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UTME SYLLABUS – PHYSICS

GENERAL OBJECTIVES
The aim of the Unified Tertiary Matriculation Examination (UTME) syllabus in Physics is to prepare the candidates for the Board’s examination. It is designed to test their achievement of the course objectives, which are to:

(1) sustain their interest in physics;
(2) develop attitude relevant to physics that encourage accuracy, precision and objectivity;
(3) interpret physical phenomena, laws, definitions, concepts and other theories;
(4) demonstrate the ability to solve correctly physics problems using relevant theories and concepts.

DETAILED SYLLABUS

 TOPICS/CONTENTS/NOTES OBJECTIVES 1. MEASUREMENTS AND UNITS (a) Length, area and volume: Metre rule, Venier calipers Micrometer Screw-guage, measuring cylinder. (b) Mass (i) unit of mass; (ii) use of simple beam balance; (iii) concept of beam balance. (c) Time (i) unit of time; (ii) time-measuring devices. (d) Fundamental physical quantities (e) Derived physical quantities and their units (i) Combinations of fundamental quantities and determination of their units; (f) Dimensions (i) definition of dimensions (ii) simple examples (g) Limitations of experimental measurements (i) accuracy of measuring instruments; (ii) simple estimation of errors; (iii) significant figures; (iv) standard form. (h) Measurement, position, distance and displacement (i) concept of displacement; (ii) distinction between distance and displacement; (iii) concept of position and coordinates; (iv) frame of reference. Candidates should be able to: i. identify the units of length, area and volume; ii. use different measuring instruments; iii. determine the lengths, surface areas and volume of regular and irregular bodies; iv. identify the unit of mass; v. use simple beam balance, e.g Buchart’s balance and chemical balance; vi. identify the unit of time; vii. use different time-measuring devices; viii. relate the fundamental physical quantities to their units; ix. deduce the units of derived physical quantities; x. determine the dimensions of physical quantities; xi. use the dimensions to determine the units of physical quantities; xii. test the homogeneity of an equation; xiii. determine the accuracy of measuring instruments; xiv. estimate simple errors; xv. express measurements in standard form. Candidates should be able to: i. use strings, meter ruler and engineering calipers, vernier calipers and micrometer, screw guage; ii. note the degree of accuracy; iii. identify distance travel in a specified direction; iv. use compass and protractor to locate points/directions; v. use Cartesians systems to locate positions in x-y plane; vi. plot graph and draw inference from the graph. 2. Scalars and Vectors (i) definition of scalar and vector quantities; (ii) examples of scalar and vector quantities; (iii) relative velocity; (iv) resolution of vectors into two perpendicular directions including graphical methods of solution. Candidates should be able to: i. distinguish between scalar and vector quantities; ii. give examples of scalar and vector quantities; iii. determine the resultant of two or more vectors; iv. determine relative velocity; v. resolve vectors into two perpendicular components; vi. use graphical methods to solve vector problems. 3. Motion (a) Types of motion: translational, oscillatory, rotational, spin and random (b) Relative motion (c) Causes of motion (d) Types of force (i) contact (ii) force field (e) linear motion (i) speed, velocity and acceleration; (ii) equations of uniformly accelerated motion; (iii) motion under gravity; (iv) distance-time graph and velocity time graph; (v) instantaneous velocity and acceleration. (f) Projectiles: (i) calculation of range, maximum height and time of flight from the ground and a height; (ii) applications of projectile motion. (g) Newton’s laws of motion: (i) inertia, mass and force; (ii) relationship between mass and acceleration; (iii) impulse and momentum; (iv) force – time graph; (v) conservation of linear momentum (Coefficient of restitution not necessary). (h) Motion in a circle: (i) angular velocity and angular acceleration; (ii) centripetal and centrifugal forces; (iii) applications. (i) Simple Harmonic Motion (S.H.M): (i) definition and explanation of simple harmonic motion; (ii) examples of systems that execute S.H.M; (iii) period, frequency and amplitude of S.H.M; (iv) velocity and acceleration of S.H.M; (iii) simple treatment of energy change in S.H.M; (iv) force vibration and resonance (simple treatment). Candidates should be able to;i. identify different types of motion; ii. solve numerical problem on collinear motion; iii. identify force as cause of motion; iv. identify push and pull as forms of force; v. identify electric and magnetic attractions, gravitational pull as forms of field forces; vi. differentiate between speed, velocity and acceleration; vii. deduce equations of uniformly accelerated motion; viii. solve problems of motion under gravity; ix. interpret distance-time graph and velocity-time graph; x. compute instantaneous velocity and acceleration; xi. establish expressions for the range, maximum height and time of flight of projectiles; xii. solve problems involving projectile motion; xiii. solve numerical problems involving impulse and momentum; xiv. interpretation of area under force – time graph; xv. interpret Newton’s laws of motion; xvi. compare inertia, mass and force; xvii. deduce the relationship between mass and acceleration; xviii. interpret the law of conservation of linear momentum and application; xix. establish expression for angular velocity, angular acceleration and centripetal force; xx. solve numerical problems involving motion in a circle; xxi. establish the relationship between period and frequency; xxii. analyse the energy changes occurring during S.H.M; xxiii. identify different types of forced vibration; xxiv. enumerate applications of resonance. 4. Gravitational field (i) Newton’s law of universal gravitation; (ii) gravitational potential; (iii) conservative and non-conservative fields; (iv) acceleration due to gravity; (v) variation of g on the earth’s surface; (vi) distinction between mass and weight escape velocity; (vii) parking orbit and weightlessness. Candidates should be able to: i. identify the expression for gravitational force between two bodies; ii. apply Newton’s law of universal gravitation; iii. give examples of conservative and non- conservative fields; iv. deduce the expression for gravitational field potentials; v. identify the causes of variation of g on the earth’s surface; vi. differentiate between mass and weight; vii. determine escape velocity. 5. Equilibrium of Forces (a) equilibrium of particles: (i) equilibrium of coplanar forces; (ii) triangles and polygon of forces; (iii) Lami’s theorem. (b) principles of moments (i) moment of a force; (ii) simple treatment and moment of a couple (torgue); (iii) applications. (c) conditions for equilibrium of rigid bodies under the action of parallel and non- parallel forces (i) resolution and composition of forces in two perpendicular directions; (ii) resultant and equilibrant. (d) centre of gravity and stability (i) stable, unstable and neutral equilibra. Candidates should be able to: i. apply the conditions for the equilibrium of coplanar forces to solve problems; ii. use triangle and polygon laws of forces to solve equilibrium problems;iii. use Lami’s theorem to solve problems; iv. analyse the principle of moment of a force; v. determine moment of a force and couple; vi. describe some applications of moment of a force and couple; vii. apply the conditions for the equilibrium of rigid bodies to solve problems; viii. resolve forces into two perpendicular directions; ix. determine the resultant and equilibrant of forces; x. differentiate between stable, unstable and neutral equilibra. 6. (a) Work, Energy and Power (i) definition of work, energy and power; (ii) forms of energy; (iii) conservation of energy; (iv) qualitative treatment between different forms of energy; (v) interpretation of area under the force- distance curve. (b) Energy and society (i) sources of energy; (ii) renewable and non-renewable energy e.g. coal, crude oil etc.; (iii) uses of energy; (iv) energy and development; (v) energy diversification; (vi) environmental impact of energy e.g. global warming, greenhouse effect and spillage; (vii) energy crises; (viii) conversion of energy; (ix) devices used in energy production. (c) Dams and energy production (i) location of dams (ii) energy production (d) nuclear energy (e) solar energy (i) solar collector; (ii) solar panel for energy supply. Candidates should be able to: i. differentiate between work, energy and power; ii. compare different forms of energy, giving examples; iii. apply the principle of conservation of energy; iv. examine the transformation between different forms of energy; v. interpret the area under the force – distance curve. vi. solve numerical problems in work, energy and power.Candidates should be able to: i. itemize the sources of energy; ii. distinguish between renewable and non- renewable energy, examples should be given; iii. identify methods of energy transition; iv. explain the importance of energy in the development of the society; v. analyze the effect of energy use to the environment; vi. identify the impact of energy on the environment; vii. identify energy sources that are friendly or hazardous to the environment; viii. identify energy uses in their immediate environment; ix. suggests ways of safe energy usex. state different forms of energy conversion.

