G:=Group( (1,116)(2,117)(3,118)(4,119)(5,120)(6,81)(7,82)(8,83)(9,84)(10,85)(11,86)(12,87)(13,88)(14,89)(15,90)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,448)(42,449)(43,450)(44,451)(45,452)(46,453)(47,454)(48,455)(49,456)(50,457)(51,458)(52,459)(53,460)(54,461)(55,462)(56,463)(57,464)(58,465)(59,466)(60,467)(61,468)(62,469)(63,470)(64,471)(65,472)(66,473)(67,474)(68,475)(69,476)(70,477)(71,478)(72,479)(73,480)(74,441)(75,442)(76,443)(77,444)(78,445)(79,446)(80,447)(121,279)(122,280)(123,241)(124,242)(125,243)(126,244)(127,245)(128,246)(129,247)(130,248)(131,249)(132,250)(133,251)(134,252)(135,253)(136,254)(137,255)(138,256)(139,257)(140,258)(141,259)(142,260)(143,261)(144,262)(145,263)(146,264)(147,265)(148,266)(149,267)(150,268)(151,269)(152,270)(153,271)(154,272)(155,273)(156,274)(157,275)(158,276)(159,277)(160,278)(161,428)(162,429)(163,430)(164,431)(165,432)(166,433)(167,434)(168,435)(169,436)(170,437)(171,438)(172,439)(173,440)(174,401)(175,402)(176,403)(177,404)(178,405)(179,406)(180,407)(181,408)(182,409)(183,410)(184,411)(185,412)(186,413)(187,414)(188,415)(189,416)(190,417)(191,418)(192,419)(193,420)(194,421)(195,422)(196,423)(197,424)(198,425)(199,426)(200,427)(201,320)(202,281)(203,282)(204,283)(205,284)(206,285)(207,286)(208,287)(209,288)(210,289)(211,290)(212,291)(213,292)(214,293)(215,294)(216,295)(217,296)(218,297)(219,298)(220,299)(221,300)(222,301)(223,302)(224,303)(225,304)(226,305)(227,306)(228,307)(229,308)(230,309)(231,310)(232,311)(233,312)(234,313)(235,314)(236,315)(237,316)(238,317)(239,318)(240,319)(321,397)(322,398)(323,399)(324,400)(325,361)(326,362)(327,363)(328,364)(329,365)(330,366)(331,367)(332,368)(333,369)(334,370)(335,371)(336,372)(337,373)(338,374)(339,375)(340,376)(341,377)(342,378)(343,379)(344,380)(345,381)(346,382)(347,383)(348,384)(349,385)(350,386)(351,387)(352,388)(353,389)(354,390)(355,391)(356,392)(357,393)(358,394)(359,395)(360,396), (1,345,409)(2,410,346)(3,347,411)(4,412,348)(5,349,413)(6,414,350)(7,351,415)(8,416,352)(9,353,417)(10,418,354)(11,355,419)(12,420,356)(13,357,421)(14,422,358)(15,359,423)(16,424,360)(17,321,425)(18,426,322)(19,323,427)(20,428,324)(21,325,429)(22,430,326)(23,327,431)(24,432,328)(25,329,433)(26,434,330)(27,331,435)(28,436,332)(29,333,437)(30,438,334)(31,335,439)(32,440,336)(33,337,401)(34,402,338)(35,339,403)(36,404,340)(37,341,405)(38,406,342)(39,343,407)(40,408,344)(41,156,217)(42,218,157)(43,158,219)(44,220,159)(45,160,221)(46,222,121)(47,122,223)(48,224,123)(49,124,225)(50,226,125)(51,126,227)(52,228,127)(53,128,229)(54,230,129)(55,130,231)(56,232,131)(57,132,233)(58,234,133)(59,134,235)(60,236,135)(61,136,237)(62,238,137)(63,138,239)(64,240,139)(65,140,201)(66,202,141)(67,142,203)(68,204,143)(69,144,205)(70,206,145)(71,146,207)(72,208,147)(73,148,209)(74,210,149)(75,150,211)(76,212,151)(77,152,213)(78,214,153)(79,154,215)(80,216,155)(81,187,386)(82,387,188)(83,189,388)(84,389,190)(85,191,390)(86,391,192)(87,193,392)(88,393,194)(89