Slope-Intercept Form Developing Concepts: Fill & Download for Free

A Short Introduction to Black Holes The basic idea of a black hole is simply an object whose gravity is so strong that light cannot escape from it. It is black because it does not reflect light, nor does its surface emit any light. Before Princeton Physicist John Wheeler coined the term black hole in the mid-1960s, no one outside of the theoretical physics community really paid this idea much attention. In 1798, the French mathematician Pierre Laplace first imagined such a body using Newton's Laws of Physics (the three laws plus the Law of Universal Gravitation). His idea was very simple and intuitive. We know that rockets have to reach an escape velocity in order to break free of Earth's gravity. For Earth, this velocity is 11.2 km/sec (40,320 km/hr or 25,000 miles/hr). Now let's add enough mass to Earth so that its escape velocity climbs to 25 km/sec…2000 km/sec…200,000 km/sec, and finally the speed of light: 300,000 km/sec. Because no material particle can travel faster than light, once a body is so massive and small that its escape velocity equals light-speed, it becomes dark. This is what Laplace had in mind when he thought about “black stars.” This idea was one of those idle speculations at the boundary of mathematics and science at the time, and nothing more was done with the idea for over 100 years. Once Albert Einstein had completed developing his Theory of General Relativity in 1915, the behavior of matter and light in the presence of intense gravitational fields was revisited. This time, Newton's basic ideas had to be extended to include situations in which time and space could be greatly distorted. There was an intense effort by mathematicians and physicists to investigate all of the logical consequences of Einstein's new theory of gravity and space. It took less than a year before one of the simplest kinds of bodies was thoroughly investigated through complex mathematical calculations. The German mathematician Karl Schwarzschild investigated what would happen if all the matter in a body were concentrated at a mathematical point. In Newtonian physics, we call this the center of mass of the body. Schwarzschild chose a particularly simple body: one that was a perfect sphere and not rotating at all. Mathematicians such as Roy Kerr, Hans Reissner, and Gunnar Nordstrom would later work out the mathematical details for other kinds of black holes. Schwarzschild black holes are actually very simple. Mathematicians even call them elegant because their mathematics is so compact, exact, and beautiful. They have a geometric feature called an “event horizon” (Problem 1) that mathematically distinguishes the inside of the black hole from the outside. These two regions have very different geometric properties for the way that space and time behave. The world outside the event horizon is where we live and contains our universe, but inside the event horizon, space and time behave in very different ways entirely (Problem 9). Once inside, matter and light cannot get back out into the rest of the universe. This horizon has nothing to do, however, with the Newtonian idea of an escape velocity. By the way, these statements sound very qualitative and vague to students, but the mathematics that goes into making these statements is both complex and exact. With this in mind, there are four basic kinds of black hole solutions to Einstein's equations: Space Math Welcome to Space Math @ NASA ! x Schwarzschild: These are spherical and do not rotate. They are defined only by their total mass. Reissner-Nordstrom: These possess mass and charge but do not rotate. Kerr: These rotate and are flattened at the poles, and only described by their mass and amount of spin (angular momentum). Kerr-Nordstrom: These possess mass and charge, and they rotate. There are also other types of black holes that come up when quantum mechanics is applied to understanding gravity or when cosmologists explore the early history of the universe. Among these are Planck-Mass: These have a mass of 0.00000001 kilograms and a size that is 100 billion billion times smaller than a proton. Primordial: These can have a mass greater than 10 trillion kilograms and were formed soon after the big bang and can still exist today. Smaller black holes have longsince vanished through evaporation in the time since the big bang. A Common Misconception Black holes cannot suck matter into them except under certain conditions. If the sun instantly turned into a black hole, Earth and even Mercury would continue to orbit the new object and not fall in. There are two common cases in the universe in which matter can be dragged into a black hole. Case 1: If a body orbits close to the event horizon in an elliptical orbit, it emits gravitational radiation, and its orbit will eventually decay in millions of years. Case 2: A disk of gas can form around a black hole, and through friction, matter will slowly slide into the black hole over time. How Black Holes are Formed Black holes can come in any size, from microscopic to supermassive. In today's universe, massive stars detonate as supernovae and this can create stellar-mass black holes (1 solar mass = 1.9×1030 kg). When enough of these are present in a small volume of space, like the core of a globular cluster, black holes can absorb each other and in principle, can grow to several hundred times the mass of the sun. If there is enough matter (i.e., gas, dust, and stars) for a black hole to “eat,” it can grow even larger. There is a black hole in the star-rich core of the Milky Way that has a mass equal to nearly 5 million suns. The cores of more massive and distant galaxies have supermassive black holes containing the equivalent of 100 million to as much as 10 billion suns. Astronomers are not entirely sure how these supermassive black holes evolved so quickly to their present masses given that the universe is only 14 billion years old. Currently, there are no known ways to create black holes with masses less than about 0.1 times the sun's mass, and through a speculative process called Hawking Radiation, black holes less than 1 trillion kg in mass would have evaporated by now if they had formed during the Big Bang. xi A Short List of Known Black Holes Stellar-Mass Name Constellation Distance Mass (Light years) (in solar units) Cygnus X-1 Cygnus 7,000 16 SS 433 Aquila 16,000 11 Nova Mon 1975 Monocerous 2,700 11 Nova Persi 1992 Perseus 6,500 5 IL Lupi Lupus 13,000 9 Nova Oph 1977 Ophiuchus 33,000 7 V4641 Sgr Sagittarius 32,000 7 Nova Vul 1988 Vulpecula 6,500 8 V404 Cygni Cygnus 8,000 12 Note: The mass is the sum of the companion star and the black hole masses. '16' means 16 times the mass of the sun. Galactic - Mass Name Constellation Distance Mass (Light years) (in solar units) NGC-205 Andromeda 2,300,000 90,000 Messier-33 Triangulum 2,600,000 50,000 SgrA* Sagittarius 27,000 4,600,000 Messier-31 Andromeda 2,300,000 45,000,000 NGC-1023 Canes Venatici 37,000,000 44,000,000 Messier-81 Ursa Major 13,000,000 68,000,000 NGC-3608 Leo 75,000,000 190,000,000 NGC-4261 Virgo 100,000,000 520,000,000 Messier-87 Virgo 52,000,000 3,000,000,000 Note: The first three are called 'Intermediate-mass' black holes. The remainder are called 'supermassive'. Space Math Welcome to Space Math @ NASA ! The Nearest Stellar Black Holes 1 An artist's concept of Cygnus X-1 shows hot gas from the giant blue star flowing toward the black hole, forming a bright accretion disk. [ESA/Hubble & ESA Information Centre (Kornmesser & Christensen)] How close is the nearest black hole to our own sun? Because our Milky Way galaxy is a very flat disk of stars, we can use Cartesian coordinates to map out where the nearest black holes are! Black holes are created when very massive stars explode as supernova. Fortunately, this does not happen very often in our corner of the Milky Way, so black holes are actually very far apart ! The table below gives the coordinates of the seven nearest black holes to our sun and solar system. The mass of each black hole is given in terms of solar mass units so that ‘16’ means that the mass of the black hole is 16 times that of our sun. All distances (X, Y and D) are given in light years, where 1 light year = 9.6 trillion kilometers. Name Mass X Y D A) Cygnus X-1 16 6600 -2400 B) SS-433 11 8000 -14000 C) Nova Monocerotes 1975 11 -1400 2400 D) Nova Persi 1992 5 5600 3300 E) IL Lupi 9 6500 -11000 F) Nova Vulpeculi 1988 8 2200 -6100 G) V404 Cygni 12 6900 -4000 Problem 1 – Create a Cartesian coordinate grid with coordinate intervals of 5, 10, 15 representing distances of 5000, 10000, 15000, light years, with the sun at the Origin. On this grid, plot the location of each black hole shown in the table above. Problem 2 – Using a ruler, measure the distance between the sun and each black hole, convert this to its true distance rounded to the nearest thousands of light years, and enter the result in the last column of the table. Problem 3 – What is the mean, median and mode distance between stellar black holes in the neighborhood of our sun rounded to the nearest thousands of light years? Space Math Welcome to Space Math @ NASA ! Answer Key 1 Problem 1 – Create a Cartesian coordinate grid with coordinate intervals of 5, 10, 15 representing distances of 5000, 10000, 15000, light years, with the sun at the Origin. On this grid, plot the location of each black hole shown in the table above. Answer: See diagram above. Problem 2 – Using a ruler, measure the distance between the sun and each black hole, convert this to its true distance rounded to the nearest thousands of light years, and enter the result in the last column of the table. Answer: In order from top to bottom in the table: 7000, 16000, 3000, 7000, 14000, 7000, 8000. Note: Students may also use the Pythagorean Theorem. Problem 3 –What is the mean, median and mode distance between stellar black holes in the neighborhood of our sun to the nearest thousands of light years? Answer: Mean = (7000+16000 +3000 +7000+ 14000+7000+ 8000)/ 7 Mean = 9,000 light years. Median= 7,000 light years. Mode=7,000 light