 TOPICS/CONTENTS/NOTES OBJECTIVES 7. Friction (i) static and dynamic friction; (ii) coefficient of limiting friction and its determination; (iii) advantages and disadvantages of friction (iv) reduction of friction; (v) qualitative treatment of viscosity and terminal velocity; (vi) Stoke’s law. Candidates should be able to: i. differentiate between static and dynamic friction; ii. determine the coefficient of limiting friction; iii. compare the advantages and disadvantages of friction; iv. suggest ways by which friction can be reduced; v. analyse factors that affect viscosity and terminal velocity; vi. apply Stoke’s law. 8. Simple Machines (i) definition of simple machines; (ii) types of machines; (iii) mechanical advantage, velocity ratio and efficiency of machines. Candidates should be able to: i. identify different types of simple machines; ii. solve problems involving simple machines. 9. Elasticity (i) elastic limit, yield point, breaking point, Hooke’s law and Young’s modulus; (ii) the spring balance as a device for measuring force; (iii.) work done per unit volume in springs and elastic strings; Candidates should be able to: i. interpret force-extension curves; ii. interpret Hooke’s law and Young’s modulus of a material; iii use spring balance to measure force; iv. determine the work done in spring and elastic strings. 10. Pressure (a) Atmospheric Pressure (i) definition of atmospheric pressure; (ii) units of pressure (S.I) units (Pa); (iii) measurement of pressure; (iv) simple mercury barometer; aneroid barometer and manometer; (v) variation of pressure with height; (vi) the use of barometer as an altimeter. (b) Pressure in liquids (i) the relationship between pressure, depth and density (P = ρgh) (ii) transmission of pressure in liquids (Pascal’s Principle) (iii) application Candidates should be able to: i. recognize the S.I units of pressure (Pa); ii. identify pressure measuring instruments; iii. relate the variation of pressure to height; iv. use a barometer as an altimeter; v. determine the relationship between pressure depth and density; vi apply the principle of transmission of pressure in liquids to solve problems; vii. determine and apply the principle of pressure in liquid. 11. Liquids At Rest (i) determination of density of solids and liquids (ii) definition of relative density (iii) upthrust on a body immersed in a liquid (iv) Archimedes’ principle and law of floatation and applications, e.g. ships and hydrometers. Candidates should be able to: i. distinguish between density and relative density of substances; ii. determine the upthrust on a body immersed in a liquid; iii. apply Archimedes’ principle and law of floatation to solve problems. 12. Temperature and Its Measurement (i) concept of temperature (ii) thermometric properties (iii) calibration of thermometers (iv) temperature scales –Celsius and Kelvin. (v) types of thermometers (vi) conversion from one scale of temperature to another Candidates should be able to: i. identify thermometric properties of materials that are used for different thermometers; ii. calibrate thermometers;iii. differentiate between temperature scales e.g. Celsius and Kelvin; iv. compare the types of thermometers; vi. convert from one scale of temperature to another.

 TOPICS/CONTENTS/NOTES OBJECTIVES 13. Thermal Expansion (a) Solids (i) definition and determination of linear, volume and area expansivities; (ii) effects and applications, e.g. expansion in building strips and railway lines; (iii) relationship between different expansivities. (b) Liquids (i) volume expansivity; (ii) real and apparent expansivities; (iii) determination of volume expansivity; (iv) anomalous expansion of water. Candidates should be able to: i. determine linear and volume expansivities; ii. assess the effects and applications of thermal expansivities; iii. determine the relationship between different expansivities; iv. determine volume, apparent, and real expansivities of liquids; v. analyse the anomalous expansion of water. 14. Gas Laws (i) Boyle’s law (isothermal process) (ii) Charle’s law (isobaric process) (iii) Pressure law (volumetric process) (iv) absolute zero of temperature (v) general gas equation: ( PV = constant ) T (vi) ideal gas equation e.g. Pv = nRT (iv) Van der waal gas Candidates should be able to: i. interpret the gas laws; ii. use expression of these laws to solve numerical problems; iii. interpret Van der waal equation for one mole of a real gas. 15. Quantity of Heat (i) heat as a form of energy; (ii) definition of heat capacity and specific heat capacity of solids and liquids; (iii) determination of heat capacity and specific heat capacity of substances by simple methods e.g. method of mixtures and electrical method and Newton’s law of cooling Candidates should be able to: i. differentiate between heat capacity and specific heat capacity; ii. determine heat capacity and specific heat capacity using simple methods; iii. solve numerical problems. 16. Change of State (i) latent heat; (ii) specific latent heats of fusion and vaporization; (iii) melting, evaporation and boiling; (iv) the influence of pressure and of dissolved substances on boiling and melting points; (v) application in appliances. Candidates should be able to: i. differentiate between latent heat and specific latent heats of fusion and vaporization; ii. differentiate between melting, evaporation and boiling; iii. examine the effects of pressure and of dissolved substance on boiling and melting points. iv. solve numerical problems. 17. Vapours (i) unsaturated and saturated vapours; (ii) relationship between saturated vapour pressure (S.V.P) and boiling; (iii) determination of S.V.P by barometer tube method; (iv) formation of dew, mist, fog, and rain; (v) study of dew point, humidity and relative humidity; (vi) hygrometry; estimation of the humidity of the atmosphere using wet and dry bulb hygrometers. Candidates should be able to: i. distinguish between saturated and unsaturated vapours; ii. relate saturated vapour pressure to boiling point; iii. determine S.V.P by barometer tube method; iv. differentiate between dew point, humidity and relative humidity; vi. estimate the humidity of the atmosphere using wet and dry bulb hygrometers; vii. solve numerical problems. 18. Structure of Matter and Kinetic Theory (a) Molecular nature of matter (i) atoms and molecules; (ii) molecular theory: explanation of Brownian motion, diffusion, surface tension, capillarity, adhesion, cohesion and angles of contact e.tc; (iii) examples and applications. (b) Kinetic Theory (i) assumptions of the kinetic theory (ii) using the theory to explain the pressure exerted by gas, Boyle’s law, Charles’ law, melting, boiling, vapourization, change in temperature, evaporation, etc. Candidates should be able to: i. differentiate between atoms and molecules; ii. use molecular theory to explain Brownian motion , diffusion, surface, tension, capillarity, adhesion, cohesion and angle of contact; iii. examine the assumptions of kinetic theory; iv. interpret kinetic theory, the pressure exerted by gases, Boyle’s law, Charles’s law, melting, boiling, vaporization, change in temperature, evaporation, etc.

 TOPICS/CONTENTS/NOTES OBJECTIVES 19. Heat Transfer (i) conduction, convection and radiation as modes of heat transfer; (ii) temperature gradient, thermal conductivity and heat flux; (iii) effect of the nature of the surface on the energy radiated and absorbed by it; (iv) the conductivities of common materials; (v) the thermos flask; (vi) land and sea breeze; (vii) engines. Candidates should be able to: i. differentiate between conduction, convection and radiation as modes of heat transfer; ii. solve problems on temperature gradient, thermal conductivity and heat flux; iii. assess the effect of the nature of the surface on the energy radiated and absorbed by it; iv. compare the conductivities of common materials; v. relate the component part of the working of the thermos flask; vi. differentiate between land and sea breeze; vii. analyse the principles of operating internal combustion jet engines, rockets. 20. Waves (a) Production and Propagation (i) wave motion; (ii) vibrating systems as source of waves; (iii) waves as mode of energy transfer; (iv) distinction between particle motion and wave motion; (v) relationship between frequency, wavelength and wave velocity (V=f λ); (vi) phase difference, wave number and wave vector; (vii) progressive wave equation e.g. Y = A sin 2 vt    (b) Classification (i) types of waves; mechanical and electromagnetic waves; (ii) longitudinal and transverse waves; (iii) stationary and progressive waves; (iv) examples of waves from springs, ropes, stretched strings and the ripple tank. (c) Characteristics/Properties (i) reflection, refraction, diffraction and plane polarization; (ii) superposition of waves e.g. interference (iii) Beats; (iv) Doppler effects (qualitative treatment only). Candidates should be able to: i. interpret wave motion; ii. identify vibrating systems as sources of waves; iii use waves as a mode of energy transfer; iv distinguish between particle motion and wave motion; v. relate frequency and wave length to wave velocity; vi. determine phase difference, wave number and wave vector; vii. use the progressive wave equation to compute basic wave parameters; viii. differentiate between mechanical and electromagnetic waves; ix. differentiate between longitudinal and transverse waves; x. distinguish between stationary and progressive waves; xi. indicate the example of waves generated from springs, ropes, stretched strings and the ripple tank; 21. Propagation of Sound Waves (i) the necessity for a material medium; (ii) speed of sound in solids, liquids and air; (iii) reflection of sound; echoes, reverberation and their applications; (iv) disadvantages of echoes and reverberations. Candidates should be able to: i. determine the need for a material medium in the propagation of sound waves; ii. compare the speed of sound in solids, liquids and air; iii. relate the effects of temperature and pressure to the speed of sound in air; iv. solve problem on echoes, reverberation and speed; v. compare the disadvantages and advantages of echoes. vi. solve problems on echo, reverberation and speed of sound. 22. Characteristics of Sound Waves (i) noise and musical notes; (ii) quality, pitch, intensity and loudness and their application to musical instruments; (iii) simple treatment of overtones produced by vibrating strings and their columns Fo= 1 T (  m / ?) 2L  Candidates should be able to: i. differentiate between noise and musical notes; ii. analyse quality, pitch, intensity and loudness of sound notes; iii. evaluate the application of (ii) above in the construction of musical instruments; iv. identify overtones by vibrating stings and air columns; iv. itemize acoustical examples of resonance; vi. determine the frequencies of notes emitted by air columns in open and closed pipes in relation to their lengths. 23. Light Energy (a) Sources of Light (i) natural and artificial sources of light; (ii) luminous and non-luminous objects. (b) Propagation of light (i) speed, frequency and wavelength of light; (ii) formation of shadows and eclipse; (iii) the pin-hole camera. Candidates should be able to: i. compare the natural and artificial sources of light; ii. differentiate between luminous and non luminous objects; iii. relate the speed, frequency and wavelength of light; iv. interpret the formation of shadows and eclipses; v. solve problems using the principle of operation of a pin-hole camera. 24. Reflection of Light at Plane and Curved Surfaces (i) laws of reflection; (ii) application of reflection of light; (iii) formation of images by plane, concave and convex mirrors and ray diagrams; (iv) use of the mirror formula: $\frac{1}{f}$ = $\frac{1}{u}$ + $\frac{1}{v}$(v) linear magnification. Candidates should be able to: i. interpret the laws of reflection; ii. illustrate the formation of images by plane, concave and convex mirrors; iii. apply the mirror formula to solve optical problems; iv. determine the linear magnification; v. apply the laws of reflection of light to the working of periscope, kaleidoscope and the sextant.