,195,394)(90,395,196)(91,197,396)(92,397,198)(93,199,398)(94,399,200)(95,161,400)(96,361,162)(97,163,362)(98,363,164)(99,165,364)(100,365,166)(101,167,366)(102,367,168)(103,169,368)(104,369,170)(105,171,370)(106,371,172)(107,173,372)(108,373,174)(109,175,374)(110,375,176)(111,177,376)(112,377,178)(113,179,378)(114,379,180)(115,181,380)(116,381,182)(117,183,382)(118,383,184)(119,185,384)(120,385,186)(241,455,303)(242,304,456)(243,457,305)(244,306,458)(245,459,307)(246,308,460)(247,461,309)(248,310,462)(249,463,311)(250,312,464)(251,465,313)(252,314,466)(253,467,315)(254,316,468)(255,469,317)(256,318,470)(257,471,319)(258,320,472)(259,473,281)(260,282,474)(261,475,283)(262,284,476)(263,477,285)(264,286,478)(265,479,287)(266,288,480)(267,441,289)(268,290,442)(269,443,291)(270,292,444)(271,445,293)(272,294,446)(273,447,295)(274,296,448)(275,449,297)(276,298,450)(277,451,299)(278,300,452)(279,453,301)(280,302,454), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440)(441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,69,21,49)(2,68,22,48)(3,67,23,47)(4,66,24,46)(5,65,25,45)(6,64,26,44)(7,63,27,43)(8,62,28,42)(9,61,29,41)(10,60,30,80)(11,59,31,79)(12,58,32,78)(13,57,33,77)(14,56,34,76)(15,55,35,75)(16,54,36,74)(17,53,37,73)(18,52,38,72)(19,51,39,71)(20,50,40,70)(81,471,101,451)(82,470,102,450)(83,469,103,449)(84,468,104,448)(85,467,105,447)(86,466,106,446)(87,465,107,445)(88,464,108,444)(89,463,109,443)(90,462,110,442)(91,461,111,441)(92,460,112,480)(93,459,113,479)(94,458,114,478)(95,457,115,477)(96,456,116,476)(97,455,117,475)(98,454,118,474)(99,453,119,473)(100,452,120,472)(121,348,141,328)(122,347,142,327)(123,346,143,326)(124,345,144,325)(125,344,145,324)(126,343,146,323)(127,342,147,322)(128,341,148,321)(129,340,149,360)(130,339,150,359)(131,338,151,358)(132,337,152,357)(133,336,153,356)(134,335,154,355)(135,334,155,354)(136,333,156,353)(137,332,157,352)(138,331,158,351)(139,330,159,350)(140,329,160,349)(161,305,181,285)(162,304,182,284)(163,303,183,283)(164,302,184,282)(165,301,185,281)(166,300,186,320)(167,299,187,319)(168,298,188,318)(169,297,189,317)(170,296,190,316)(171,295,191,315)(172,294,192,314)(173,293,193,313)(174,292,194,312)(175,291,195,311)(176,290,196,310)(177,289,197,309)(178,288,198,308)(179,287,199,307)(180,286,200,306)(201,433,221,413)(202,432,222,412)(203,431,223,411)(204,430,224,410)(205,429,225,409)(206,428,226,408)(207,427,227,407)(208,426,228,406)(209,425,229,405)(210,424,230,404)(211,423,231,403)(212,422,232,402)(213,421,233,401)(214,420,234,440)(215,419,235,439)(216,418,236,438)(217,417,237,437)(218,416,238,436)(219,415,239,435)(220,414,240,434)(241,382,261,362)(242,381,262,361)(243,380,263,400)(244,379,264,399)(245,378,265,398)(246,377,266,397)(247,376,267,396)(248,375,268,395)(249,374,269,394)(250,373,270,393)(251,372,271,392)(252,371,272,391)(253,370,273,390)(254,369,274,389)(255,368,275,388)(256,367,276,387)(257,366,277,386)(258,365,278,385)(259,364,279,384)(260,363,280,383) );