years Space Math Welcome to Space Math @ NASA ! The Nearest Supermassive Black Holes 2 Artist image of a supermassive black hole. Courtesy (NASA/Hubble/Dana Berry) Within the dense cores of most galaxies, lurk black holes that have grown over the eons into supermassive objects containing millions of times the mass of a stellar black hole. Some rare galaxies have two or three of these black holes, but far more have only one. Black holes may never lose mass. They steadily gain mass over the millennia by consuming interstellar gas, and even entire stars that are unfortunate enough to become trapped by their colossal gravity. The table below gives the distances and locations of the ten closest supermassive black holes to the Milky Way galaxy. The mass of each supermassive black hole is given in terms of solar mass units so that ’90,000’ means that the mass of the supermassive black hole is 90,000 times that of our sun. All distances (X, Y) are given in millions of light years, where 1 light year = 9.6 trillion kilometers. Name Mass X Y D NGC-205 90,000 1.1 2.0 Messier-33 50,000 0.5 2.6 Sagittarius A* 3 million 0 0 Messier-31 45 million 1.2 2.0 NGC-1023 44 million -35.0 12.7 Messier-81 68 million 6.5 -11.3 NGC-3608 190 million -70.5 -25.7 NGC-4261 520 million -64.3 -76.7 Messier-87 3 billion -33.4 -39.8 Problem 1 – Use the 2-point distance formula to determine the distance, in millions of light years, between the Milky Way (0,0) and each of the nearby supermassive black holes. Enter your answer in the ‘D’ column. Problem 2 – Which supermassive black hole is the closest to Messier-87? Problem 3 – From the location of NGC-3608 as the new origin (0,0) what would be the new coordinates of the other supermassive black holes? Space Math Welcome to Space Math @ NASA ! Answer Key 2 Name Mass X Y D NGC-205 90,000 1.1 2.0 2 Messier-33 50,000 0.5 2.6 3 Sagittarius A* 3 million 0 0 0 Messier-31 45 million 1.2 2.0 2 NGC-1023 44 million -35.0 12.7 37 Messier-81 68 million 6.5 -11.3 13 NGC-3608 190 million -70.5 -25.7 75 NGC-4261 520 million -64.3 -76.7 100 Messier-87 3 billion -33.4 -39.8 52 Problem 1 – Use the 2-point distance formula to determine the distance, to the nearest million light years, between the Milky Way (0,0) and each of the nearby supermassive black holes. Enter your answer in the ‘D’ column. Problem 2 – Which supermassive black hole is the closest to Messier-87? Answer: NGC-1023 is located D2 = (-33.4 – (-35.0))2 + (-39.8 – (12.7)2 so D = 52 million light years; Problem 3 – From the location of NGC-3608 as the new origin (0,0) what would be the new coordinates of the other supermassive black holes? Answer: Subtract the coordinate (-70.5, -25.7) from the other coordinates to get: Name Mass X Y NGC-205 90,000 71.6 27.7 Messier-33 50,000 71.0 28.3 Sagittarius A* 3 million 70.5 25.7 Messier-31 45 million 71.7 27.7 NGC-1023 44 million 35.5 38.4 Messier-81 68 million 77.0 14.4 NGC-3608 190 million 0 0 NGC-4261 520 million 6.4 -51.0 Messier-87 3 billion 37.1 -14.1 Space Math Welcome to Space Math @ NASA ! Exploring the Size and Mass of a Black Hole 3 Black holes are so incredibly dense that enormous amounts of matter can be compressed into their very small volumes. No known physical event can make black holes smaller than the mass of a small star. But because black holes are a product of gravity, at least theoretically, there is no limit to how big or how small they can be. The table below gives the predicted radius of black holes containing various amounts of matter. None of these black holes have been observed, but their sizes have been determined from their stated masses. The masses are all given in terms of the mass of our Earth, 5.97x1024 kilograms so that ‘2.0’ means a black hole with twice the mass of our Earth. Mass Radius 1.0 0.88 cm 2.0 1.76 cm 3.2 2.82 cm 5.0 4.40 cm 7.5 6.60 cm 8.7 7.76 cm 11.0 9.68 cm Problem 1 – Graph the data in the table. Problem 2 – From the graph, use any method to calculate the slope, S, of the data. What are the physical units for the value of this slope? Problem 3 – From the table, calculate the slope, S, of the data. Problem 4 – Write a linear equation of the form R(M) = R0 + S M that expresses the black hole Mass-Radius Law. Problem 5 – To the nearest tenth of a meter, what would you predict as the radius of a black hole with the mass of the planet Jupiter, if the mass of Jupiter is 318 times the mass of Earth? Space Math Welcome to Space Math @ NASA ! Answer Key 3 0 2 4 6 8 10 12 0 2 4 6 8 10 1 Black Hole Mass (Earth=1.0) Radius (cm) 2 Problem 1 – Graph the data in the table. (see above) Problem 2 – From the graph, use any method to calculate the slope of the data. What are the physical units for the value of this slope? Answer: Select any two points: (2,1.76) and (5,4.4). Draw a line between the points and the slope is just the change in the y values (4.4-1.76) divided by the change in the x values (5-2) so S = 2.64/3 = 0.88. The units are centimeters per Earth mass. Problem 3 – From the table, calculate the slope of the data. Answer: From the table, if M = 2.0, R = 1.76 cm, and if M=5.0, R = 4.4 cm. The slope is just S = (4.4-1.76)/(5.0-2.0) = 2.64/3.0 = 0.88 cm/Earth mass. Problem 4 – Write a linear equation of the form y = b + S x that expresses the black hole Mass-Radius Law. Answer: We know that the slope, S = 8.4. If we substitute the coordinates for one of the points into this equation (2.1.76) we get 1.76 = b + 0.88 (2). Then solving for the y intercept we get b= 0.0, so the formula reads R(M) = 0.88 M Problem 5 – To the nearest tenth of a meter, what would you predict as the radius of a black hole with the mass of the planet Jupiter, if the mass of Jupiter is 318 times the mass of Earth? Answer: For M = 318 Earths, R(318) = 0.88 x 318 = 280 centimeters or 0.28 meters. Space Math Welcome to Space Math @ NASA ! The Moon as a Black Hole! 4 Suppose that a group of hostile aliens passed through our solar system and decided to convert our moon into a black hole! A body with the mass of our moon (about 7 million trillion tons!) would be compressed into a black hole with a diameter of only 0.2 millimeters! Problem 1 – In the space below, draw a black disk 0.2 millimeters in diameter to represent the size of Black Hole Moon. Problem 2 - The Earth as a black hole would have a radius of 8.7 millimeters. In the space below, draw a circle the size of Black Hole Earth. Problem 3 - If the distance to the moon is 356,000 kilometers, how far from our Black Hole Earth would the new Black Hole Moon be located if its diameter were only 0.2 millimeters? Space Math Welcome to Space Math @ NASA ! Answer Key 4 Problem 1 - Moon black hole shown as the following dot . Problem 2 - Earth black hole shown above. It has a diameter of 17 millimeters, or about the same diameter as a dime. Problem 3 - Answer: It would still be 356,000 kilometers because this is NOT a scaled drawing of the black holes sizes, but an illustration of their actual sizes, so the distance between the black disks above would be 356,000 kilometers! Space Math Welcome to Space Math @ NASA ! Exploring Black Holes 5 All bodies produce gravity. The more mass a body has, the more gravity it creates. It is also true that the smaller you make a body by compressing it, the more intense its gravity is at its surface. Suppose you made a body that had such an intense gravity that even light could not escape from it. That body would be called a black hole, because anything falling into it, even light, could never escape from it again. Black holes can come in all imaginable sizes. Suppose that some aliens could turn the planets and moons in our solar system into black holes. How big would they be? On a black piece of paper, use a ruler and a compass to make circles that are as large as the black holes mentioned in each of the following problems. Cut these circles out, and make a black hole mobile of the smaller bodies in the solar system! Problem 1 - Mercury is a black hole with a radius of 0.5 millimeters. Problem 2 - Venus is a black hole with a radius of 7 millimeters Problem 3 - Earth is a black hole with a radius of 9 millimeters Problem 4 - The moon is a black hole with a radius of 0.1 millimeters Problem 5 - Mars is a black hole with a radius of 1.0 millimeter Problem 6 - Pluto is a black hole with a radius of 0.02 millimeters The giant planets will need black circles that are much bigger! Problem 7 - Jupiter is a black hole with a radius of 280 centimeters Problem 8 - Saturn is a black hole with a radius of 83 centimeters Problem 9 - Uranus is a black hole with a radius of 13 centimeter Problem 10 - Neptune is a black hole with a radius of 15 centimeter Space Math Welcome to Space Math @ NASA ! Answer Key (approximate sizes) 5 Mercury Diameter = 1 mm Venus Diameter= 14 mm Earth Diameter= 18 mm . Moon Diameter= 0.2 mm Mars Diameter= 2 mm Pluto Diameter= 0.04 mm . Space Math Welcome to Space Math @ NASA ! Exploring a Full-Sized Black Hole 6 This black ball shown below is the exact size of a black hole with a diameter of 9.0 centimeters. Such a black hole would have a mass of 5 times the mass of our Earth. All of this mass would be INSIDE the ball below. Although it looks pretty harmless, if this black hole were at arms-length, you would already be dead. In fact, if you were closer to it than the distance from New York to San Francisco, 1 150-pound person would weigh 3 tons and would be crushed by their own weight! Suppose that you could survive being crushed to death as you got closer to the black hole shown above. To stay in an orbit around the black hole so that you did not fall in, you have to be traveling at a specific speed V, in kilometers per second, that depends on your distance R, in meters from the center of the black hole, is given below: 44,700 V R = Problem 1 - If you were orbiting at the distance of the Space Shuttle (R=6,800 km) from the center of this black hole, what would your orbital speed be in A) kilometers/sec? B) kilometers/hour? C) miles per hour (1 mile = 1.6 km). Problem 2 - If a small satellite were orbiting 20 centimeters away from the center of the black hole