 TOPICS/CONTENTS/NOTES OBJECTIVES 25. Refraction of Light Through at Plane and Curved Surfaces (i) explanation of refraction in terms of velocity of light in the media; (ii) laws of refraction; (iii) definition of refractive index of a medium; (iv) determination of refractive index of glass and liquid using Snell’s law; (v) real and apparent depth and lateral displacement; (vi) critical angle and total internal reflection. (b) Glass Prism (i) use of the minimum deviation formula: (ii) type of lenses; (iii) use of lens formula: $\frac{1}{f}$ = $\frac{1}{u}$ + $\frac{1}{v}$ and Newton’s formular (F2 = ab) (iv) magnification Candidates should be able to: i. interpret the laws of reflection; ii. determine the refractive index of glass and liquid using Snell’s law; iii. determine the refractive index using the principle of real and apparent depth; iv. determine the conditions necessary for total internal reflection; v. examine the use of periscope, prism, binoculars, optical fibre; vi. apply the principles of total internal reflection to the formation of mirage; vii. use of lens formula and ray diagrams to solve optical numerical problems; viii. determine the magnification of an image; ix. calculate the refractive index of a glass prism using minimum deviation formula. 26. Optical Instruments (i) the principles of microscopes, telescopes, projectors, cameras and the human eye (physiological details of the eye are not required); (ii) power of a lens; (iii) angular magnification; (iv) near and far points; (v) sight defects and their corrections. Candidates should be able to: i. apply the principles of operation of optical instruments to solve problems; ii. distinguish between the human eye and the cameras; iii. calculate the power of a lens; iv. evaluate the angular magnification of optical instruments; v. determine the near and far points; vi. detect sight defects and their corrections. 27. (a) Dispersion of light and colours (i) dispersion of white light by a triangular Prism; (ii) production of pure spectrum; (iii) colour mixing by addition and subtraction; (iv) colour of objects and colour filters; (v) rainbow. (b) Electromagnetic spectrum (i) description of sources and uses of various types of radiation. Candidates should be able to: i. identify primary colours and obtain secondary colours by mixing; ii. understand the formation of rainbow; iii. deduce why objects have colours; iv. relate the expression for gravitational force between two bodies; v. apply Newton’s law of universal gravitation; vi. analyse colours using colour filters; vii. analyse the electromagnetic spectrum in relation to their wavelengths, sources, detection and uses. 28. Electrostatics (i) existence of positive and negative charges in matter; (ii) charging a body by friction, contact and induction; (iii) electroscope; (iv) Coulomb’s inverse square law, electric field and potential; (v) electric field intensity and potential difference; (vi) electric discharge and lightning. Candidates should be able to: i. identify charges; ii. examine uses of an electroscope; iii. apply Coulomb’s square law of electrostatics to solve problems; iv. deduce expressions for electric field intensity and potential difference; v. identify electric field flux patterns of isolated and interacting charges; vi. analyse the distribution of charges on a conductor and how it is used in lightening conductors 29. Capacitors (i) types and functions of capacitors; (ii) parallel plate capacitors; (iii) capacitance of a capacitor; (iv) the relationship between capacitance, area separation of plates and medium between the platesC = $\frac{EA}{d}$(v) capacitors in series and parallel; (vi) energy stored in a capacitor. Candidates should be able to: i. determine uses of capacitors; ii. analyse parallel plate capacitors; iii. determine the capacitance of a capacitor; iv. analyse the factors that affect the capacitance of a capacitor; v. solve problems involving the arrangement of a capacitor; vi. determine the energy stored in capacitors. 30. Electric Cells (i) simple voltaic cell and its defects; (ii) Daniel cell, Leclanche cell (wet and dry); (iii) lead –acid accumulator and Nickel-Iron (Nife) Lithium lron and Mercury cadmium; (iv) maintenance of cells and batteries (detail treatment of the chemistry of a cell is not required); (v) arrangement of cells; (vi) efficiency of a cell. Candidates should be able to: i. identify the defects of the simple voltaic cell and their correction; ii. compare different types of cells including solar cell; iii. compare the advantages of lead-acid and Nikel iron accumulator; iv. solve problems involving series and parallel combination of cells.

 TOPICS/CONTENTS/NOTES OBJECTIVES 31. Current Electricity (i) electromagnetic force (emf), potential difference (p.d.), current, internal resistance of a cell and lost Volt; (ii) Ohm’s law; (iii) measurement of resistance; (iv) meter bridge; (v) resistance in series and in parallel and their combination; (vi) the potentiometer method of measuring emf, current and internal resistance of a cell. (i) electrical networks. Candidates should be able to: i. differentiate between emf, p.d., current and internal resistant of a cell; ii. apply Ohm’s law to solve problems; iii. use metre bridge to calculate resistance; iv. compute effective total resistance of both parallel and series arrangement of resistors; v. determine the resistivity and the conductivity of a conductor; vi. measure emf. current and internal resistance of a cell using the potentiometer; vii. identify the advantages of the potentiometer; viii. apply Kirchoff’s law in electrical networks. 32. Electrical Energy and Power (i) concepts of electrical energy and power; (ii) commercial unit of electric energy and power; (iii) electric power transmission (v) heating effects of electric current; (vi) electrical wiring of houses; (vii) use of fuses. Candidates should be able to: i. apply the expressions of electrical energy and power to solve problems; ii. analyse how power is transmitted from the power station to the consumer; iii. identify the heating effects of current and its uses; iv. identify the advantages of parallel arrangement over series; v. determine the fuse rating. 33. Magnets and Magnetic Fields (i) natural and artificial magnets; (ii) magnetic properties of soft iron and steel; (iii) methods of making magnets and demagnetization; (iv) concept of magnetic field; (v) magnetic field of a permanent magnet; (vi) magnetic field round a straight current carrying conductor, circular wire and solenoid; (vii) properties of the earth’s magnetic field; north and south poles, magnetic meridian and angle of dip and declination; (viii) flux and flux density; (ix) variation of magnetic field intensity over the earth’s surface (x) applications: earth’s magnetic field in navigation and mineral exploration. Candidates should be able to: i. give examples of natural and artificial magnets; ii. differentiate between the magnetic properties of soft iron and steel; iii. identify the various methods of making magnets and demagnetizing magnets; iv. describe how to keep a magnet from losing its magnetism; v. determine the flux pattern exhibited when two magnets are placed together pole to pole; vi. determine the flux of a current carrying conductor, circular wire and solenoid including the polarity of the solenoid; vii. determine the flux pattern of a magnet placed in the earth’s magnetic fields; viii. identify the magnetic elements of the earth’s flux; ix. determine the variation of earth’s magnetic field on the earth’s surface; x. examine the applications of the earth’s magnetic field. 34. Force on a Current-Carrying Conductor in a Magnetic Field (i) quantitative treatment of force between two parallel current-carrying conductors; (ii) force on a charge moving in a magnetic field; (iii) the d. c. motor; (iv) electromagnets; (v) carbon microphone; (vi) moving coil and moving iron instruments; (vii) conversion of galvanometers to ammeters and voltmeter using shunts and multipliers; (viii) sensitivity of a galvanometer. Candidates should be able to: i. determine the direction of force on a current carrying conductor using Fleming’s left-hand rule; ii. interpret the attractive and repulsive forces between two parallel current- carrying conductors using diagrams; iii. determine the relationship between the force, magnetic field strength, velocity and the angle through which the charge enters the field; iv. interpret the working of the d. c. motor; v. analyse the principle of electromagnets and give examples of its application; vi. compare moving iron and moving coil instruments; vii. convert a galvanometer into an ammeter or a voltmeter; viii. identify the factors affecting the sensitivity of a galvanometer. 35. (a) Electromagnetic Induction (i) Faraday’s laws of electromagnetic induction; (ii) factors affecting induced emf; (iii) Lenz’s law as an illustration of the principle of conservation of energy; (iv) a.c. and d.c generators; (v) transformers; (vi) the induction coil.(b) Inductance (i) explanation of inductance; (ii) unit of inductance; (iii) energy stored in an inductor:(iv) application/uses of inductors. (c) Eddy Current (i) reduction of eddy current (ii) applications of eddy current Candidates should be able to: i. interpret the laws of electromagnetic induction; ii. identify factors affecting induced emf; iii. recognize how Lenz’s law illustrates the principle of conservation of energy; iv. interpret the diagrammatic set up of A.C. generators; v. identify the types of transformer; vi. examine principles of operation of transformers; vii. assess the functions of an induction coil; viii. draw some conclusions from the principles of operation of an induction coil; ix. interpret the inductance of an inductor; x. recognize units of inductance; xi. calculate the effective total inductance in series and parallel arrangement; xii. deduce the expression for the energy stored in an inductor; xiii. examine the applications of inductors; xiv. describe the method by which eddy current losses can be reduced; xv. determine ways by which eddy currents can be used. 36. Simple A. C. Circuits (i) explanation of a.c. current and voltage; (ii) peak and r.m.s. values; (iii) a.c. source connected to a resistor; (iv) a.c source connected to a capacitor- capacitive reactance; (v) a.c source connected to an inductor inductive reactance; (vi) series R-L-C circuits; (vii) vector diagram, phase angle and power factor; (viii) resistance and impedance; (ix) effective voltage in an R-L-C circuits; (x) resonance and resonance frequency: Candidates should be able to: i. identify a.c. current and d.c. voltage; ii. differentiate between the peak and r.m.s. values of a.c.; iii. determine the phase difference between current and voltage; iv. interpret series R-L-C circuits; v. analyse vector diagrams; vi. calculate the effective voltage, reactance and impedance; vii. recognize the condition by which the circuit is at resonance; viii. determine the resonant frequency of R-L-C arrangement; ix. determine the instantaneous power, average power and the power factor in a. c. circuits.