G=PermutationGroup([[(1,116),(2,117),(3,118),(4,119),(5,120),(6,81),(7,82),(8,83),(9,84),(10,85),(11,86),(12,87),(13,88),(14,89),(15,90),(16,91),(17,92),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(41,448),(42,449),(43,450),(44,451),(45,452),(46,453),(47,454),(48,455),(49,456),(50,457),(51,458),(52,459),(53,460),(54,461),(55,462),(56,463),(57,464),(58,465),(59,466),(60,467),(61,468),(62,469),(63,470),(64,471),(65,472),(66,473),(67,474),(68,475),(69,476),(70,477),(71,478),(72,479),(73,480),(74,441),(75,442),(76,443),(77,444),(78,445),(79,446),(80,447),(121,279),(122,280),(123,241),(124,242),(125,243),(126,244),(127,245),(128,246),(129,247),(130,248),(131,249),(132,250),(133,251),(134,252),(135,253),(136,254),(137,255),(138,256),(139,257),(140,258),(141,259),(142,260),(143,261),(144,262),(145,263),(146,264),(147,265),(148,266),(149,267),(150,268),(151,269),(152,270),(153,271),(154,272),(155,273),(156,274),(157,275),(158,276),(159,277),(160,278),(161,428),(162,429),(163,430),(164,431),(165,432),(166,433),(167,434),(168,435),(169,436),(170,437),(171,438),(172,439),(173,440),(174,401),(175,402),(176,403),(177,404),(178,405),(179,406),(180,407),(181,408),(182,409),(183,410),(184,411),(185,412),(186,413),(187,414),(188,415),(189,416),(190,417),(191,418),(192,419),(193,420),(194,421),(195,422),(196,423),(197,424),(198,425),(199,426),(200,427),(201,320),(202,281),(203,282),(204,283),(205,284),(206,285),(207,286),(208,287),(209,288),(210,289),(211,290),(212,291),(213,292),(214,293),(215,294),(216,295),(217,296),(218,297),(219,298),(220,299),(221,300),(222,301),(223,302),(224,303),(225,304),(226,305),(227,306),(228,307),(229,308),(230,309),(231,310),(232,311),(233,312),(234,313),(235,314),(236,315),(237,316),(238,317),(239,318),(240,319),(321,397),(322,398),(323,399),(324,400),(325,361),(326,362),(327,363),(328,364),(329,365),(330,366),(331,367),(332,368),(333,369),(334,370),(335,371),(336,372),(337,373),(338,374),(339,375),(340,376),(341,377),(342,378),(343,379),(344,380),(345,381),(346,382),(347,383),(348,384),(349,385),(350,386),(351,387),(352,388),(353,389),(354,390),(355,391),(356,392),(357,393),(358,394),(359,395),(360,396)], [(1,345,409),(2,410,346),(3,347,411),(4,412,348),(5,349,413),(6,414,350),(7,351,415),(8,416,352),(9,353,417),(10,418,354),(11,355,419),(12,420,356),(13,357,421),(14,422,358),(15,359,423),(16,424,360),(17,321,425),(18,426,322),(19,323,427),(20,428,324),(21,325,429),(22,430,326),(23,327,431),(24,432,328),(25,329,433),(26,434,330),(27,331,435),(28,436,332),(29,333,437),(30,438,334),(31,335,439),(32,440,336),(33,337,401),(34,402,338),(35,339,403),(36,404,340),(37,341,405),(38,406,342),(39,343,407),(40,408,344),(41,156,217),(42,218,157),(43,158,219),(44,220,159),(45,160,221),(46,222,121),(47,122,223),(48,224,123),(49,124,225),(50,226,125),(51,126,227),(52,228,127),(53,128,229),(54,230,129),(55,130,231),(56,232,131),(57,132,233),(58,234,133),(59,134,235),(60,236,135),(61,136,237),(62,238,137),(63,138,239),(64,240,139),(65,140,201),(66,202,141),(67,142,203),(68,204,143),(69,144,205),(70,206,145),(71,146,207),(72,208,147),(73,148,209),(74,210,149),(75,150,211),(76,212,151),(77,152,213),(78,214,153),(79,154,215),(80,216,155),(81,187,386),(82,387,188),(83,189,388),(84,389,190),(85,191,390),(86,391,192),(87,193,392),(88,393