shown above, how fast would it be traveling in A) km/second? B) percentage of the speed of light? (The speed of light = 300,000 km/sec). Problem 3) If the orbit is a circle, how long: A) would the Space Shuttle in Problem 1 take to go once around in its orbit? B) would it take the satellite in Problem 2 to go once around in its orbit? Space Math Welcome to Space Math @ NASA ! Answer Key 6 Problem 1 - If you were orbiting at the distance of the Space Shuttle (R=6,800 km) from the center of this black hole, what would your orbital speed be in A) kilometers/sec? B) kilometers/hour? C) miles per hour (1 mile = 1.6 km). Answer; A) The formula says that for R = 6,800,000 meters, V = 17 km/sec. B) 1 hour = 3600 seconds, so V = 17 km/sec x (3600 sec/1 hour) = 61,200 km/hour. C) V = 61,200 km/sec x (1 mile / 1.6 km) = 38,250 miles/hr Problem 2 - If a small satellite were orbiting 20 centimeters away from the center of the black hole shown above, how fast would it be traveling in A) km/second? B) percentage of the speed of light? (The speed of light = 300,000 km/sec). Answer; A) R = 0.2 meters, so from the formula V = 100,000 km/sec B) Speed = 100% x (100000/300000) so speed = 33% the speed of light. Problem 3) If the orbit is a circle, how long: A) would the Space Shuttle in Problem 1 take to go once around in its orbit? B) would it take the satellite in Problem 2 to go once around in its orbit? A) Orbit circumference, C = 2πR so for R = 6,800 km, C = 40,000 kilometers. The Shuttle speed is V=17 km/sec, so the time is T = C/V or 2,353 seconds. This equals about 39 minutes. B) A) Orbit circumference, C = 2πR so for R = 0.2 meters, C = 1.25 meters. The satellite speed is V=100,000 km/sec. Converting this to meters we get 100,000,000 meters/sec, so the time is T = C/V or 0.000000013 seconds. This is 13 billionths of a second! Space Math Welcome to Space Math @ NASA ! A Scale-Model Black Hole - Orbit Speeds 7 Black holes can come in all sizes, so let's build one that fits on your desk top, and explore some of its interesting properties! Get a basket ball (radius 12 cm) and paint it black. Note: An actual black hole of this size would have a mass equal to about 14 times the Earth! Most of the real weird things about black holes are hidden in the ‘numbers’ that define their properties. Distance (m) Orbit Speed (km/s) 10 33,400 20 23,600 30 19,300 40 16,700 Orbit Speed – At a distance of 18 cm from the center of the basketball black hole, the orbit speed is 300,000 km/sec. This is called the Photon Radius because only photons travel fast enough to remain in orbit at this distance. Because relativistic effects are large, it is not simple matter to calculate exact orbit speeds in the distorted geometry of space near a black hole. But at distances farther than 10 times the basketball black hole radius (1.20 meters), we can start to use Newtonian physics to get approximate answers. Problem 1 - If the speed of light is 300,000 km/sec, what is the orbit speed at 10 m from the center of this basketball black hole in terms of a percentage of the lightspeed? Problem 2 – Suppose you wanted to move the satellite from an orbit distance of 30 meters to 10 meters in order to study the event horizon more closely. By how much would you have to change the satellites speed? Space Math Welcome to Space Math @ NASA ! Answer Key 7 Problem 1 - If the speed of light is 300,000 km/sec, what is the orbit speed at 10 m from the center of this basketball black hole in terms of a percentage of the lightspeed? Answer: The table says that at 10 m the speed will be 33,400 km/s so in terms of the speed of light this is 100% x (33,400/300,000) = 11% the speed of light! Math Note: For a circular orbit V2 = GM/R so for M= 14x earth = 8.36 x 1025 kg, and R = 10 meters, we have V = 33,400 km/s to 2 significant figures. At 10 times the black hole radius (1.2 meters), the relativity correction is about (1-1/10)1/2 or 0.95. That means our Newtonian answer is good to about 95% accuracy at this distance and gets better the farther away we are

I've not studied generalized splines, but based upon my understanding of cubic splines, I can infer what seems to me should be the meaning of the term "generalized splines" or just "splines". If you want the definition word for word, you can perhaps get it from Wikipedia, but I intend to write my bit like a definition from a mathematics text that, were I to write a text or treatise on numerical methods, would be the definition I would compose. Please note that this is evidence of what I've been called before, and not always kindly: "pedantic". :)First, since I think that many people who've learned about some kind of "splines" in mathematics have actually heard of or used "cubic splines", I'll write what I'd called the definition of the term "cubic splines approximation". Then I'll tell what I'd mean by "quadratic spline approximation", "affine spline approximation" (erroneously some authors may call these "linear spline approximation", just as calculus books often misuse the adjective "linear"), and "constant spline approximation". Following that, I'll tell you what I'd mean by "spline approximation" or, equivalently, "generalized spline approximation", in a related vein, to maintain some semblance of philosophical/mathematical/pedagogical cogency in my terminology. Please feel free to look up the extant definitions in Wikipedia or in some text or treatise on numerical analysis and correct me if I've misinterpreted the intent of those authors, as numerical analysis really is not my research area.Let f be some function defined on a subinterval I of the real line, and let us assume that there is not known to us a rule for f, that is, we haven't knowledge of any formula (expression, etc) using the variable t, such that if x is in the interval I, then f(x) is equal to the given expression with x substituted for t. An example of such an expression we wish we had is something like f(t)=t^4-3t^2+t-5. If this were the correct expression for f then for any number x in I, we would compute f at x by squaring x twice, subtracting from the result thrice the square of x, adding in x, and from that result subtracting 5. But we don't have such an expression or rule. Further, if we are scientists we may believe or contend that some value in which we are interested, like, say, temperature, is "expressible as a function of position along the interval I" (for perhaps the interval I is occupied by a thin wire or filament we are heating on one end and cooling on the other). But because we, being good scientists, are presumably doing an experiment, we do not presuppose that we know even the actual values of the temperature at each number in the interval I. We only measure it at a few locations (numbers) and hope that we can discover a reasonable rule (expression) f so that we can have some confidence that the temperature T at any location x in I is close enough for our purposes to f(x). Let's say we measure the temperature at 7 locations, denoted by x_0, x_1, ..., x_6, and let's say that the values we get for our measurements are, correspondingly, T_0, T_1, ..., T_6. Geometrically, this means that if I plot the points (x_0,T_0), (x_1,T_1), ..., (x_6,T_6) in a plane, then these points lie "almost on" the graph of f, if our measurements are well made (with little or no error) and if there is such a function f that gives as output the temperature at each location along the interval I. (Even if no expression gives us the temperature distribution, there may be a function that gives the desired output for each number x in I, because there are uncountably more functions from I into the set of possible temperatures than there are expressions that can be used to define such functions.) Thus we seek a function whose graph passes through all seven of these points. One way to do this is with a cubic spline approximation to the data set{(x_0,T_0), (x_1,T_1), ..., (x_6,T_6)}.After I work out some of the details for this example, I shall tell you a more formal definition. In this example, I will use two splines. Let s_1 and s_2 be those two splines. To say that I'm using a cubic spline approximation, I must require that s_1 and s_2 both be cubic polynomial functions, but I as yet do not know their coefficients:s_1(t)=a_{0,1}+a_{1,1}t+a_{2,1}t^2+a_{3,1}t^3,and similarly,s_2(t)=a_{0,2}+a_{1,2}t+a_{2,2}t^2+a_{3,2}t^3.Moreover, the requirement that our cubic spline approximation graph pass through all of the given 7 data points leads to the following additional conditions:s_1(x_0)=T_0, s_1(x_1)=T_1, ..., s_1(x_3)=T_3,and similarly,s_2(x_3)=T_3, s_2(x_4)=T_4, ..., s_2(x_6)=T_6,which turns out to be a system of 4 linear equations in the coefficients of s_1 and then a system of 4 linear equations in the coefficients of s_2. Since both s_1 and s_2 pass through the point (x_3,T_3), it follows that the cubic spline approximation c meets our requirements and is a continuous function from I into the real line, where we define c as follows (it is not absolutely required that a cubic spline approximation be a continuous function, if your application does not require comtinuity, but most discussion of cubic spline approximation does focus on continuous ones, and hence so shall I):c(t)=(s_1(t))(H(t-x_0)-H(t-x_3))+(s_2(t))(H(t-x_3)-H(t-x_6)),where by H we mean Heavyside's step function, given byH(t)=if(t>0 or t=0,1,else(0)).(Note that if a<b in the real line, then H(t-a)-H(t-b)) describes a so-called square wave, of height 1, and by multiplying by s_1 in one case, and multiplying by s_2 in the other case yields a piece of the graph of s_1, followed by a piece of the graph of s_2, over the interval I, with one point in common between the two pieces. This use of Heavyside's step function is not uncommon in differential equations or control theory, but I'm not sure how many numerical methods presentations use it this way.) This is then an approximation to the temperature distribution in that thin wire that might be useful in various calculations after this.Now for the formal definition of cubic spline approximation:Definition: Let data points (x_0,T_0), ..., (x_n,T_n) be given, either as samples from some function f defined on I, or as points "near" the graph of such a function f. Then a corresponding continuous cubic spline approximation to f is a continuous function given by a formula such asc(t)=(s_1(t))(H(t-x_0)-H(t-x_{j_1}))+(s_2(t))(H(t-x_{j_1})-H(t-x_{j_1}))+...