 TOPICS/CONTENTS/NOTES OBJECTIVES 37. Conduction of Electricity Through (a) liquids (i) electrolytes and non-electrolyte; (ii) concept of electrolysis; (iii) Faraday’s laws of electrolysis; (iv) application of electrolysis, e.g. electroplating, calibration of ammeter etc. (b) gases (i) discharge through gases (qualitative treatment only); (ii) application of conduction of electricity through gases; Candidates should be able to: i. distinguish between electrolytes and non- electrolytes; ii. analyse the processes of electrolysis; iii. apply Faraday’s laws of electrolysis to solve problems;iv. analyse discharge through gases; v. determine some applications/uses of conduction of electricity through gases. 38. Elementary Modern Physics (i) models of the atom and their limitations; (ii) elementary structure of the atom; (iii) energy levels and spectra; (iv) thermionic and photoelectric emissions; (v) Einstein’s equation and stopping potential (vi) applications of thermionic emissions and photoelectric effects; (vii) simple method of production of x-rays; (viii) properties and applications of alpha, beta and gamma rays; (ix) half-life and decay constant; (x) simple ideas of production of energy by fusion and fission; (xi) binding energy, mass defect and Einstein’s Energy equation [∆E = ∆Mc2] (xii) wave-particle paradox (duality of matter); (xiii) electron diffraction; (xiv) the uncertainty principle. Candidates should be able to: i. identify the models of the atom and write their limitations; ii. describe elementary structure of the atom; iii. differentiate between the energy levels and spectra of atoms; iv. compare thermionic emission and photoelectric emission; v. apply Einstein’s equation to solve problems of photoelectric effect; vi. calculate the stopping potential; vii. relate some application of thermionic emission and photoelectric effects; viii. interpret the process involved in the production of x-rays; ix identify some properties and applications of x-rays; x. analyse elementary radioactivity; xi. distinguish between stable and unstable nuclei; xii. identify isotopes of an element; xiii. compare the properties of alpha, beta and gamma rays; xiv. relate half-life and decay constant of a radioactive element; xv. determine the binding energy, mass defect and Einstein’s energy equation; xvi. analyse wave particle duality; xvii. solve some numerical problems based on the uncertainty principle and wave – particle duality. 39. Introductory Electronics (i) distinction between metals, semiconductors and insulators (elementary knowledge of band gap is required); (ii) intrinsic and extrinsic semiconductors; (iii) uses of semiconductors and diodes in rectification and transistors in amplification; (iv) n-type and p-type semiconductors; (v) elementary knowledge of diodes and transistors. Candidates should be able to: i. differentiate between conductors, semi- conductors and insulators; ii. distinguish between intrinsic and extrinsic semiconductors; iii. distinguish between electron and hole carriers; iv. distinguish between n-type and p-type semiconductor; v. analyse diodes and transistor vi. relate diodes to rectification and transistor to amplification.

RECOMMENDED TEXTS

Ike, E.E. (2014). Essential Principles of Physics, Jos ENIC Publishers.

Ike, E.E. (2014). Numerical Problems and Solutions in Physics, Jos: ENIC Publishers. Nelson, M. (1977). Fundamentals of Physics, Great Britain: Hart Davis Education.

Nelson, M. and Parker … (1989). Advanced Level Physics, (Sixth Edition): Heinemann.

Okeke, P.N. and Anyakoha, M.W. (2000). Senior Secondary School Physics, Lagos: Pacific Printers.

Olumuyiwa, A. and Ogunkoya, O. O. (1992). Comprehensive Certificate Physics, Ibadan: University Press Plc.


text:
## Help with names

A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you found it. This is the forum for "non-academic" content.
GUYTU6J
Posts: 1069
Joined: August 5th, 2016, 10:27 am
Location: 中国

### Re: Help with names

Two questions:
1.Why are Catacryst and Metacatacryst are named so?Is it related to "catalyst" and "crystal"?
2.How many ways are there in which two 34P13s can share one or two common blocks if we consider different phases?And how to tell them apart besides apgcodes?What if there are more?
Lifewiki: User:GUYTU6J
---
Someone please find a use for this:

Code: Select all

x = 9, y = 7, rule = B3/S23
6bo$6bobo$5bo2bo$b2o3b2o$o2bo$bobo$2bo!
#C [[ COLOR BACKGROUND 255 200 82 COLOR ALIVE 81 143 51 ]]

mniemiec
Posts: 1113
Joined: June 1st, 2013, 12:00 am

### Re: Help with names

GUYTU6J wrote:Two questions:
1.Why are Catacryst and Metacatacryst are named so?Is it related to "catalyst" and "crystal"?
2.How many ways are there in which two 34P13s can share one or two common blocks if we consider different phases?And how to tell them apart besides apgcodes?What if there are more?
1. Most likely.
2. There are 3 ways two of these can share blocks: in a straight line, at right angles, or both simultaneously. Both oscillators can have one of 13 different phase separations between the (+0..+12), but since they are both the same, and all 3 kinds of interactions are symmetrical, symmetry reduces this to 7 distinct ones (+0..+6). The block cannot eat both oscillators simultaneously, so +0 is never viable. The other 6 (+1..+6) are viable when one block is shared, but +1 and +3 don't work when both blocks are shared, yielding 16 viable sharings:

Code: Select all

x = 170, y = 111, rule = B3/S23
8bo29bo29bo29bo29bo29bo$7bobo27bobo27bobo27bobo27bobo27bobo$$9bo29bo 29bo29bo29bo29bo6bobbo26bobbo26bobbo26bobbo26bobbo26bobbo10bo29bo29b o29bo29bo29bo10bo29bo29bo29bo29bo29bo6bo3bo25bo3bo25bo3bo25bo3bo25bo 3bo25bo3bo7b3o27b3o27b3o27b3o27b3o27b3o3boo7b3o18boo7b3o18boo7b3o18b oo7b3o18boo7b3o18boo7b3o3boo6bo3boobo14boo6bo3boobo14boo6bo3boobo14b oo6bo3boobo14boo6bo3boobo14boo6bo3boobo11bo7bo21bo7bo21bo7bo21bo7bo 21bo7bo21bo7bo11bo6bo22bo6bo22bo6bo22bo6bo22bo6bo22bo6bo12bobbo26bo bbo26bobbo26bobbo26bobbo26bobbo$$9boo28boo28boo28boo28boo28boo$9boo28b
oo28boo28boo28boo28boo$$160bo11boo27bobo5bo22bo28bo28b3o28boo5bo10b 3o4b3o20bo3bo3bo20boo4bobboo19b3obbooboo22bo4b3o22bo4boo3boo6bob3o17b oo5bo4boobo14boo3bobo3b4o15boo3bobo6boo14boobbo3bobobo17boo7boboo3boo 3bo3b4o17boobb3o5bo17boo3boo4b3o16boobb3o3boobbo15boobbo6bob3o14booboo 5booboo7b3o3bo28boo23bo4bo25bo6bo21b3o4bo22boo6bobo7boobo26bo3bo29bo 87bo9b3o29bo58bo28bo30bo9boo27bo30boo27bobo29boo27boo9boo28boo27b3o 27bobbo27bo29bobo39bo29boo57bobo27bobo8bo60bo28b3o27bobo26booboo8bo 28b3o28bo29boo28bobo8bo59bo911bo29bo29bo29bo29bo29bo10bobo27bobo27b obo27bobo27bobo27bobo$$10bo29bo29bo29bo29bo29bo$10bobbo26bobbo26bobbo 26bobbo26bobbo26bobbo$9bo29bo29bo29bo29bo29bo$9bo29bo29bo29bo29bo29bo$
9bo3bo25bo3bo25bo3bo25bo3bo25bo3bo25bo3bo$10b3o27b3o27b3o27b3o27b3o27b 3o$5b3o7boo18b3o7boo18b3o7boo18b3o7boo18b3o7boo18b3o7boo$boboo3bo6boo 14boboo3bo6boo14boboo3bo6boo14boboo3bo6boo14boboo3bo6boo14boboo3bo6boo$o7bo21bo7bo21bo7bo21bo7bo21bo7bo21bo7bo$bo6bo22bo6bo22bo6bo22bo6bo22b o6bo22bo6bo$4bobbo26bobbo26bobbo26bobbo26bobbo26bobbo$$9boo28boo28boo 28boo28boo28boo9boo28boo28boo28boo28boo28boo$$160bo$11boo27bobo5bo22b o28bo28b3o28boo5bo$10b3o4b3o20bo3bo3bo20boo4bobboo19b3obbooboo22bo4b3o
22bo4boo$3boo6bob3o17boo5bo4boobo14boo3bobo3b4o15boo3bobo6boo14boobbo 3bobobo17boo7boboo$3boo3bo3b4o17boobb3o5bo17boo3boo4b3o16boobb3o3boobb
o15boobbo6bob3o14booboo5booboo$7b3o3bo28boo23bo4bo25bo6bo21b3o4bo22boo 6bobo$7boobo26bo3bo29bo87bo$9b3o29bo58bo28bo30bo$9boo27bo30boo27bobo
29boo27boo$9boo28boo27b3o27bobbo27bo29bobo$39bo29boo57bobo27bobo$8bo 60bo28b3o27bobo26booboo$8bo28b3o28bo29boo28bobo$8bo59bo15$35bo59bo29bo
15b2o$15b2o!  otismo Posts: 303 Joined: August 18th, 2010, 1:41 pm Location: Florida Contact: ### Re: Help with names GUYTU6J wrote: mniemiec wrote: stillater - An oscillator within a still-life (?). When there is only one living cell in the rotor,there are two ways to change it into a still life. Code: Select all x = 28, y = 8, rule = LifeHistory 3.A9.A9.A$2.A.A.2A4.A.A.2A4.A.A.2A$2.A.2A.A4.A.2A.A4.A.2A.A$2A.E6.2A
8.2A$.A.A.2A4.A.A.2A4.A.AE2A$.A.A2.A4.A.A2.A4.A.A2.A$2.A2.A6.A2.A6.A 2.A$3.2A8.2A8.2A!