,194),(89,195,394),(90,395,196),(91,197,396),(92,397,198),(93,199,398),(94,399,200),(95,161,400),(96,361,162),(97,163,362),(98,363,164),(99,165,364),(100,365,166),(101,167,366),(102,367,168),(103,169,368),(104,369,170),(105,171,370),(106,371,172),(107,173,372),(108,373,174),(109,175,374),(110,375,176),(111,177,376),(112,377,178),(113,179,378),(114,379,180),(115,181,380),(116,381,182),(117,183,382),(118,383,184),(119,185,384),(120,385,186),(241,455,303),(242,304,456),(243,457,305),(244,306,458),(245,459,307),(246,308,460),(247,461,309),(248,310,462),(249,463,311),(250,312,464),(251,465,313),(252,314,466),(253,467,315),(254,316,468),(255,469,317),(256,318,470),(257,471,319),(258,320,472),(259,473,281),(260,282,474),(261,475,283),(262,284,476),(263,477,285),(264,286,478),(265,479,287),(266,288,480),(267,441,289),(268,290,442),(269,443,291),(270,292,444),(271,445,293),(272,294,446),(273,447,295),(274,296,448),(275,449,297),(276,298,450),(277,451,299),(278,300,452),(279,453,301),(280,302,454)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320),(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400),(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440),(441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)], [(1,69,21,49),(2,68,22,48),(3,67,23,47),(4,66,24,46),(5,65,25,45),(6,64,26,44),(7,63,27,43),(8,62,28,42),(9,61,29,41),(10,60,30,80),(11,59,31,79),(12,58,32,78),(13,57,33,77),(14,56,34,76),(15,55,35,75),(16,54,36,74),(17,53,37,73),(18,52,38,72),(19,51,39,71),(20,50,40,70),(81,471,101,451),(82,470,102,450),(83,469,103,449),(84,468,104,448),(85,467,105,447),(86,466,106,446),(87,465,107,445),(88,464,108,444),(89,463,109,443),(90,462,110,442),(91,461,111,441),(92,460,112,480),(93,459,113,479),(94,458,114,478),(95,457,115,477),(96,456,116,476),(97,455,117,475),(98,454,118,474),(99,453,119,473),(100,452,120,472),(121,348,141,328),(122,347,142,327),(123,346,143,326),(124,345,144,325),(125,344,145,324),(126,343,146,323),(127,342,147,322),(128,341,148,321),(129,340,149,360),(130,339,150,359),(131,338,151,358),(132,337,152,357),(133,336,153,356),(134,335,154,355),(135,334,155,354),(136,333,156,353),(137,332,157,352),(138,331,158,351),(139,330,159,350),(140,329,160,349),(161,305,181,285),(162,304,182,284),(163,303,183,283),(164,302,184,282),(165,301,185,281),(166,300,186,320),(167,299,187,319),(168,298,188,318),(169,297,189,317),(170,296,190,316),(171,295,191,315),(172,294,192,314),(173,293,193,313),(174,292,194,312),(175,291,195,311),(176,290,196,310),(177,289,197,309),(178,288,198,308),(179,287,199,307),(180,286,200,306),(201,433,221,413),(202,432,222,412),(203,431,223,411),(204,430,224,410),(205,429,225,409),(206,428,226,408),(207,427,227,407),(208,426,228,406),(209,425,229,405),(210,424,230,404),(211,423,231,403),(212,422,232,402),(213,421,233,401),(214,420,234,440),(215,419,235,439),(216,418,236,438),(217,417,237,437),(218,416,238,436),(219,415,239,435),(220,414,240,434),(241,382,261,362),(242,381,262,361),(243,380,263,400),(244,379,264,399),(245,378,265,398),(246,377,266,397),(247,376,267,396),(248,375,268,395),(249,374,269,394),(250,373,270,393),(251,372,271,392),(252,371,272,391),(253,370,273,390),(254,369,274,389),(255,368,275,388),(256,367,276,387),(257,366,277,386),(258,365,278,385),(259,364,279,384),(260,363,280,383)]])