+(s_{k-1}(t))(H(t-x_0)-H(t-x_{j_{k-1}}))+(s_k(t))(H(t-x_{j_k})-H(t-x_{j_{k-1}})),where 0<j_1<j_2<...<j_k=n, and s_1, ..., s_k are cubic polynomial functions.Notice that1. I did not require that all of the cubic pieces be determined by the same number of original data points, like I did in the first example, and that2. In the definition, I did not describe an algorithm for obtaining a specific cubic spline approximation that uses a certain number of data points per cubic "piece". This makes the definition more flexible so that one can adapt the notion as one sees fit and write an algorithm for one's own application, based upon this concept of a cubic spline. It will also facilitate the simplicity and straightforwardness of the remaining promised definitions.Now let us turn to quadratic spline approximations. One may not hear that terminology much, but their use has been in full-swing for centuries. I expect that in a numerical methods text, one would come across them by some name or other with which I am less familiar. One place discontinuous ones as well as continuous or even differentiable ones are used is in approximation of integrals by Simpson's method or by Gaussian quadrature. To begin the discussion of quadratic spline approximations, let us recall our temperature data set, but with two less data points:{(x_0,T_0), (x_1,T_1), ..., (x_4,T_4)}.In this example, as before, I will use two splines. Let s_1 and s_2 be those two splines. To say that I'm using a quadratic spline approximation, I must require that s_1 and s_2 both be quadratic polynomial functions, but I as yet do not know their coefficients:s_1(t)=a_{0,1}+a_{1,1}t+a_{2,1}t^2,and similarly,s_2(t)=a_{0,2}+a_{1,2}t+a_{2,2}t^2.The requirement that our quadratic spline approximation graph pass through all of the given 5 data points leads to the following additional conditions:s_1(x_0)=T_0, s_1(x_1)=T_1, s_1(x_2)=T_2,and similarly,s_2(x_2)=T_2, s_2(x_3)=T_3, s_2(x_4)=T_4,which turns out to be a system of 3 linear equations in the coefficients of s_1 and then a system of 3 linear equations in the coefficients of s_2. This time our continuous quadratic spline approximation to the temperature distribution in the wire is given byq(t)=(s_1(t))(H(t-x_0)-H(t-x_2))+(s_2(t))(H(t-x_2)-H(t-x_4)).Before continuing to an example of a continuous affine spline approximation and more definitions, I shall comment on why I use the adjective "affine" where others often use the adjective "linear". The ideas described here all can be developed nicely in other settings such as higher dimensional spaces, which leads one to matrix spaces, vector spaces, Hilbert spaces, Banach spaces, and a myriad of other function spaces. In those settings, very important is the concept of a linear transformation. In those contexts, a linear transformation always - always - maps the zero vector to the zero vector. The real numbers form a one dimensional vector space. Thus a linear operator defined on the real line maps zero to zero. In calculus textbooks, this is buggered by calling functions linear if they are defined by an expression of the form mx+b, where b is the intercept and m is the slope of the line which is the graph of the function in question. But if the intercept is not zero, then the function whose graph has that nonzero intercept does not map zero to zero. In the higher dimensional contexts - as well as the one dimensional case, the graph of a linear transformation is a subspace of a certain vector space - in the one dimensional case the graph is a subspace of the two dimensional plane. Translation of this subspace by the intercept produces the graph of the affine function with that intercept and the same slope as the original linear function. The lack of mention of this and myriad other more modern approaches to calculus and its applications in modern calculus texts is a plethora of pedagogical errors perpetuated by the publishing companies that just want to make another buck off of republished material that should not be revised the way it's being done, and that horrendous anti-advancement fire is fueled by the modern penchant in education to focus solely on making the "C" students happy without alienating the "D" students and without boring to death the "B" students, but completely ignoring the needs of top students, is implemented via the modern version of quality assessment at colleges and universities in which the only measurements taken involve "happiness" measures and only one or two of these extremely flawed measures is used for deciding salary increases, hiring decisions, tenure decisions, promotion decisions, and teaching assignments. One can forgive old, old, old calculus texts for this failure to be updated, but the latest ones should never be published without more modern treatments built into the pedagogy so that at least those few top studentswho actually take the time to read a few pages of the textbook will get from it something that won't be contradicted in a later course where the instructor presents a more cogent and up to date version of a course like linear algebra or differential equations or numerical analysis.Now let us turn to continuous affine spline approximations. Again, one may not hear that terminology much, but of course, they are used very much. Probably, gentle readers are more accustomed to the term "piecewise linear" in this context. (In light of my above foray into pedagogical philosophy, I'd suggest that numerical methods texts should use a term like "piecewise affine"instead.) To begin the discussion of quadratic spline approximations, let us recall our temperature data set, but with two less data points yet again:{(x_0,T_0), (x_1,T_1), (x_2,T_2)}.In this example, as before, I will use two splines. Let s_1 and s_2 be those two splines. To say that I'm using an affine spline approximation, I must require that s_1 and s_2 both be affine functions, but I as yet do not know their coefficients:s_1(t)=a_{0,1}+a_{1,1}t,and similarly,s_2(t)=a_{0,2}+a_{1,2}t.Their graphs are straight lines in the real plane, and their intercepts are a_{0,1} and a_{0,2}, respectively. Simarly, their slopes are, respectively, a_{1,1} and a_{1,2}. The requirement that our affine spline approximation graph pass through all of the given 3 data points leads to the following additional conditions:s_1(x_0)=T_0, s_1(x_1)=T_1,and similarly,s_2(x_1)=T_1, s_2(x_2)=T_2,which turns out to be a system of 2 linear equations in the coefficients of s_1 and then a system of 2 linear equations in the coefficients of s_2. This time our continuous quadratic spline approximation to the temperature distribution in the wire is given bya(t)=(s_1(t))(H(t-x_0)-H(t-x_1))+(s_2(t))(H(t-x_1)-H(t-x_2)).The graph of a is a polygonal curve in the real plane with at most one point of nondifferntiability, at the point (x_1,T_1).Enterprising readers may already have anticipated the formal definitions, but here they are for your entertainment:Definition: Let data points (x_0,T_0), ..., (x_n,T_n) be given, either as samples from some function f defined on I, or as points "near" the graph of such a function f. Then a corresponding continuous quadratic spline approximation to f is a continuous function given by a formula such asc(t)=(s_1(t))(H(t-x_0)-H(t-x_{j_1}))+(s_2(t))(H(t-x_{j_1})-H(t-x_{j_1}))+...+(s_{k-1}(t))(H(t-x_0)-H(t-x_{j_{k-1}}))+(s_k(t))(H(t-x_{j_k})-H(t-x_{j_{k-1}})),where 0<j_1<j_2<...<j_k=n, and s_1, ..., s_k are quadratic polynomial functions.Definition: Let data points (x_0,T_0), ..., (x_n,T_n) be given, either as samples from some function f defined on I, or as points "near" the graph of such a function f. Then a corresponding continuous affine spline approximation to f is a continuous function given by a formula such asc(t)=(s_1(t))(H(t-x_0)-H(t-x_{j_1}))+(s_2(t))(H(t-x_{j_1})-H(t-x_{j_1}))+...+(s_{k-1}(t))(H(t-x_0)-H(t-x_{j_{k-1}}))+(s_k(t))(H(t-x_{j_k})-H(t-x_{j_{k-1}})),where 0<j_1<j_2<...<j_k=n, and s_1, ..., s_k are affine functions.I've decided to reneg on one promise and leave it as an exercise:Exercise: Adapt the above examples and definitions to develop the notion of a continuous constant spline approximation. Which functions agree with at least one of their continuous constant spline approximations?Now, as promised, I will explain what should be meant by the term "generalized spline approximation", but as before, I will restrict my attention to the continuous version.Let us return to our original data set:{(x_0,T_0), (x_1,T_1), ..., (x_6,T_6)}.In this example, as before, I will use two splines. Let s_1 and s_2 be those two splines. To say that I'm using a general spline approximation, I only require that s_1 and s_2 both be functions, but I as yet do not know their functional forms, if even they are given by expressions. However, I will point out in a bit where it would be good to presume certain functional forms for my two splines. The requirement that our spline approximation graph pass through all of the given 7 data points leads to the following additional conditions:s_1(x_0)=T_0, s_1(x_1)=T_1, ..., s_1(x_3)=T_3,and similarly,s_2(x_3)=T_3, s_2(x_4)=T_4, ..., s_2(x_6)=T_6,which turns out to be a system of 4 (possibly nonlinear) equations. Accordingly, we will require that our splines s_1 and s_2 be describable in terms of 4 parameters each, and we will attempt (and assume successfully so) to avoid choosing s_1 or s_2 so that the systems of equations are not both consistent and we will attempt also to avoid choosing s_1 or s_2 so that the equations above are underdetermined relative to the parameters describing them. That is, we assume the following:Among our chosen candidates for s_1, there is a unique function s_1 such thats_1(x_0)=T_0, s_1(x_1)=T_1, ..., s_1(x_3)=T_3,and among our chosen candidates for s_2, there is a unique function s_2 such thats_2(x_3)=T_3, s_2(x_4)=T_4, ..., s_2(x_6)=T_6,We solve the above systems of equations to find the unique solutions, thereby finding the unique s_1 and the unique s_2 that satisfy those systems, respectively. Since both s_1 and s_2 pass through the point (x_3,T_3), it follows that the general spline approximation g meets our requirements and, if s_1 and s_2 are continuous, is a continuous function from I into the real line, where we define g as followsg(t)=(s_1(t))(H(t-x_0)-H(t-x_3))+(s_2(t))(H(t-x_3)-H(t-x_6)).Now I will give a formal definition, and follow that with an example using our data set consisting of seven hypothetical temperature measurements.Definition: Let data points (x_0,T_0), ..., (x_n,T_n) be given, either as samples from some function f defined on I, or as points "near" the graph of such a function f, and let S be a family of (continuous) functions with the following property:There exists some positive integer n and there is some subcollection K of S such that K includes only n of the functions of the family S, and for each number x in I, there is some function s in K with x in the domain of s.Then a corresponding continuous S spline approximation to f is a continuous function given by a formula such asc(t)=(s_1(t))(H(t-x_0)-H(t-x_{j_1}))+(s_2(t))(H(t-x_{j_1})-H(t-x_{j_1}))+...