None of these have appeared on Catagolue yet
I would name them, if I were you.....
In the Game of Life there are no winners there are no losers there are only SURVIVORS.
.....**
...****
.******
*( ͡° ͜ʖ ͡°)*

Code: Select all

#CXRLE Pos=0,0
x = 26, y = 34, rule = B3/S23
23bo$23bobo$23b2o9$2b3o2$o5bo$o5bo$o5bo2$2b3o5b2o$10b2o3$2b2o$2b2o10$15b2o$15b2o!

Moosey
Posts: 3285
Joined: January 27th, 2019, 5:54 pm
Location: A house, or perhaps the OCA board. Or [click to not expand]
Contact:

### Re: Help with names

otismo wrote:
GUYTU6J wrote:
mniemiec wrote: stillater - An oscillator within a still-life (?).
When there is only one living cell in the rotor,there are two ways to change it into a still life.

Code: Select all

x = 28, y = 8, rule = LifeHistory
3.A9.A9.A$2.A.A.2A4.A.A.2A4.A.A.2A$2.A.2A.A4.A.2A.A4.A.2A.A$2A.E6.2A 8.2A$.A.A.2A4.A.A.2A4.A.AE2A$.A.A2.A4.A.A2.A4.A.A2.A$2.A2.A6.A2.A6.A
2.A$3.2A8.2A8.2A!  None of these have appeared on Catagolue yet I would name them, if I were you..... None of them really have the intention to name those uninteresting SLs. Even useful SLs like 31.4 rarely get names EDIT: What is the etymology of your term OpenSezMe? otismo wrote: I am going to call it OpenSezMe I am a prolific creator of many rather pathetic googological functions My CA rules can be found here Also, the tree game Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?" Ian07 Posts: 553 Joined: September 22nd, 2018, 8:48 am ### Re: Help with names Out of all the (generally accepted) object names I've seen, none have been nearly as weird as$rats. Where did this name come from?

otismo
Posts: 303
Joined: August 18th, 2010, 1:41 pm
Location: Florida
Contact:

### Re: Help with names

GUYTU6J wrote:The name "Laputa" comes from the famous novel Gulliver's Travels.
Why is the name given to an oscillator instead of a spaceship that really looks like a floating island?
Lilly Petunias

comes from Gulliver's Travels

( I assume Holy Matrimony )

the five P "flowers" that are really weeds that you do NOT want in your garden :

Petunias spit

Poppys pop off

Pansys are now "gay"

Poseys posers and wannabes

Peonies peons ( duh )
In the Game of Life there are no winners there are no losers there are only SURVIVORS.
.....**
...****
.******
*( ͡° ͜ʖ ͡°)*

Code: Select all

#CXRLE Pos=0,0
x = 26, y = 34, rule = B3/S23
23bo$23bobo$23b2o9$2b3o2$o5bo$o5bo$o5bo2$2b3o5b2o$10b2o3$2b2o$2b2o10$15b2o$15b2o!

format: plain


text:
Benchmarking insertion sort

More than once I claimed that using binary search doesn't improve performance of the insertion sort. For example, see answer here and comments here). Now I have time to substantiate my claim.

The only practical application of the insertion sort, where we actually do care about performance, is sorting almost sorted data; that is such data where each element is within a fixed distance from its final destination. Only this scenario is benchmarked.

First, the implementations of insertion sorts (insertion_sort.h)

#include <algorithm>

template<typename It>
void straight_insertion_sort(It first, It last) {
for (auto cur = first + 1; cur < last; ++cur) {
auto val = *cur;
auto it = cur;
if (val < *first) {
for (it = cur; it > first; --it) {
*it = *(it - 1);
}
} else {
for (it = cur; val < *(it - 1); --it) {
*it = *(it - 1);
}
}
*it = val;
}
}

template<typename It>
void binary_insertion_sort(It first, It last) {
for (auto cur = first + 1; cur < last; ++cur) {
auto val = *cur;
auto insertion_point = std::lower_bound(first, cur - 1, *cur);
std:: copy_backward(insertion_point, cur - 1, cur);
*insertion_point = val;
}
}

The benchmarks shall run against an almost sorted data. This is how the testcases are prepared. (incomplete_qsort.h, the code is adapted from std::partition) example; cutoff is added to make the array almost sorted. After a call to incomplete_qsort every element is at most cutoff away from where it is supposed to be. NB: this is not really for a review, but only for completeness.

Note: I need here. does not allow auto as an argument to lambda.

#include <algorithm>

template<typename It>
void incomplete_qsort(It first, It last, size_t cutoff) {
if (std::distance(first, last) < cutoff) {
return;
}
auto pivot = *first;
auto mid1 = std::partition(first, last,
[pivot](const auto& em) {return em < pivot; });
auto mid2 = std::partition(mid1, last,
[pivot](const auto& em) {return !(pivot < em); });
incomplete_qsort(first, mid1, cutoff);
incomplete_qsort(mid2, last, cutoff);
}

This is the driver (benchmark.cpp):

#include "incomplete_qsort.h"
#include "insertion_sort.h"
#include <chrono>
#include <iostream>
#include <iomanip>
#include <iostream>
#include <numeric>
#include <random>
#include <vector>

using iter = std::vector<int>::iterator;

using sorter = void (*)(iter, iter);
double run_benchmark(std::vector<int>& data, sorter s) {
auto start = std::chrono::system_clock::now();
s(data.begin(), data.end());
auto end = std::chrono::system_clock::now();
std::chrono::duration<double> diff = end - start;
return diff.count();
}

int main(int argc, char ** argv)
{
std::random_device rd;
std::mt19937 g(rd());

for (int i = 12; i < 25; i++) {
auto size = 1 << i;
std::vector<int> data1(size);
std::vector<int> data2(size);
std::iota(data1.begin(), data1.end(), 0);
std::shuffle(data1.begin(), data1.end(), g);
incomplete_qsort(data1.begin(), data1.end(), 16);
std::copy(data1.begin(), data1.end(), data2.begin());

double duration1 = run_benchmark(data1, straight_insertion_sort);
double duration2 = run_benchmark(data2, binary_insertion_sort);
std::cout << std::setw(8) << size << ": "
<< std::setw(8) << duration1
<< std::setw(8) << duration2
<< " (" << duration2 / duration1 << ")"
<< '\n';
}
}

And finally the results, compiled with -O3:

    4096:  5.2e-05 0.000158 (3.03846)
8192:  9.1e-05 0.000269 (2.95604)
16384: 0.000161 0.000494 (3.06832)
32768: 0.000275 0.000968 (3.52)
65536: 0.000555 0.001823 (3.28468)
131072: 0.001171 0.003686 (3.14774)
262144: 0.002084 0.007765 (3.72601)
524288: 0.004457 0.015087 (3.38501)
1048576: 0.008304 0.030951 (3.72724)
2097152: 0.017204 0.063931 (3.71605)
4194304: 0.033697 0.132659 (3.93682)
8388608:  0.06833 0.277166 (4.05629)
16777216: 0.136164 0.569059 (4.17922)

• Sorry, but is there a specific question here, i.e., what in particular are you looking for in an answer?
– Juho
May 16, 2020 at 11:11
• @Juho According to the chapter of this exchange, any facet of the code presented is a fair game for a reviewer. I am interested in everything people may come up with. Style, methodology, omissions, outright errors, etc.
– vnp
May 16, 2020 at 15:44
• Ok, thanks for the clarification!
– Juho
May 16, 2020 at 15:55
• Correct me if I'm wrong, but the results seem to indicate that binary search is still an aver of 3 times faster than insertion sort. May 16, 2020 at 17:49
• @pacmaninbw Umm... I don't see it. The last column is binary / straight.
– vnp
May 17, 2020 at 0:18

Your initial claim sounds about right to me, since for each iteration, checking at most cutoff elements for the insertion_point in the straight version (due to the restriction on the input) should become increasingly faster than checking logarithmic many in the binary version. Of course there is a lot more to consider like cache locality, but computational complexity should be the dominating factor in this case. That being said, I see some potential to improve your benchmark.

Benchmarking

Verify that your implementations are correct

A testsuite would of course be best practice, but the absolute minimum is to make sure that your algorithms return the same result as std::sort. The binary insertion sort you provided has an off-by-one-error, thus rendering your results useless. For the following two lines, the shown fix was to increase all end-iterators by one:

auto insertion_point = std::lower_bound(first, cur, *cur);
std::copy_backward(insertion_point, cur, cur + 1);

Choose a suitable baseline

Without any generally accepted baseline for the runtime of the algorithms, it is hard to argue whether the results are significant in any way. Again, std::sort does the job.

Test against (somewhat) equally optimized implementations

I am not an expert in optimization, but managed to shave off about 30% of the runtime of the binary version by adding an early return and using std::upper_bound instead of std::lower_bound, both of which indirectly happen in your straight version:

for (auto cur = first + 1; cur < last; ++cur) {
if (*(cur - 1) < *cur) { continue; }
auto val = *cur;
auto insertion_point = std::upper_bound(first, cur, *cur);
std::copy_backward(insertion_point, cur, cur + 1);
*insertion_point = val;
}

The change from std::lower_bound to std::upper_bound does not change much due to the input format, which leads us to the next chapter.

Use realistic data

In your benchmark, you simply shuffle the numbers from 0 to n and partially sort them again, which means that there are no duplicates in the input. This is a rather strict constraint and probably allows for even more optimized algorithms (e.g. bucket sort). An input vector where each element is drawn from a chosen probability distribution (and then again partially sorted) should yield more representative results.