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 2 2 20 20 60 60 2 2 2 2 2 6 6 6 6 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + - + + + + - - + + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 Q16 D10 D10 C3⋊D4 C3⋊D4 D20 D20 Dic20 C3⋊Q16 S3×D5 C3⋊D20 C2×S3×D5 C3⋊D20 C3⋊Dic20 kernel C2×C3⋊Dic20 C3⋊Dic20 C10×C3⋊C8 C6×Dic10 C2×Dic30 C2×Dic10 C60 C2×C30 C2×C3⋊C8 Dic10 C2×C20 C30 C3⋊C8 C2×C12 C20 C2×C10 C12 C2×C6 C6 C10 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 16 2 2 2 2 2 8

Matrix representation of C2×C3⋊Dic20 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 1 240
,
 214 228 0 0 0 0 13 33 0 0 0 0 0 0 14 199 0 0 0 0 42 29 0 0 0 0 0 0 166 62 0 0 0 0 228 75
,
 162 45 0 0 0 0 113 79 0 0 0 0 0 0 133 210 0 0 0 0 42 108 0 0 0 0 0 0 171 140 0 0 0 0 101 70

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[214,13,0,0,0,0,228,33,0,0,0,0,0,0,14,42,0,0,0,0,199,29,0,0,0,0,0,0,166,228,0,0,0,0,62,75],[162,113,0,0,0,0,45,79,0,0,0,0,0,0,133,42,0,0,0,0,210,108,0,0,0,0,0,0,171,101,0,0,0,0,140,70] >;

C2×C3⋊Dic20 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes {\rm Dic}_{20}
% in TeX

G:=Group("C2xC3:Dic20");
// GroupNames label

G:=SmallGroup(480,395);
// by ID

G=gap.SmallGroup(480,395);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,176,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^40=1,d^2=c^20,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽


text:
## Chain or Compound Percentages

If £50 is increased by 10%, we calculate 10% of £50 and add the answer.

£50+£5=£55

We could have obtained the same answer by multiplying £50 by

$1+10\%=1+\frac{10}{100}=1.1$

This method is especially convenient when an amount is increased several times by different percentages.

Suppose £50 is increased in turn by 10%, 15%, 20% and 25% successively.

To increase by 10% multiply by

$1+10\%=1+\frac{10}{100}=1.1$

To increase by 15% multiply by

$1+15\%=1+\frac{15}{100}=1.15$

To increase by 20% multiply by

$1+20\%=1+\frac{10}{100}=1.2$

To increase by 25% multiply by

$1+25\%=1+\frac{10}{100}=1.25$

To increase by 10% then 15% then 20% then 25%, multiply by


text:
# Solve RMQ (Range Minimum Query) by finding LCA (Lowest Common Ancestor)

Given an array A[0..N-1]. For each query of the form [L, R] we want to find the minimum in the array A starting from position L and ending with position R. We will assume that the array A doesn’t change in the process, i.e. this article describes a solution to the static RMQ problem

Here is a description of an asymptotically optimal solution. It stands apart from other solutions for the RMQ problem, since it is very different from them: it reduces the RMQ problem to the LCA problem, and then uses the Farach-Colton and Bender algorithm, which reduces the LCA problem back to a specialized RMQ problem and solves that.