+(s_{k-1}(t))(H(t-x_0)-H(t-x_{j_{k-1}}))+(s_k(t))(H(t-x_{j_k})-H(t-x_{j_{k-1}})),where 0<j_1<j_2<...<j_k=n, and s_1, ..., s_k are functions chosen from the family S, such that if x is in the interval [x_j,x_{j+1}], where j is a nonnegative integer between 0 and n, inclusive, then x is in the domain of s_j.For example, if S is the family of all exponential functions s defined by expressions of the forms(t)=(a)exp((t-m)/q),where a, m, and q are real numbers, and if our temperatures are all positive, then we can find a continuous S spline approximation, using three members of S, for our temperature data, of the formc(t)=((a_1)exp((t-m_1)/q_1)(H(t-x_0)-H(t-x_2))+((a_2)exp((t-m_2)/q_2)(H(t-x_2)-H(t-x_4))+((a_3)exp((t-m_3)/q_3)(H(t-x_4)-H(t-x_6)).We needed 3 functions instead of just two from S because S is not a 4 parameter family of functions but only a three parameter family. Thus if we used only two functions from S, the equations to guarantee continuity would be too many per spline.Hopefully by typing this on my iPod, I have not failed to catch any errors, but if there are errors, feel free to let me know. I'll try to edit it if needed.

The Japanese still viewed the Soviets as weak as they had been in the 1905 war. Originally the Japanese thought they could get their oil from the Soviet Union by invading the Soviet Union before the start of WWII. They were repulsed and decided that the Dutch East Indies and to consider a preemptive strike on Pearl Harbor.“General Georgi Zhukov had arrived at the Mongolia-Manchuria border in the early morning hours of June 5, 1939, after a grueling three-day trip from Moscow. He insisted on immediately questioning the Soviet defenders and touring the site of the recent border clashes with Japanese troops. Peering through his field glasses at the small figures scurrying about on the east bank of the Halha River and after tossing out sharply worded questions, Zhukov came to the belief that this was not another mere border clash with the Japanese. For years, Japanese troops had been probing the nearly 3,000-mile-long border that separated the Union of Soviet Socialist Republics and Soviet-protected Mongolia from the Japanese protectorate of Manchukuo, which was created shortly after the Japanese invaded Manchuria in 1931.”“The energetic Zhukov filed a report at the end of his first day, saying this looked to be the beginning of a major escalation by the Japanese and the forces of Manchukuo. The Soviet 57th Corps did not appear up to the task of stopping the Japanese in Zhukov’s assessment. He recommended a temporary holding action to protect the bridgehead on the east bank of the Halha River, which was called Khalkhin Gol by the Soviets, until substantial reinforcements could be mustered for a counteroffensive.”“One day after submitting his report to Moscow, the Soviet high command responded by naming the 42-year-old Zhukov to head the military effort, succeeding former General Nikolai Feklenko. The Soviets had tired of the Japanese incursions and were determined to make a point in the East as war clouds continued to build over Western Europe.”“Both sides were about to square off in a decisive struggle that became known as the Nomonhan Incident, which lasted several months with upward of 50,000 killed or wounded. More than 3,000 miles from Moscow, the small undeclared war went largely unnoticed in the West, but it was to have a profound influence on the coming world war. The defeat at Nomonhan caused the Japanese to turn southward toward the oil-rich East Indies and prompted the Imperial Japanese Navy to consider a preemptive strike on the U.S. Navy at Pearl Harbor. The use of combined arms and overwhelming force under Zhukov was later to play a significant role in the eventual Soviet repulse of the Nazi thrusts at Moscow and Stalingrad.”Zhukov’s Opportunity to Prove Himself“In reviewing the situation at Nomonhan, Zhukov realized that he had some skin in the game. Stalin’s purge of the Red Army had just come to an end, with its leadership cadre, including Commander in Chief Marshal Mikhail Tukhachevsky and more than half of its senior commanders were executed. This void opened opportunities for younger, talented, and savvy men like Zhukov if they could prove themselves and survive the rough and tumble atmosphere created by the bloody purges.”“The bright, ambitious Zhukov, the son of peasants born some 60 miles east of Moscow, was determined to succeed. Once his June 5 report was filed, the Soviets provided substantial reinforcements to the young commander, including the 36th Mechanized Infantry Division; 7th, 8th, and 9th Mechanized Brigades; 11th Tank Brigade; and the 8th Cavalry Division. In addition, Zhukov’s newly named First Army Group received a heavy artillery regiment and a tactical air wing with more than 100 planes and a group of 21 experienced pilots, who had won combat citations while fighting in Spain.”“The Soviets were aided by the fact that their military buildup occurred at Tamsag Bulak, 80 to 90 miles west of the Halha River and away from Japanese aerial observation. However, the nearest Soviet rail line was at Borzya, some 400 miles west of the Halha River. Japanese self-confidence and racist stereotypes of their Soviet opponents also played a role. Their railhead was only 50 miles away, and they believed that the Soviets simply could not concentrate a large combined armed force so far from their nearest railhead. Supplies would have to be transported by truck over dirt roads or over flat, open territory, making the convoys vulnerable to air attack if matters escalated. Japanese commanders also rejected the idea that the Soviets could adapt themselves to defeat Japanese tactics, which had proven so successful earlier in Korea, Manchuria, and China.”“In short, the Japanese did not believe the Soviets would rise to the challenge over such a small sliver of territory in such a faraway place. While the Japanese claimed the border to the Halha River, which flows northwest into Lake Buir Nor, the Soviets contended the border ran through the mud-brick hamlet of Nomonhan, some 10 miles east of the river. At its widest point, the disputed backwater area was less than 12 miles wide and approximately 30 miles long.”Tensions Rise on the Japanese-Soviet Border“Japanese-Soviet antagonism ran deep, dating back even before the Russo-Japanese war of 1904-1905 as the two powers struggled for years to assert dominance over parts of China, Korea, and Manchuria. After their defeat in the Russo-Japanese War, the Russian czars were forced to recognize Japan’s interests in Korea, and they ceded the Liaoutung Peninsula, renamed Kwantung by the Japanese. Kwantung contained the port of Dairen and the important naval base of Port Arthur. A few years later the Japanese established a special force, the Kwantung Army, to administer the area. The army was to later spearhead Japanese expansion on the Asian mainland.”“For its part, the Russian bear managed to paw Outer Mongolia from China in 1911, making it a protectorate called the Mongolian People’s Republic (MPR). World War I and the subsequent Russian Revolution caused considerable internal strife as the Bolsheviks took power in Russia. The resulting power vacuum and a weak China enabled the Japanese to take Manchuria. The poorly defined borders created additional uncertainty. In June 1937, Japanese forces fired on Soviet gunboats, killing 37 sailors in the Amur River that flowed between the USSR and Manchukuo. Border problems arose again in the summer of 1938 where the poorly defined borders of Korea, Manchukuo, and the Soviet Union met. The skirmishes at Changkufeng Hill provided a reminder to Stalin that the Imperial Japanese Army still threatened his eastern flank as matters heated up in Europe.”First Clashes on the Border“Although the reports differ considerably, the initial conflict at Nomonhan began on May 11, 1939, when a Japanese-backed force of provincial Manchukuo cavalry clashed with a Mongolian-Soviet patrol north of the Holsten River that flows westward to the Halha River. Japan’s Kwantung Army, which operated with some autonomy from Tokyo, decided to take matters into its own hands despite the fact that its eight divisions faced 30 Soviet divisions along the mutual border running from Lake Baikal to Vladivostok on the Pacific. Matters continued to escalate, and the next day a flight of Japanese light bombers attacked an MPR border post on the west bank of the Halha River, indisputably within the borders of the Soviet-backed country. MPR troops then crossed the fast-flowing 100- to 150-meter-wide Halha River and took up new positions between the river and Nomonhan as combat resumed.”