Additionally, you should always put some thought into the type of elements you are sorting, e.g. for int copying is fine, but for larger classes the benchmark needs to be adapted towards utilizing std::move.

Run tests multiple times

This is especially important for micro optimizations, so small size in our case, and the reason why microbenchmark support libraries like google/benchmark exist. If you’re not willing to put up with the hazzle of integrating it in your project, quick-bench.com allows for easy online benchmarking.

I quickly threw together an example using your code and the fixed algorithm, you can find it here.

Specify your compiler version and hardware

This is not as relevant for proving a general point, but of course results will differ when using compilers of different development levels (or even using your own homecrafted one). Here, websites like quick-bench come in handy again.

Code quality

Naming

As mentioned by others, duration1 and duration2 as well as data1 and data2 are quite unhelpful. Also, iterators are usually named begin and end instead of first and last. Other than that, your naming is expressive enough.

Creating the input vector

You initialize two vectors of the needed size, thus default initializing all the elements. Then you fill the first one and copy the partially sorted result back to the other. Preferably, one would reserve an empty vector and then use a custom function like iota_n (example) to back-insert all the elements. Once they are shuffled and partially sorted, simply use

auto data_copy = initial_data;

instead of calling std::copy.

Also, you included <iostream> twice.

Insertion sort

Whereas binary_insertion_sort is readable and reasonably easy to grasp, it took me a while longer for straight_insertion_sort. The if-case can only occur in the beginning of the range to sort and does nothing but catch an edge-case. It can be simplified to

for (auto cur = first + 1; cur < last; ++cur) {
if (*cur < *first) {
std::rotate(first, cur, cur + 1);
}
else {
auto val = *cur;
auto it = cur;
for (it = cur; val < *(it - 1); --it) {
*it = *(it - 1);
}
*it = val;
}
}

, which actually seems to be a bit faster. I tried making the else-case more readable whilst preserving speed by using std::rotate once more, but failed to do so.

For both algorithms, you use < to compare iterators, where usually != is used, see this SO thread. It doesn’t make any difference speedwise.

• Welcome to Code Review. This is a very good first contribution! May 17, 2020 at 18:55
• Great review, thanks.
– vnp
May 17, 2020 at 22:41
• Naming

As pointed in comments, duration1 and duration2 are bad names since they lead to a confusion. duration_straight and duration_binary seem to be better choice.


text:
# Distribution of the Difference of Uniformly Random Elements

In the search to decision reduction of 'On Ideal Lattices and Learning with Errors over Rings', the authors implicitly use the fact that the difference of distinct, uniformly random elements of a (finite) field is again uniformly random (Lemma 5.9). This is again used in the reductions of 'Worst-case to Average-case Reductions for Module Lattices', for Module Learning with Errors.

• Firstly, why is this true?
• Secondly, what happens when you relax the algebraic conditions on the sample space? For example, in an integral domain, one might not have inverses at all; is there a different sufficient and necessary condition for the difference of distinct, uniformly random elements in an integral domain to be again uniformly random? What about simple modules? Euclidean domains?

Firstly, why is this true?

This is easy to see, if we consider the finite field as a finite group with the addition operation (and ignore the multiplication operation)

If we consider the value $$X - Y$$, where $$X, Y$$ are uniformly and independently distributed elements of the group, then there are $$n^2$$ equiprobable $$X, Y$$ pairs possible (each with a probability $$n^{-2}$$.

For any group value $$Z$$, we have $$Z = X - Y$$ for $$n$$ possible $$X, Y$$ pairs (if we select any value for $$X$$, we see that the unique value of $$Y$$ that them sum to $$Z$$ is $$X-Z$$); each possible pair has probability $$n^{-2}$$, and so the probability for the sum to be $$Z$$ is $$n^{-1}$$. This holds for all group members, and so the sum is equidistributed.

Secondly, what happens when you relax the algebraic conditions on the sample space?

As long as the relaxation still leaves us with a ring (that is, the addition operation is still a group), the above argument still holds.

It's worth mentioning that the conditions needed for $$f(X_0, X_1)$$ to be uniformly random based off the distributions of $$X_0, X_1$$ are quite mild usually. In particular what you need is:

1. $$X_0$$ and $$X_1$$ to be independent
2. At least one of $$X_0, X_1$$ to be uniformly random (say that it is $$X_0$$)
3. $$f(\cdot, X_1) : G\to G$$ to be a bijection [1] for each choice of $$X_1$$ (where $$X_1$$ is the potentially non-uniform random variable).

Then $$f(X_0, X_1)$$ will be uniformly distributed. The proof is fairly easy, so I'll include a sketch of it below:

1. Start by looking at $$\Pr_{(X_0, X_1)}[f(X_0, X_1) = k]$$ for $$k\in G$$
2. Rewrite this as $$\sum_{g\in G}\Pr_{(X_0, X_1)}[f(X_0, X_1) = k\mid X_1 = g]\Pr_{X_1}[X_1 = g]$$
3. Use independence to write $$\Pr_{(X_0, X_1)}[f(X_0, X_1) = k\mid X_1 = g] = \Pr_{X_0}[f(X_0, g) = k]$$
4. Use that bijections "preserve" the property of being uniformly random (so $$f(X_0, g)$$ is uniformly random, meaning $$\Pr_{X_0}[f(X_0, g) = k] = 1/|G|$$)
5. Collect all of the relevant terms and simplify to show that $$\Pr_{(X_0, X_1)}[f(X_0, X_1) = k] = 1/|G|$$

A common source of bijections of the desired form are group operations. In particular, if $$g\in G$$ is a fixed group element, then the operation $$x\mapsto x + g$$ (where $$+$$ is the group operation in an arbitrary group) will always be a bijection. This includes when the fixed group element is the "inverse" of another element, meaning the function $$x \mapsto x + (-g)$$, which is your situation.

The above also includes the "obvious" caveat that $$|G| < \infty$$ for the uniform distribution to even make sense. One can work with larger groups by using the "Haar measure" rather than the "Uniform distribution", but given that you cannot even store arbitrary elements of such groups this is not a useful point for cryptography.

As for the question of what happens when we relax the algebraic conditions on the sample space, you may note that the way I formulated it above actually requires no assumptions of a group structure on $$G$$. It may be the case that the family of bijections $$\{f(\cdot, g)\}_{g\in G}$$ itself gives $$G$$ a group structure (the composition of two bijections is a bijection, bijections can be inverted, etc), although more properly I expect this would only show that $$G$$ is a subset (which may not be a subgroup!) of some group, where the group structure may be non-obvious or "complicated".

[1] One can weaken this further if $$f(\cdot X_1) : G_1\to G_2$$. The property that you need from a bijection is that it is a "regular" map, in the sense that there exists some constant $$c\in\mathbb{N}$$ such that $$\forall g\in G_2$$, $$|f^{-1}(g)| = c$$ (so all preimages are the same size). Bijections are an easy source of this (where $$c = 1$$), but other such maps exist (say $$f : \mathbb{Z}_4\to \mathbb{Z}_2$$ given by $$x\mapsto x\bmod 2$$, where $$c = 2$$).


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# Diffusion¶

By default, diffusion coefficients must be specified for all contributions, i.e. for each component in each phase (looping automatically through all phases for each consecutive component).

In case of many components and many phases, the number of inputs required can be very large, even if for many (e.g. intermetallic) phases no diffusion coefficients are available or need be specified. Therefore, like in section Phase Interactions, a terse mode is available in which only the positively defined diffusion data are included. The terse mode is automatically invoked if the first input in the diffusion data block is preceded by a phase and component number. In this case, the end of the diffusion data input has to be defined by the keyword end_diffusion_data. Terse mode must be used if extra line options are needed. These options can be specified at any place after the definition of the corresponding diffusion term, beginning with the same combination of phase and component number (see below).

For each contribution (phase-component pair), several options for diffusion data input are available. Those data refer to the diffusion flux of the corresponding element in the given phase, which may be caused only by the composition gradient of the same element (diagonal term) or by the gradients of all elements (one line of the diffusion matrix).

There are three main keywords (diagonal, diagonal_dilute, multi and multi_plus) and various specifiers which consist of single characters (n, d, g, l, z, i, I, or f) with the following meaning:

• diagonal: only the diagonal term is taken into account (Fick's equation). Correspondingly, only one character of n, d, g, l, z, i, I, or f has to be specified after diagonal.

Example

diagonal i

* diagonal_dilute: The line of the diffusion matrix (otherwise specified by multi or multi_plus) is reduced to a diagonal term only by using the dilute limit for the flux component. By default, the concentration of the flux component is reduced to a small value while increasing the concentration of the matrix component. By additionally using the extra line option dilute_reference, other elements may be chosen instead of the matrix. This leads to always positive diagonal terms while all the off-diagonal terms are vanishing. The option can be used for approximately diagonalizing the diffusion matrix for sake of simplicity or numerical stability. A gradual dilute limit can be achieved using the extra line option dilute followed by a factor (0<f<1) by which the composition of the flux component shall be multiplied when calculating a line of the diffusion matrix using options multi or multi_plus with specifiers g, l or z.

• multi: the flux of the given element is calculated using the gradients of all elements (Fick-Onsager). A string of N characters of n, d, g, l, z, or f for each dissolved element has to be specified after multi. If no characters are specified at all, g is assumed for all elements.

Example

multi ggnggg

• multi_plus: same as multi with the difference that all off-diagonal terms are normalized with the composition of the flux component when calculated from the database. During simulation, the normalized coefficients are multiplied by the local composition. Thus, the intrinsic linear dependency of the off-diagonal coefficients on the flux component is taken into account automatically.