## 1. Algorithm

We construct a Cartesian tree from the array A. A Cartesian tree of an array A is a binary tree with the min-heap property (the value of parent node has to be smaller or equal than the value of its children) such that the in-order traversal of the tree visits the nodes in the same order as they are in the array A.

In other words, a Cartesian tree is a recursive data structure. The array A will be partitioned into 3 parts: the prefix of the array up to the minimum, the minimum, and the remaining suffix. The root of the tree will be a node corresponding to the minimum element of the array A, the left subtree will be the Cartesian tree of the prefix, and the right subtree will be a Cartesian tree of the suffix.

In the following image you can see one array of length 10 and the corresponding Cartesian tree.

The range minimum query [l, r] is equivalent to the lowest common ancestor query [l', r'], where l' is the node corresponding to the element A[l] and r' the node corresponding to the element A[r]. Indeed the node corresponding to the smallest element in the range has to be an ancestor of all nodes in the range, therefor also from l' and r'. This automatically follows from the min-heap property. And is also has to be the lowest ancestor, because otherwise l' and r' would be both in the left or in the right subtree, which generates a contradiction since in such a case the minimum wouldn’t even be in the range.

In the following image you can see the LCA queries for the RMQ queries [1, 3] and [5, 9]. In the first query the LCA of the nodes A[1] and A[3] is the node corresponding to A[2] which has the value 2, and in the second query the LCA of A[5] and A[9] is the node corresponding to A[8] which has the value 3.

Such a tree can be built in $O(N)$ time and the Farach-Colton and Benders algorithm can preprocess the tree in $O(N)$ and find the LCA in $O(1)$.

## 2. Construction of a Cartesian tree

We will build the Cartesian tree by adding the elements one after another. In each step we maintain a valid Cartesian tree of all the processed elements. It is easy to see, that adding an element s[i] can only change the nodes in the most right path – starting at the root and repeatedly taking the right child – of the tree. The subtree of the node with the smallest, but greater or equal than s[i], value becomes the left subtree of s[i], and the tree with root s[i] will become the new right subtree of the node with the biggest, but smaller than s[i] value.

This can be implemented by using a stack to store the indices of the most right nodes.

vector<int> parent(n, -1);
stack<int> s;
for (int i = 0; i < n; i++) {
int last = -1;
while (!s.empty() && A[s.top()] >= A[i]) {
last = s.top();
s.pop();
}
if (!s.empty())
parent[i] = s.top();
if (last >= 0)
parent[last] = i;
s.push(i);
}


text:
# Cauchy principal value integral

Doing numerical integration I encountered the following calculation of the Cauchy principal value of the following integral:

$$\int_0^{\pi/2}d\varphi\frac{1}{\cos(\varphi)-c},$$

with $0<c<1$.

What is the strategy to solve it?

-
You should try $\psi = \tan(\varphi/2)$. See here for related formulas. –  vanna Aug 1 '12 at 18:26