“The Japanese saw this as a direct challenge to the Kwantung Army’s 23rd Division, an undermanned and poorly supported unit assigned to the backwater area. The 23rd had thinly armored tankettes armed only with a machine gun, and 40 percent of its 60 artillery pieces were Type 38 short-range 75mm guns dating back to 1907. The artillery regiment had a dozen 120mm howitzers, and most of the artillery was horse drawn. In addition, each infantry regiment had four rapid-fire 37mm guns and four 75mm mountain guns, but the division lacked high-velocity, low-trajectory guns appropriate to the terrain and necessary to handle oncoming tanks.”The Japanese Repulsed at the Halha River“On May 20, Japanese aerial reconnaissance discovered the Soviet buildup near Tamsag Bulak. The Japanese pulled together a force of 2,000 men to attack the enemy troops that had crossed the Halha River, starting with a two-pronged drive 40 miles south from Kanchuerhmiao along the east bank of the Halha. Four infantry companies and additional troops from Manchukuo were to push west from Nomonhan to help pin and destroy an approximately 400-man force of Soviet-MPR troops. That should have worked, except the Japanese failed to consider the possibility that the Soviet units spotted at Tamsag Bulak would be committed.”“The Japanese had substantially greater numbers, and they surprised the Soviet-Mongolian forces near Nomonhan. The Mongolian cavalry was routed and forced back, causing the Soviets to pullback as well toward the Halha River. As they neared the river, the fighting intensified and Soviet artillery and armored cars forced the Japanese forces to dig in on a low hill several miles east of the Halha. Because of faulty radio equipment, a 220-man Japanese unit under Lt. Col. Azuma was not kept apprised of the changing picture near the bridgehead. He continued south, unaware that the main Japanese force had been deflected and forced to dig in well away from the river crossing. Azuma’s force was spotted, and additional Soviet artillery was brought east across the Halha to further protect the bridgehead.”“The main Japanese force was now bogged down, dug in and under fire some two to three miles away, while Azuma’s lightly armed force was trapped by heavy fire. Azuma was nearly surrounded, with enemy infantry and cavalry attacking while heavy bombardment came from both sides of the Halha River. His cavalry dismounted and struggled to dig defensive positions in the sand. He could retreat northward, but doing so without orders would be a criminal offense punishable by death under the Japanese system. The artillery barrage continued and destroyed Azuma’s remaining trucks and the unit’s small reserve of ammunition. Only four men in his command managed to escape that night with the rest killed or captured. The main Japanese unit was not able to make progress, and three nights later it pulled back to Kanchuerhmiao. Japanese casualties neared 500, with one-fourth attributed to the main unit.”The Soviets and Japanese Reinforce“Although the Soviets had forced the enemy from the field, little mention of that fact was made at the time by either side. The Kwantung Army continued to assure Tokyo that it planned to avoid prolonged conflict, telling superiors that the Soviets would not be able to deploy large ground forces around Nomonhan. The Japanese forces, however, did request river-crossing equipment and craft, which should have alerted Tokyo that their army might be planning forays west across the Halha River into mutually recognized Mongolian territory. Moscow was growing suspicious of Japanese actions, and Zhukov was pulled from his duties as deputy commander of the Belarusian Military District in Minsk and sent east to investigate the disputed border area.”“With Zhukov now on the scene, the Soviet buildup continued at Tamsag Bulak. On the other side, Japanese confidence was high, and the Imperial forces remained largely unaware of either the size of the buildup or the change in the enemy’s command with the arrival of Zhukov. The Soviet-MPR bridgehead east of the Halha was gradually expanded during the first half of June with no response from the Japanese. Soviet strafing raids east of the river did cause concern at Japanese headquarters. The Japanese decided to respond in kind to the air attacks to send a strong message to the Soviets, a decision that dramatically escalated the situation. Japan’s 7th Division was to be pressed into service to assist the relatively new and understrength 23rd Division. The 23rd was to be reinforced with 180 planes, a strong strike force of two regiments of medium and light tanks, an artillery regiment, and an infantry regiment. Japanese strength would grow to some 15,000 men, 120 artillery and antitank guns, 70 tanks, and 180 aircraft.”“The Japanese remained exceptionally confident—so much so that they cut back the aerial reconnaissance west of the Halha so as not to alert the enemy. They were unaware that they were now facing a Soviet force of some 12,500 men, 109 artillery and antitank guns, 186 tanks, 266 armored cars, and more than 100 planes. Although the Japanese may have had a modest advantage in some categories, it was more than offset by the Soviets’ six-to-one armor advantage, which proved crucial in the relatively flat, open terrain.”Seizing Air Superiority“The Japanese plan of attack was rather straightforward. The main body of the 23rd Division would seize the Fui Heights, a pancake-shaped raised area located on the east bank of the Halha some 11 miles north of the confluence of the Halha and Holsten Rivers, where much of the fighting would occur. The 23rd Division and related units would cross the Halha near the Fui Heights on a freshly built pontoon bridge and move southward along the west bank of the Halha toward the Soviet bridge. At the same time, another force under Lt. Gen. Yasuoka Masaomi would move south along the east side of the Halha to engage the Soviet and MPR units and pin them between the two advancing forces near the Soviet-built bridge located near the confluence of the two rivers.”“The Japanese planned to neutralize Soviet airpower with a preemptive strike at the Soviet base near Tamsag Bulak, located well inside the MPR. The Kwantung Army kept this component of the plan secret from Tokyo, concerned that higher ups would not approve. The general staff in Tokyo did learn of the planned air attack and went on record opposing it, but the semi-independent Kwantung Army actually elected to move the air attack two days forward to June 27. The Japanese surprise air attack caught a group of newly arrived Soviet airmen flat-footed. Returning pilots claimed 98 Soviet planes destroyed and 51 damaged, while the Japanese reportedly lost only one bomber, two fighters, and a scout plane. In short, the Japanese had achieved air supremacy over the Halha at the start of the Kwantung Army’s July offensive.””The Kwantung Army was ecstatic over the results of the air attack, but it received a severe rebuke from Tokyo. The Kwanting commanders resented the desk jockeys at the general staff and continued to believe in a need to maintain its dignity and maintain the border against what it perceived to be Soviet incursions. The Japanese plans went forward, and on July 1 the Japanese took the Fui Heights east of the Halha. Then, on the moonless night of July 2-3, the Japanese managed to build a pontoon bridge on the Halha River across from the Fui Heights. By early morning, the 26th Regiment and the 71st and 72nd Infantry Regiments began the slow crossing on the narrow bridge. The heavy armored vehicles had to remain behind, but the 18 37mm antitank guns, 12 75mm mountain guns, eight 75mm field guns, and four 120mm howitzers made it across undetected with all the infantry by nightfall.”“The Soviets were now vulnerable with the enemy moving southward undetected on the west bank toward their bridgehead while facing a Japanese tank and infantry force on the east bank. Again the Japanese caught the Soviets by surprise when Yasuoka’s tanks attacked in the early morning hours of July 3, with guns blazing amid a passing thunderstorm. The startled Soviet 149th Infantry Regiment scattered before the tanks.”“Zhukov, still apparently unaware of the Japanese presence on the west bank of the Halha, ordered his 11th Tank Brigade and related units north toward a hill called Bain Tsagan, located on the west bank. In the early morning hours of July 3, the Japanese, with their rapid-firing 37mm antitank guns and armor-piercing shells, mauled the Soviets. Zhukov, now realizing that a large Japanese force had crossed the Halha and was threatening his position, ordered the remainder of the 11th Tank Brigade, 7th Brigade, 24th Regiment, and an armored battalion of the 8th Mongolian Cavalry Division against the southward-moving Japanese force.”“The Soviet armor had little infantry support, enabling the Japanese infantry to swarm over the Soviet vehicles, prying open hatch covers and destroying many of the tanks with gasoline bombs. Successive Soviet attacks were dealt with by the Japanese, who quickly found that the Russian gasoline tank engines could be taken out by gunners and gasoline bombs. However, by that afternoon unrelenting Soviet counterattacks and ranged-in artillery forced the Japanese to begin digging defensive positions on the west bank of the Halha just south of Bain Tsagan.”An Untenable Position“Zhukov then massed more than 450 tanks and armored cars in the area against a Japanese force that had left its armored vehicles behind before crossing the shaky pontoon bridge over the Halha. The only hope then lay with Yasuoka’s force, which had proceeded southward on the east side of the Halha. If that force succeeded, it would relieve pressure on Komatsubara’s embattled unit on the west bank. Yasuoka’s initial efforts were successful in crushing the first lines of Soviet artillery, but successive lines of Soviet infantry, tanks, and artillery slowed them and inflicted heavy losses. The Japanese Type 89 medium tank with its low-velocity 57mm cannon and relatively thin 17mm armor proved a poor match against the Soviet antitank guns and BT 5/7 tanks and armored cars. And the Soviets had another trick up their sleeves east of the Halha in the form of Japanese-produced piano wire coiled nearly invisibly as part of the defensive works. The wire entangled the gears and wheels of the Japanese light tanks like butterflies in a web, enabling Soviet artillery to zero in and finish off the tanks.”