> **Example**

multi_plus glgggg

• d: This specifier indicates that the corresponding term (diagonal or off-diagonal term of the diffusion matrix) will be used and specified directly by the user. If d is selected, the pre-exponential factor and the activation energy (which can be set to 0 for no temperature dependence) of the diffusion coefficient have to be specified.

• n: No diffusion flux will be simulated for the element (and gradient) in the given phase . diagonal n is default in the terse input mode.

• i: Infinite diffusion is assumed for the term (diagonal terms only!), so that the composition in the phase is mixed completely. Use this option instead of a high explicit value for the diffusion coefficient in order to improve performance (high values of the diffusion coefficients imply very small internal diffusion time steps of the explicit solver, leading to high computational effort). Using i with multi makes no sense and therefore is not allowed. diagonal i corresponds to the former infinite in the old notation.

Important: In i mode, no distinction between touching and non-touching grains is made. Instead, complete mixing for the specified component between all grains of this phase is performed.

• I: Complete mixing of the component is performed for each grain individually. Contacting grains interchange rapidly through the common interface region, so that nearly infinite diffusion is achieved between all grains of this phase which are in direct or indirect contact. How effective the interchange between contacting grains is depends on the contact geometry and the actual phase-field time-step. I is only allowed with diagonal and corresponds to the former infinite_restricted (old notation).

• g: The diffusion coefficients will be calculated by Thermo-Calc™ (via the TQ interface) using diffusion data from database. This is only possible if a GES-file is used which contains mobility data (see How to create a GES file). g stands for global and means that only one global value for the diffusion coefficient and the effective activation energy will be calculated, based on the average temperature and composition in the given phase. This temperature dependent description will be used for the whole simulation domain. Updates are calculated at a certain time interval which is defined by the user at the end of the concentration data input for all contributions.

• l: Same as g, but locally defined diffusion coefficients are calculated. For performance reasons, these local values cannot be directly calculated from the diffusion database. Instead, the values are interpolated from the local composition dependence with respect to all elements inside the phase. Using l, diffusion of the diagonal term is described more precisely compared to the g option, but higher calculation times are required for retrieval of the data from database as well as for interpolation.

Important: Using the l option can easily lead to demixing diffusion matrixes and divergence of the diffusion solver if extrapolation is going too far! This is especially the case if composition is close to spinoidal like it is typically the case for γ and γ' in Ni-base alloys. Furthermore, the diffusion time step must be reduced correspondingly, leading to longer simulation times. Use maxfactor_local to control extrapolation limits (see below for more details).

• z: Same as g, but instead of one global value for the diffusion coefficient a user-specified number of equidistant segments is created along the z-direction of the simulation domain. For each of these segments a global value for the diffusion coefficient and the effective activation energy is calculated. The number of segments has to be specified at the end of the diffusion data input. Use this option in case of strong temperature gradients where large temperature intervals exist inside the domain, or if strong concentration gradients are present in z direction (e.g. diffusion couple).

• f: Read diffusion coefficient temperature-dependent from file. The file name which contains the diffusion data (first column: temperature, second column: diffusion coefficient) has to be given in an extra line

The described keywords are valid from MICRESS® 6.2 onward. The old less systematic keywords (diff, no_diff, database_global, database_local, infinite, infinite_restricted) should be avoided but are still recognized.

A grain boundary diffusion model is available to consider an increased diffusivity of a phase in the interface region to other phases. This increase is expressed as a reduction of the activation energy, leading to a temperature-dependent factor on the diffusion coefficients. The user has to specify this activation energy correction in J/mol as well as the physical interface thickness which is needed to scale to some real thickness. Thus, grain boundary diffusion is only possible for elements in phases which have also diffusion in the bulk and not e.g. for stoichiometric elements in intermetallics without solubility range. The input works only in the terse mode, i.e. if each diffusion term is defined with a proceeding element and phase number. First, bulk diffusion has to be defined for this contribution. Afterwards, for the same contribution, grain boundary diffusion can be added by using the keyword +b, followed by a string with length number of phases + 1 consisting of n or b. This string defines for each phase whether in grain boundaries to the actual phase enhanced diffusion is to be considered. For each b character, an activation energy difference and a physical interface thickness has to be entered in extra lines

The advanced extra line option cushion is available to get more control over the internal diffusion time stepping. Normally, an automatic criterion is used which determines the time-stepping per line in the diffusion matrix. The option cushion is only available with terse mode (see above). Adding an extra input line (in arbitrary order with respect to the other diffusion input data) consisting of two integers for the phase and component number, the keyword cushion and a cushion factor (real value) leads to a modification of the diffusion time step for this contribution by the user defined cushion factor. This option is useful only in case of numerical problems (divergence of diffusion with formation of "chess pattern") as well as for debugging purposes. The specified value should be \<1 to ensure numerical stability.

The keywords infinite_limit and maxfactor_local are also only available in terse mode as they specify further information for diffusion terms which must be specified before. infinite_limit allows to automatically switch to restricted infinite diffusion (I) if the smallest value of the diagonal term in the simulation domain exceeds the value given in the same line after the keyword. maxfactor_local allows specifying the maximal factor for extrapolation of diffusion coefficients when using the option l. For diagonal terms, this factor specifies the allowed extrapolation range as well for increasing as for decreasing the local diffusion coefficient, while for off-diagonal terms decreasing is allowed down to 0. Default value for this factor is 10. The resulting maximum ranges are given in the .diff output for those diffusion terms which use diffusion data from database (g, l, z) and where local extrapolation (l) has been specified.

The option factor finally allows for a correction of the diffusivity itself by a specified factor. If the diffusivity has been defined before as multi, this factor applies to all terms. Thus, it acts as if it were a factor on the mobility of the diffusing species. Please note that the factor also applies to diffusion data specified by d and f.

If any of the diffusion contributions specified by the user is making use of mobility data included in the .GES5 file (g, l, z), an update interval for these data from the .GES5 file (in seconds) has to be specified. Small intervals typically lead to a more exact simulation of the diffusion processes but increase simulation time, especially if local extrapolation is used. The interval can be either specified as a constant value or be read from a file as time-dependent.

Furthermore, in case of any z specifier having been used, the number of diffusion segments in z-direction has to be given.

###### Example 1¶

Diffusion data

...
# Diffusion
# =========
# ["Terse Mode": Each line starts with component number and phase number]
# Options:   diagonal|diagonal_dilute [x]   multi|multi_plus [y(1..k)]
#  x: one of the characters "n", "d", "g", "l", "z", "i", "I", or "f"
#  y: chain of "n", "d", "g", "l", "z", or "f" (for each component)
#  default: "g"  resp. "gggg..."
#  Rem: "n":no diffusion, "d": input, "f": T-dep. from file
#       "i":infinite, "I": infinite in each grain
#       from database: "g": global, "l": local, "z" global z-segmented
# Extra line option: [+b] for grain-boundary diffusion
# Extra line option (prefactor on time step): cushion <0-1>
# Extra line option: infinite_limit_[d|t] (in cm**2/s|K)
# Extra line option: maxfactor_local [real > 1.0] (default: 10.0)
# Extra line option: factor [real > 0.]
# Extra line option: dilute [real >= 0.] (default:1.0)
# Finish input of diffusion data with 'end_diffusion_data'.
#
# How shall diffusion of component CR in phase LIQUID be solved?
1 0 diagonal d
# Diff.-coefficient:
# Prefactor? (real) [cm**2/s]
2.00000E-04
# Activation energy? (real) [J/mol]
0.0000
# How shall diffusion of component CO in phase LIQUID be solved?
multi g
...


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# Strategy Library

## Paired Switching

### Introduction

Paired switching is a strategy where in its simplest form, investors pick two assets that are negatively correlated and periodically switch position based on their relative performance. The idea behind this strategy is that if the assets are negatively correlated, then a traditional mixed portfolio might lead to a lower return than the return for the individual assets. If the negative correlation exists, switching positions could improve the performance of a portfolio where two assets are statically weighted.

### Method

After two negatively correlated assets are determined, we will analyze them on a quarterly basis. The analysis will consist of retrieving historical prices for the two assets and calculating their performances over the prior quarter. We will buy the asset that yields a higher return during the period. The position is held for one quarter, and then the analysis is repeated.

 def Rebalance(self):
self.months +=1
if(self.months%3==0):
history_call = self.History(self.Securities.Keys,timedelta(days=90))
if not history_call.empty:
first_bars = history_call.loc[self.first.Symbol.Value]
last_p1 = first_bars["close"].iloc[0]
second_bars = history_call.loc[self.second.Symbol.Value]
last_p2 = second_bars["close"].iloc[0]
first_performance = (float(self.Securities[self.first.Symbol].Price) - float(last_p1))/(float(self.Securities[self.first.Symbol].Price))
second_performance = (float(self.Securities[self.second.Symbol].Price) - float(last_p2))/(float(self.Securities[self.second.Symbol].Price))
if(first_performance > second_performance):
if(self.Securities[self.second.Symbol].Invested==True):
self.Liquidate(self.second.Symbol)
self.SetHoldings(self.first.Symbol,1)
else:
if(self.Securities[self.first.Symbol].Invested==True):
self.Liquidate(self.first.Symbol)
self.SetHoldings(self.second.Symbol,1)

### Source

You can also see our Documentation and Videos. You can also get in touch with us via Chat.