The indefinite integral is elementary, can be found using half-angle substitution: $$\int \frac{\mathrm{d} \varphi}{\cos(\varphi)-c} = \frac{2}{\sqrt{1-c^2}} \operatorname{arctanh}\left(\frac{1+c}{\sqrt{1-c^2}} \tan\frac{\varphi}{2} \right)+C =: F(\varphi)$$ The Cauchy principal value is found as a symmetric limit: $$\operatorname{P.V.} \int_0^\frac{\pi}{2} \frac{\mathrm{d} \varphi}{\cos(\varphi)-c} = \lim_{\epsilon\to 0} \left(F\left(\frac{\pi}{2}\right) - F\left( \arccos(c)+\epsilon\right) + F\left( \arccos(c)-\epsilon\right) - F(0)\right)$$ Easily $$F\left(\frac{\pi}{2}\right) - F(0) = \frac{2}{\sqrt{1-c^2}} \operatorname{arctanh}\left(\sqrt{\frac{1+c}{1-c}}\right) = \frac{2}{\sqrt{1-c^2}} \operatorname{arctanh}\left(\sqrt{\frac{1-c}{1+c}}\right) - \frac{i \pi}{\sqrt{1-c^2}}$$ and, using $\tan\left(\frac{1}{2} \arccos(c)\right) = \sqrt{\frac{1-c}{1+c}}$ $$\lim_{\epsilon\to 0} \left( F\left( \arccos(c)-\epsilon\right) - F\left( \arccos(c)+\epsilon\right) \right) = \frac{2}{\sqrt{1-c^2}} \lim_{\epsilon \to 0} \left( \operatorname{arctanh}\left(1+\frac{\epsilon}{\sqrt{1-c^2}}\right) - \operatorname{arctanh}\left(1-\frac{\epsilon}{\sqrt{1-c^2}}\right) \right) = \frac{i \pi}{\sqrt{1-c^2}}$$ Combining, $$\operatorname{P.V.} \int_0^\frac{\pi}{2} \frac{\mathrm{d} \varphi}{\cos(\varphi)-c} = \frac{2}{\sqrt{1-c^2}} \operatorname{arctanh}\left(\sqrt{\frac{1-c}{1+c}}\right) = \frac{1}{\sqrt{1-c^2}} \log \frac{1+\sqrt{1-c^2}}{c}$$
Thanks a lot! However, how to proceed in case of $$\int_0^{\pi/2}d\varphi\frac{f(\varphi)}{\cos(\varphi)-c},$$ where $f(\varphi)$ has no divergencies? –  pawel_winzig Aug 1 '12 at 20:56
The problem is that $$f(\varphi)$$ is very complicated, so there is no chance to get the indefinite integral. But I think the trafo $$\psi=\tan(\varphi/2)$$ is helpful. In the end I have to use gsl_integration_qawc. –  pawel_winzig Aug 1 '12 at 21:15


text:
# Measurement of charged jet suppression in Pb-Pb collisions at √sNN=2.76 TeV

Submission Date:
18/11/2013
Article Information
Submission Form
System:
Pb-Pb
Energy:
2.76 TeV
Abstract Plain Text:

The ALICE collaboration at the LHC reports a measurement of the transverse momentum spectra of jets in Pb–Pb collisions at √sNN = 2.76 TeV. Jets are reconstructed from charged particles using the anti-kT jet algorithm with jet resolution parameters R of 0.2 and 0.3 in pseudo-rapidity |eta| < 0.5. The transverse momentum of charged particles is measured down to 0.15 GeV/c which gives access to the low pT fragments of the jet. Jets found in heavy-ion collisions are corrected event-by-event for average background density and on an inclusive basis (via unfolding) for residual background fluctuations. A strong suppression of jet production in central events with respect to peripheral events is observed. The suppression is found to be similar to the suppression of charged hadrons, which suggests that substantial energy is radiated at angles larger than the R considered in the analysis. The fragmentation bias on jets introduced by requiring a high pT leading particle, which rejects jets with a soft fragmentation pattern, is similar for central and peripheral events. The ratio of jets with R=0.2 and R = 0.3 is found to be similar in pp and Pb–Pb events, indicating no strong broadening of the radial jet structure.


text:
# And with minitab please. 2. In a metallurgy lab, there are two different analytical tests that...

