1 10/10/01 Fermat’s Little Theorem From the Multinomial Theorem Thomas J. Osler (osler@rowan.edu) Rowan University‚ Glassboro‚ NJ 08028 Fermat’s Little Theorem [1] states that n p −1 − 1 is divisible by p whenever p is prime and n is an integer not divisible by p. This theorem is used in many of the simpler tests for primality. The so-called multinomial theorem (described in [2]) gives the expansion of a multinomial to an integer power p > 0‚ (a1 + a2 + ⋅⋅⋅ + an ) p = p k1 k2 kn a1 a2
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Kirchhoff’s Law Kirchhoff’s current law (KCL) imposes constraints on the currents in the branches that are attached to each node of a circuit. In simplest terms‚ KCL states that the sum of the currents that are entering a given node must equal the sum of the currents that are leaving the node. Thus‚ the set of currents in branches attached to a given node can be partitioned into two groups whose orientation is away from (into) the node. The two groups must contain the same net current. In general
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“The Arrow impossibility theorem and its implications for voting and elections” Arrow’s impossibility theorem represents a fascinating problem in the philosophy of economics‚ widely discussed for insinuating doubt on commonly accepted beliefs towards collective decision making procedures. This essay will introduce its fundamental assumptions‚ explain its meaning‚ explore some of the solutions available to escape its predictions and finally discuss its implications for political
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a NOR gate. DeMorgan’s theorems state the same equivalence in "backward" form: that inverting the output of any gate results in the same function as the opposite type of gate (AND vs. OR) with inverted inputs De Morgan’s theorem is used to simplify a lot expression of complicated logic gates. For example‚ (A + (BC)’)’. The parentheses symbol is used in the example. _ The answer is A BC. Let’s apply the principles of DeMorgan’s theorems to the simplification of
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A Brief History of the Pythagorean Theorem Just Who Was This Pythagoras‚ Anyway? Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece‚ and did much traveling through Egypt‚ learning‚ among other things‚ mathematics. Not much more is known of his early years. Pythagoras gained his famous status by founding a group‚ the Brotherhood of Pythagoreans‚ which was devoted to the study of mathematics. The group was almost cult-like in that it had symbols‚ rituals and prayers. In addition
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Describe the role of courts in the criminal justice process. The courts serve as the venue where disputes are then settled and justice is administered. In Australian courts the adversarial system of trial is used to determine guilt‚ this is two sides‚ one representing the accused the other the state‚ debate over the guilt of the accused; this is mediated by a neutral third party‚ the judge or magistrate. Guilt is determined by the judge‚ magistrate or a jury. Punishment for the accused if found
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THE WIZARD OF OZ 2 The Wizard of Oz Scarecrow’s Speech on Pythagorean Theorem The Pythagorean theorem is one of the earliest theorems known to ancient civilization. The well-known theorem is named after the Greek mathematician and philosopher‚ Pythagoras. In the Wizard of Oz‚ after the Scarecrow gets a brain‚ he states the Pythagorean theorem. However‚ he mistakenly says it applies to an isosceles triangle when it applies to a right triangle. He not only says the wrong triangle
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central The Central Limit Theorem A long standing problem of probability theory has been to find necessary and sufficient conditions for approximation of laws of sums of random variables. Then came Chebysheve‚ Liapounov and Markov and they came up with the central limit theorem. The central limit theorem allows you to measure the variability in your sample results by taking only one sample and it gives a pretty nice way to calculate the probabilities for the total ‚ the average and the proportion
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Application There are some applications of Thevenin’s Theorem in our daily lives. Thevenin’s Theorem is very useful to reduce a network with several voltage sources and resistors to an equivalent circuit composed a single voltage source and a single resistance connected to a load only. It is used in simplifying and analysing complex linear networks power systems and circuits where one particular where a particular load resistor‚ RL in the circuit is subject to change‚ and recalculation of the circuit
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Fermat’s last theorem Currently holding the world record for longest standing math problem ever‚ Fermat’s last theorem went unsolved for 365 years. Fermat’s last theorem was one of the largest white whales in the study of math. Over the centuries‚ thousands were puzzled by the impossible problem. From its conception to its solution‚ Fermat’s last theorem was one of the most difficult to solve yet easy to understand problems in mathematics. First‚ I will discuss the theorem and how it was introduced
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