“Making matters worse for the Japanese, Yasuoka’s infantry units were not able to catch up with the tanks, so the two forces fought separately and less effectively. By evening, his forces slowed, with the infantry dug in well short of the Soviet bridgehead. At that time, only 50 percent of the Japanese tanks were operable and able to withdraw to the initial jumping off point. Additional Soviet aircraft had also appeared over the combat zone, engaging the Japanese in fierce struggles for air supremacy.”“The Japanese commanders now realized they were indeed in an untenable position, with their forces divided on both sides of the river and with a more powerful than imagined enemy force between them. The Japanese commander realized his forces had no additional bridge-building materials on hand, making it imperative that his forces west of the Halha be withdrawn immediately. Most of the forces made it back north and across the Halha that night and reestablished themselves by the morning of July 4 on the Fui Heights. The covering Japanese unit managed to make the crossing the following night before destroying the pontoon bridge.”“The Kwantung Army made the fateful decision to pull its two tank regiments from the combat zone. The Soviets had actually suffered heavier tank losses, but they had begun with a significant numerical advantage in armor and were able to continue delivering improved tanks to the battlefield.”Lessons of the First Battles“In both the May 28 and the early June battles, the Japanese lacked sufficient military intelligence, and they had seriously underestimated the enemy’s strength and determination. Hubris and their racist sense of inherent superiority over the Soviets and their MPR allies played a role as well. Most could not contemplate the idea that Soviet firepower could overcome the Japanese fighting spirit. The events also gave the Soviets reason to pause. The movement of some 10,000 enemy troops over the Halha River had gone undetected despite heightened awareness, and the Soviets might have been able to pin and destroy the units west of the Halha had they moved faster.”“Zhukov was no fool. In fact, he proved to be a quick learner throughout his career. In those two days at Nomonhan, Zhukov learned to use large tank formations as an independent attack force, rather than simply as support for infantry. This was far different from conventional views, and the concept would be proven again and on a much larger scale by the German panzer divisions early in World War II. The Soviets had learned the importance of hatch covers that could be locked from the inside, frustrating enemy infantry attempts to put tanks out of commission by opening the hatches. They also found that gasoline-powered tanks, with their exposed ventilation grills and exhaust manifolds, could be easily set afire. The combat at Bain Tsagan clearly showed the importance of overwhelming force and close integration of tanks, motorized infantry, artillery, and air power in defeating an enemy. The need for improved aerial reconnaissance was also apparent. These were important lessons for Zhukov and the Red Army as World War II loomed.”An 800-Mile Supply Loop“Moscow now agreed to send additional reinforcements to Zhukov. Thousands of men and machines were sent east, requiring additional trucks and transports to shuttle the men and equipment from the railhead at Borzya to the front lines. The effort resulted in a continuous 800-mile, five-day, round-trip shuttle that put both men and machines to the test.”“The Soviets by now had managed to build seven bridges across the Halha and Holsten Rivers to support their operations, including one invisible bridge built with its surface some 10-12 inches below the water level so it would not be seen by Japanese pilots. Between July 8-12, Japanese probing continued making gradual progress against the Soviets although they suffered substantial losses to superior Soviet artillery. The fighting became especially fierce in the early morning hours of July 12 as the Japanese forces pushed to within 1,500 yards of the primary Soviet bridgehead over the Halha. As the day wore on, Soviet counterattacks with two infantry battalions and some 150 armored vehicles coupled with strong artillery support managed to push the Japanese back to their starting point. The Soviet artillery especially had taken its toll and the Japanese decided to suspend their night attacks that had resulted in 85 deaths and three times that in wounded personnel in one regiment alone.”“The officers of the Kwantung Army were determined to save face and push the invaders back to the west side of the Halha. Other divisions were stripped of their heavy artillery, and the 3rd Heavy Field Artillery Brigade was shipped from Japan to Nomonhan. That brigade came equipped with 16 relatively modern 150mm howitzers and 16 100mm artillery pieces, all pulled by tractors rather than horses like the artillery in the 23rd Division. Army officials hoped to mount a large-scale artillery duel, force the enemy back to the west bank, and then withdraw after saving face.”“The Soviets were not standing pat, and reinforcements and supplies continued to pour into Zhukov’s First Army Group, including two additional artillery regiments and literally tons of artillery shells.”Looking For Conflict Resolution“The Japanese fired the next salvos in the conflict with a sustained barrage on July 23. The Soviets rose to the challenge, with artillery rounds falling on the Japanese positions from artillery located on both sides of the Halha. The Soviets wrested air supremacy from the Japanese, their fighters strafing enemy positions and shooting down two Japanese balloons used for artillery spotting. The intense artillery duel went on for two days with the Soviets demonstrating that they had plentiful ammunition, better artillery, and more of it. The heavy Soviet guns were deployed out of range of the Japanese guns, and the Russian 152mm artillery was deadly beyond 15,000 yards.”“The Kwantung Army ended its ill-fated artillery attack on July 25, and the general staff in Tokyo began pursuing a diplomatic resolution. That, in turn, irritated the Kwantung Army, which wanted to fight on to save face. The army’s General Isogai was called to Tokyo and told point blank that the Kwantung Army was to maintain its defensive position east of the Soviet positions while the government worked to resolve matters through diplomacy.”“Amazingly, the Kwantung Army chose to ignore the directive, but it was the Soviets under Zhukov who were to take the next step, based in part on events in both the West and the East. German demands on Poland were heating things up in the West, and both the Germans and the British were attempting to obtain an alliance with the Soviets. In addition, Richard Sorge, the Soviet super spy in Japan, confirmed that the Japanese high command wanted a diplomatic resolution to the problems at Nomonhan.”“In light of these developments, Soviet Premier Josef Stalin decided to order massive combat operations against the Japanese. Zhukov’s First Army Group was further strengthened by two infantry divisions, the 6th Tank Brigade, 212th Airborne Brigade, additional smaller units, and two Mongolian cavalry divisions. Zhukov’s air support also was considerably strengthened. The Japanese remained unaware of the additional buildup. They had only one and a half divisions in place.”Zhukov’s Maskirovka“Zhukov kept up his aerial reconnaissance and scouting efforts to gather information on the opposing forces. He planned to lead his central force in a frontal assault against the main Japanese positions several miles east of the Halha. Zhukov’s northern and southern forces—with the bulk of the Soviet armor—would turn the enemy’s flanks, creating an envelopment of the Japanese. Overwhelming force and tactical surprise would be keys to the plan.”“During World War II the Soviets were to become masters of maskirovka—or military deception—but Zhukov was to prove his near mastery of the concept even at this early stage. Trucks carrying men and matériel on the long journey from the staging area at Tamsag Bulak traveled only at night with their vehicles’ lights blacked out. Aware that the Japanese were tapping their telephone lines and intercepting their radio messages, the Soviets sent a series of easily deciphered messages concerning the building of defensive positions and preparations for a long winter campaign. Well before the attack, the Soviets nightly broadcasted the recorded sounds of tank and aircraft engines along with construction sounds. The Japanese became accustomed to the broadcasts and were unconcerned by the actual noise as the Soviets moved their tanks and equipment into position on the night of the attack.”“Zhukov cleverly directed a series of minor attacks on August 7-8 to expand his bridgehead east of the Halha by some three miles. He made sure these were contained by the Kwantung Army, further lulling the enemy into believing that the Soviets were relatively weak and poorly led. Under the cover of darkness on the night of August 19-20, the major force of his First Army Group crossed to the east bank of the Halha into the expanded Soviet enclave. Massive amounts of artillery on both sides of the river were available to support the assault.”Stalin’s Offensive Begins“The attack began with Soviet bombers pounding the Japanese lines just before 6 am on August 20, followed by nearly three full hours of heavy artillery fire before the bombers attacked again. “The shock and vibration of incoming bombs and artillery rounds” caused Japanese “radio-telegraph keys to chatter so uncontrollably that the frontline troops could not communicate with the rear, compounding their confusion and helplessness,” reported one observer.”