text:
Categories

# circumference to diameter calculator

Increase in Circumference. Diameter d of a circle is a straight line connecting two points of a circle and passing through the center. To calculate the Circumference of a circle with a diameter of 5, you need to multiply the diameter by π: C = d ∙ π C = 5 x 3.14 C = 15.7. You just measured your trees circumference. How to calculate the circumference of the Earth The used formula for calculations are indicated below the form. Circumference to Diameter Calculator. If you have to perform several calculations within less time, it can prove to be a hassle if less time is available. $\text{Circumference}=\pi\times d$ Let's understand the concept with the help of an example. 15.9 = 14.13 (*) (*) 14.14 inches exactly or limited to de precision of this calculator (13 decimal places). Circumference of a circle is defined as the distance around it. Questions this calculator can answer: How to find the perimeter of a circle with a radius of 1 meters? Using the knowledge that the diameter is 4.3m on the diagram and knowing that C = πd, we can calculate the circumference. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. What is its perimeter? You just measured your trees circumference. Measuring Circumference and Diameter. This is also known as the longest chord of the circle. This circumference calculator will find the area, circumference and diameter of a circle. The constant pi, designated by the Greek letter π, is the ratio of the circumference to the diameter of a circle. To calculate the circumference of a circle, use the formula C = πd, where "C" is the circumference, "d" is the diameter, and π is 3.14. P = πD, where D is the diameter. There are two formulas. P = 2πR, where R is the radius. Four versions of a circumference calculator to solve geometric problems. Circumference of a circle is the distance around it. What is its circumference? Both calculations are designed to round up. Learn more about pi, or explore hundreds of other calculators addressing finance, math, fitness, health, and more. Area and circumference of circle calculator uses radius length of a circle, and calculates the perimeter and area of the circle. This calculator calculates the diameter from circumference value. Diameter (d) See also: Area Calculator; Area, circumference, angles of a triangle . Circumference of a circle formula. Where: C represents the circumference,. To calculate the circumference of a circular shape when the diameter is given we need to use the formula for circumference. What is the relationship between the diameter and the circumference of a circle? Setting Up the Formula. You can enter the circumference and also compute radius and diameter in mils, inches, feet, yards, miles, millimeters, centimeters, meters and kilometers.. Compute area measured by square mils, square inches, square feet, square yards, square miles, acres, hectares, square millimeters, square centimeters, square meters, and square kilometers. Wrap the measuring tape around the tree at breast height (4.5 Feet Above The Ground). r is the radius of the circle, and. Try your hand at solving a few interesting practice questions at the end of the page. Formulas for Circumference. Circumference calculator is a free tool used to calculate the circumference of a circle when the radius is given. Circumference To Diameter Calculator. BYJU’S online diameter calculator tool makes the calculation faster, and it displays the diameter in a fraction of seconds. Radius (in) : CALCULATE Diameter (in) : Area (in²) : Circumference (in) : Whenever you are learning about circumferences at school, you absolutely need to try out our Circumference Calculator. Utility to calculate diameter, radius, circumference and area of a circle. At the high school and college levels, geometry is one of the major branches of mathematics. Calculate the diameter if the circumference is 32 mm? Apart from being used to find the area and circumference of a circle , pi is used in most calculations for construction, communications, music theory, medical procedures, air travel, and space flight, just to name a few. To calculate the diameter of a circle with circumference 31, use the formla of C = πd, or d = C/π, where C is the circumference and d is the diameter. Circumference calculator. Circumference formula . Diameter to Circumference Calculator. Diameter (This is basically double the value of the radius). Stepwise solutions are provided. Such a calculator works in all directions, which is convenient. 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text:
## 9.4.3 User Inputs for the Pressure-Based Solver

If you are using the pressure-based solver, in order to calculate a spatially periodic flow field with a specified mass flow rate or pressure derivative, you must first create a grid with translationally periodic boundaries that are parallel to each other and equal in size. You can specify translational periodicity in the Periodic panel, as described in Section  7.15. (If you need to create periodic boundaries, see Section  6.8.4.)

Once the grid has been read into FLUENT, you will complete the following inputs in the Periodic Conditions panel (Figure  9.4.3):

Define Periodic Conditions...

1.   Select either the specified mass flow rate ( Specify Mass Flow) option or the specified pressure gradient ( Specify Pressure Gradient) option. For most problems, the mass flow rate across the periodic boundary will be a known quantity; for others, the mass flow rate will be unknown, but the pressure gradient ( in Equation  9.4-3) will be a known quantity.

2.   Specify the mass flow rate and/or the pressure gradient ( in Equation  9.4-3):

• If you selected the Specify Mass Flow option, enter the desired value for the Mass Flow Rate. You can also specify an initial guess for the Pressure Gradient, but this is not required.

 For axisymmetric problems, the mass flow rate is per radians.

• If you selected the Specify Pressure Gradient option, enter the desired value for Pressure Gradient.

3.   Define the flow direction by setting the X,Y,Z (or X,Y in 2D) point under Flow Direction. The flow will move in the direction of the vector pointing from the origin to the specified point. The direction vector must be parallel to the periodic translation direction or its opposite.

4.   If you chose in step 1 to specify the mass flow rate, set the parameters used for the calculation of . These parameters are described in detail below.

After completing these inputs, you can solve the periodic velocity field to convergence.

Setting Parameters for the Calculation of

If you choose to specify the mass flow rate, FLUENT will need to calculate the appropriate value of the pressure gradient . You can control this calculation by specifying the Relaxation Factor and the Number of Iterations, and by supplying an initial guess for . All of these inputs are entered in the Periodic Conditions panel.

The Number of Iterations sets the number of sub-iterations performed on the correction of in the pressure correction equation. Because the value of is not known a priori, it must be iterated on until the Mass Flow Rate that you have defined is achieved in the computational model. This correction of occurs in the pressure correction step of the SIMPLE, SIMPLEC, or PISO algorithm. A correction to the current value of is calculated based on the difference between the desired mass flow rate and the actual one. The sub-iterations referred to here are performed within the pressure correction step to improve the correction for before the pressure correction equation is solved for the resulting pressure (and velocity) correction values. The default value of 2 sub-iterations should suffice in most problems, but can be increased to help speed convergence. The Relaxation Factor is an under-relaxation factor that controls convergence of this iteration process.

You can also speed up convergence of the periodic calculation by supplying an initial guess for in the Pressure Gradient field. Note that the current value of will be displayed in this field if you have performed any calculations. To update the Pressure Gradient field with the current value at any time, click on the Update button.

Previous: 9.4.2 Theory
Up: 9.4 Periodic Flows
Next: 9.4.4 User Inputs for


text:
Welcome

Further Integration (videos, notes, quizzes with correction) - Welcome To Class

Further Integration (videos, notes, quizzes with correction): Introduction Topics

Introduction

• 1 - Welcome To Class

: 0:03:44 mins

• 2 - Presentation of Objectives

: 0:05:36 mins

• View PDF Notes Take Quiz

Lessons: Further Integration

• 1 - Presentation of Objectives

: 0:05:36 mins

• 2 - INTRODUCTION AND RECALLS OF BASIC INTEGRALS

: 0:19:41 mins

• 3 - 3-BASIC INTEGRALS (PART2)

: 0:07:38 mins

• 4 - BASIC INTEGRATION TECHNIQUES-INTEGRATION BY PARTS

: 0:18:58 mins

• 5 - BASIC INTEGRATION TECHNIQUES -SUBSTITUTION METHOD (PART 1)

: 0:07:15 mins

• 6 - BASIC INTEGRATION TECHNIQUES- SUBSTITUTION METHOD (PART2)

: 0:05:51 mins

• 7 - GCE TYPE QUESTIONS (JUNE 2020 QUESTION 3)

: 0:17:43 mins

• 8 - GCE TYPE QUESTIONS (JUNE 2018 QUESTION 4)

: 0:11:40 mins

• 9 - GCE TYPE QUESTIONS (JUNE 2018 QUESTION 4 part 2)

: 0:13:47 mins

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• Nov. 12, 2022, 7:59 p.m.

Please I have a problem on series and sequence

• Nov. 12, 2022, 7:59 p.m.

Please I have a problem on series and sequence

• May 26, 2022, 9:05 a.m.
Lynn

question 31 june 2020 has a problem, it is not showing

• May 27, 2022, 8:19 p.m.
Yuyen

Thanks for th update lynn, did you propose something with respect to it????

• April 24, 2022, 5:31 p.m.
Nkatow Elvis

Take note

• April 16, 2022, 12:54 a.m.
KENG ElSON

Kernel:

It is a vector that maps unto the zero vector under the transformation. It is otherwise called the nulls space of the transformation. Let's consider an example below.... Find the kernel of the transformation T whose matrix is  $M&space;=&space;\begin{bmatrix}&space;2&&space;1\\&space;-1&&space;4&space;\end{bmatrix}$

Solution

Let the kernel be the vector  $X=&space;\binom{x}{y}$. Then we have that

$MX&space;=&space;0_2$

$\Rightarrow&space;\begin{bmatrix}&space;2&1&space;\\&space;-1&&space;4&space;\end{bmatrix}\binom{x}{y}&space;=&space;\binom{0}{0}$

$\Rightarrow&space;\left\{\begin{matrix}&space;2x&space;+y&space;&=0&space;\\&space;-x&space;+4y&&space;=&space;0&space;\end{matrix}\right.$

$\Rightarrow&space;x&space;=&space;0,&space;y&space;=&space;0$

Hence  $X&space;=&space;ker(T)&space;=&space;\binom{0}{0}$

• April 16, 2022, 12:24 a.m.
Boum paul

Sir can you explain to me the concept of kernel and image in transformation

• April 16, 2022, 12:26 a.m.
KENG ElSON

Hello boul Paul. Be patient, your explanation is coming

• March 12, 2022, 2:55 p.m.
KENG ElSON

Good luck to everyonen Further Maths 1

• March 8, 2022, 4:37 a.m.
Zulay Jouonang

Please i face a lot of difficulties with moment of inertia urgennntttt!!!!!!

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