###### Question:

and with minitab please.
2. In a metallurgy lab, there are two different analytical tests that can be used to determine the impurity level in steel alloys. The engineer believe that Test 1 will capture more impurities in the specimen than Test 2. Eight specimens are tested using both procedures, and the results are shown in the following tabulation. Using 5% significance, investigate if the engineer's assumption if correct. Specimen Test 1 Test 2 1.7 2.0 1.6

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text:
A point estimator is any function $W(\mathbf{X})$ of a sample. That is, any statistic is a point estimator.

## Methods of Finding Estimators

### Substitution estimators

1. Empirical Distribution Function
2. Method of Moments Estimators

### Maximum likelihood estimators

1. Likelihood Function
2. Maximum Likelihood
3. Properties of MLE
4. Techniques for finding/verifying MLEs
5. List of MLEs

### Bayesian estimation

Notes on Bayesian estimation

1. The Bayesian approach to statistics
2. Bayes estimators

In a broad sense, MLE and Bayesian estimation are both model selection methods, and they are really similar. While the former is comparatively easier to implement, the latter is more robust to assumptions.

MLE calculates the likelihood function of a given sample, then takes the model with the maximum score. With likelihood being the objective function, MLE chooses the model that best justifies your available observations, which is a really strong assumption.

Bayesian estimation only provides a probability distribution over a probabilistic model subspace, rather than specify a specific probabilistic model. Bayesian estimation takes not only random sample as input, but also a prior distribution over the model subspace. The output of Bayesian estimation called a posterior, the normalized product of prior and likelihood function. The posterior typically provides sharper prediction than the prior, if the prior is close to the likelihood function. The only issue with Bayesian method is how to justify your prior.

## Methods of Evaluating Estimators

1. luck (Cor)
2. solve (Cor)
3. conditional (L-S)
4. information inequality
5. linear dependence

Def: unbiased

Def: UMVU

### Completeness & Sufficiency

Notes on Completeness & Sufficiency

Thm: (Rao-Blackwell)

Thm: (Lehmann-Scheffe)

Cor:

### Information Inequality (Cramer-Rao bound)

Notes on Information Inequality

Def: Score function

Def: Fisher information

Thm: (Information inequality)


text:
# latex' arrow tip with arrows.meta library

Consider the following MWE

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows,decorations.pathreplacing, decorations.markings,arrows.meta}
\tikzset{
big arrow/.style={
decoration={markings,mark=at position 1 with {\arrow[scale=2.4]{latex'}}},
postaction={decorate,draw}
}
}

\begin{document}
\begin{tikzpicture}
\draw [big arrow] (0,0) -- (3,0) node[below] {$x$};
\draw [-{Latex[scale=2]}] (0,0) -- (0,3) node [left] {$y$};
\end{tikzpicture}
\end{document}

I would like to use the arrows.meta library to draw an error tip exactly like latex'. I read the documentation and apparently there is no Latex' in arrows.meta.

So, how can I draw in the y axis an arrow like the one in the x axis using the arrows.meta library ? (i.e avoiding using the decoration/postaction trick).

• Since, as you said, arrows.new is superseded by arrows.meta, I'm removing my answer. Hopefully that will also give the question more visibility. – summer Mar 20 '15 at 8:02
• Similar to, not 100% duplicate of, this post. As I said there: Cannot one just copy the definition \pgfarrowsdeclare{latex'}{latex'}... from pgflibraryarrows.code.tex? There are nothing but Bézier curves. – Symbol 1 Mar 21 '15 at 2:32
• @Symbol1 I guess I can copy the definition however I wonder why weren't stealth' and latex' ported to arrows.meta. Is it possible to contact the developers for them to include these arrow styles? – petobens Mar 27 '15 at 17:37
• Probably because some tips are time-consuming for both compiler and PDF viewer. Image that if every to construction ends up with 7 curves instead of 1 curve + 6 lines, it will be like 6.9x complex or something. Or image that there are teens of tips edition containing different combination of tips (just like Adobe Creative Suite)....... – Symbol 1 Mar 27 '15 at 17:59
• @Symbol1 I opened a feature request here. – petobens Mar 29 '15 at 21:53