“At 9 am, Soviet infantry and armor began moving forward with artillery supporting their efforts. An early morning fog along the river assisted the advancing troops, and the damaged communications lines prevented the Japanese artillery from participating until a bit after 10 am when many of the forward positions had already been overrun. Japanese resistance had stiffened by noon, and combat raged over a 40-mile front. The southern Soviet force, consisting of MPR Cavalry, armor, and mechanized infantry, pushed the southern Japanese force northward and inward by eight miles during the first day’s fighting. The northern Soviet force, consisting of two MPR cavalry units, supporting armor, and mechanized infantry, pushed the northern flank back two miles to the Fui Heights. Zhukov’s central force pushed some 750 yards forward against resistance so fierce that the Japanese dared not move troops to reinforce the flanks.”“Over the next two days, the southern Soviet force broke through the Japanese lines and then proceeded to encircle and eliminate the enemy in small pockets. Once the heavy Japanese weapons were eliminated, Soviet artillery and armor tightened the ring further with flame-throwing tanks and infantry. By August 23, the southern thrust had reached Nomonhan where it could block a possible Japanese retreat.”“Japanese airpower tangled with Soviet planes, which had a two to one advantage. The Soviets had recently brought forward updated I-16 fighter planes with thicker armor plating and a strengthened windshield that could withstand the 7.7mm machine-gun fire of the Japanese Type-97 fighters. The Japanese upgunned some of their planes, but this was quickly offset once the Soviets discovered that the enemy’s fighters had unprotected fuel tanks and began filling the sky with burning Japanese planes.”Eliminating the Remaining Pockets of Japanese Resistance“On the night of August 22, the Japanese planned a counterattack against the Soviet forces that were crushing their southern flank. The 26th and 28th Regiments of the 7th Division and the 71st and 72nd Regiments were pressed into action. Only the 28th was at full strength, although its men had marched 25 miles to the front just a day earlier. The units were deployed on the night of August 23, and many units did not reach their assigned positions by the next morning. Those that did arrive scurried into position before fully reconnoitering the enemy positions.”“Using the early morning fog to mask its movements on August 24, the 72nd Regiment made for a distant stand of scrub pines only to discover rather late in the going that the pines consisted of a well-camouflaged force of Soviet tanks. The Japanese, in fact, had stumbled into a massive Soviet tank force equipped with prototype models of the superb T-34 tank with its high-velocity 76mm gun and thick sloping armor. Later models of that versatile and well-designed tank were to cause nightmares for Nazi forces in World War II. The Soviets also had updated models of the BT-7 tanks, which were now equipped with diesel rather than gasoline engines and had protection over grills and exhaust manifolds to help protect them from Japanese fire.”“The attacking Japanese southern force suffered nearly 50 percent casualties and was ordered to withdraw shortly after sunset. The Japanese forces on the northern end of the line at Fui Heights had sustained three days of hammering by the Soviets, but the 800 well-entrenched men inflicted heavy losses on the advancing Soviets while disrupting Zhukov’s time table for the entire operation.”“The ambitious Zhukov was not pleased. He got on the horn, fired the commander of the northern force, and then fired his replacement as well before sending up a member of his own staff to lead the faltering force. That night a renewed Soviet attack led by flame-throwing tanks and supported by heavy artillery managed to quash the remaining Japanese artillery on the heights. Supplies of ammunition and food were cut off as the encircling Soviets tightened the ring over the next two days. On the night of August 24-25, the remaining Japanese withdrew from Fui Heights without orders. The Soviets counted more than 600 Japanese bodies when they occupied the heights the next morning.”“Zhukov’s northern force moved on, pushing south and east toward Nomonhan. Within a day the northern and southern forces had linked up near Nomonhan. The encircled Japanese fought on, with many of their howitzers seizing up from heated overuse. The artillery units were destroyed by tank and heavy artillery fire or overrun by Soviet forces. By August 27, a Japanese relief force of two infantry regiments and an artillery regiment reached the northeast portion of the Soviet ring. The 5,000-man force could not break through, and it retired four miles east of the Soviet-claimed border at Nomonhan. Over the course of the next two days, Soviet planes, artillery, and armor continued to destroy remaining pockets of Japanese resistance within the solid ring created by Zhukov’s forces. A few hundred lucky Japanese managed to escape to relative safety east of Nomonhan.”Resolving the Japanese-Soviet Border War“Zhukov declared the disputed territory to be enemy free on August 31. The fighting had been the worst modern Japanese defeat up to that time with the Sixth Army losing between 18,000 and 23,000 men killed or wounded between May and September. The Kwantung Army lost nearly 150 aircraft and many of its tanks and artillery. The Soviets lost more than 25,600 killed or wounded according to a fairly recent Russian report. While the Soviets suffered more deaths and injuries, their repeated overpowering attacks forced the enemy from the field.”“The Molotov-Ribbentrop pact, signed on August 23, 1939, also helped to weaken the hawks in the Kwantung Army. Close military cooperation with the Germans against the Soviets—as the militants had long advocated—was no longer a realistic possibility. Following the disaster at Nomonhan, the Kwantung Army had submitted an aggressive plan for an escalation of the war, but authorities in Tokyo were determined to finally bring the Kwantung Army under control. General Nakajima of the headquarters staff flew from Tokyo to direct the Japanese forces to hold their positions and to ensure a quick, diplomatic resolution. Once on the ground, he was unbelievably swayed by the Kwantung Army’s arguments and had to be sent back again by Tokyo to convince the Kwantung Army that enough was enough. The imperial order was to be obeyed. General headquarters then cleaned house, relieving some of the Kwantung command and transferring other officers so that diplomatic efforts could move forward.”“While the world was focused on the September 1, 1939, German invasion of Poland, the Japanese and the Soviets quietly worked on an agreement signed September 15. Both sides recognized their troop positions as the temporary border. It approximated the earlier Soviet-MPR border claim with a joint commission to later formalize the boundary. Prisoners were exchanged and both the new Kwantung Army leaders and the Red Army leaders closely followed the agreement, bringing the war to a close.”“Stalin waited until the agreement was concluded, and then on September 17—when he was sure he did not face the prospect of a two-front war—sent the Red Army across the Polish border per his earlier agreement with Hitler on the partition of Poland.”The Remarkable Strategic Impact of the Nomonhan Incident“The Nomonhan Incident provided quite a learning curve for all involved. The Japanese came away with a black eye and a healthy respect for the Soviet military, which many in the international community had viewed as weak and lethargic in the wake of Stalin’s extensive purges. Japanese attention was turned southward toward the oil-rich East Indies and subsequently to the only major force that might hamper Japan’s expansionistic plans—the U.S. Navy and its base at Pearl Harbor. Japanese officials knew that U.S. industrial strength was much greater than the Soviet Union’s, and it superseded Japan’s strength by a factor of 10 to one. Japanese militants believed that a surprise attack would provide time for the Japanese to seize the resource-rich areas to the south and then build a strong defensive perimeter enabling them to negotiate a settlement with an America focused primarily on Europe.”“The Japanese elected to move southward, rebuffing German overtures to strike the Soviet Union in the Far East as the war progressed. Maj. Gen. Eugene Ort, the German ambassador to Japan, reported in late 1941 that Nomonhan had left a lasting impression on the Japanese, who “considered participation in the war against the Soviet Union too risky and too unprofitable.”“Soviet master spy Richard Sorge remained active in Tokyo. Later, his reliable reports assured Stalin that he could safely transport thousands of Siberian troops westward to deal with the German onslaught after the beginning of Operation Barbarossa, especially around Moscow and later Stalingrad. Zhukov and the winter-hardened Siberian troops used the combined arms tactics and maskirovka (deception) that had proven so successful at Nomonhan, but on a much larger scale with much more at stake.”Could the Soviet Union Have Fought a Two-Front War?T”he Soviet archives have been opened in recent years, and many military analysts now assert that the Soviet Union simply could not have survived a two-front war. Even in 1941, Maj. Gen. Arkady Kozakovtsev, then head of the Red Army in the Far East, reported quietly to a trusted confidant that the Soviet cause was hopeless “if the Japanese enter the war [against the Soviet Union] on Hitler’s side.”“The Soviet Union, in fact, was the only major power in World War II that did not have to contend with an energy-draining and resource-depleting two-front war. The early success at Nomonhan meant the Soviets could later focus solely on the Nazis at the front gate, using the hard-fought lessons learned in the East to help the Allies subdue and defeat Nazi Germany.”Soviet Surprise: How Imperial Japan Was Beat at the Battle of Nomonh An