126 (2001) MATHEMATICA BOHEMICA No. 1‚ 237–246 BLOCK DIAGONALIZATION J. J. Koliha‚ Melbourne (Received June 15‚ 1999) Abstract. We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution double commutes with the given matrix. Applications include a new characterization of an eigenprojection
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wrong. This essay will commence by defining the criteria for social welfare maximization. It will then discuss the relationship between competitive market and Pareto efficient before explaining the implication from the first and the second welfare theorem. It will then talk about equity issue arise from the topic and social welfare functions. Finally‚ the essay will conclude that being at a competitive equilibrium does not equal to social welfare maximization. Efficient‚ Equity and Social Optimum
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Chapter 8 Exercise 21 What is sampling error? It is the difference between the sample mean and the population mean Could the value of the sampling error be zero? Yes it is possible to have a zero sampling error. However‚ it is very low probability that this could happen. If it were zero‚ what would this mean? This means that the population is uniform and the sample mean and the population mean are equal. Exercise 22 List the reasons for sampling. Give an example of each reason
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Proclus‚ the last major Greek philosopher‚ who lived around 450 AD wrote (see [1] or [9] or many other sources):- Not much younger than these [pupils of Plato] is Euclid‚ who put together the "Elements"‚ arranging in order many of Eudoxus’s theorems‚ perfecting many of Theaetetus’s‚ and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes‚ who followed closely upon the first
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-- 501 B.C. circa 1850 B.C.: The Babylonians possess knowledge of what will later be known as "The Pythagorean Theorem‚" an equation that relates the sides of right triangles whereby the sum of the squares of the two "legs" (the shortest sides) of the right triangle equal the square of the
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impression that mathematics is not actually all that important. The percentage of the world’s population‚ or even of the world’s university-educated population‚ who could accurately state a single mathematical theorem proved in the last fifty years‚ is small‚ and smaller still if Fermat’s last theorem is excluded. If you ask a mathematician to explain what he or she works on‚ you will usually be met with a sheepish grin and told that it is not possible to do so in a short time. If you ask whether this
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On Cop and Robber Games with a Tunneling Robber Abhishek Parthasarathy Gautham Srinivas Mithraa Varun S Sreevatsan Vaidyanthan Sri Raghavan Prof.Nagarajan Krishnamurthy Abstract Let G be a finite connected graph. Two players‚ called Cop C and Robber R‚ play a game on G according to the following rules.First C then R occupy some vertex of G. After that they move alternately along the edges of G. The cop C wins if he succeeds in putting himself on top of the robber R‚ otherwise R wins. Both C and
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0 1 x π/2 π/2 −π/2 −1 0 1 x 0 x Review: Definitions and properties Theorem For all x ∈ [−1‚ 1] the following identities hold‚ arccos(x) + arccos(−x) = π‚ arccos(x) + arcsin(x) = π . 2 Proof: y arccos(−x) 1 1 x = sin(π/2−θ) arccos(x) θ −x = cos (π−θ) π−θ θ x = cos (θ) π/2 − θ θ arccos(x) y arcsin(x) x x = cos (θ) x Review: Definitions and properties Theorem For all x ∈ [−1‚ 1] the following identities hold‚ arcsin(−x) = − arcsin(x)‚ arctan(−x) = −
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grammar‚ Regular grammar for FA. Proving languages to be non−regular − Pumping Lemma‚ and its applications. Some closure properties of Regular languages − Closure under Boolean operations‚ reversal homomorphism‚ inverse homomorphism‚ etc. M hill−Nerode Theorem. DFA Minimization Some decision properties of Regular languages − emptiness‚ finiteness‚ membership‚ equivalence of two DFAs or REs‚ Finite automata with output. Context−free Grammars (CFGs) − Formal definition‚ sentential forms‚ leftmost and rightmost
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If human history has been able to mass-produce one thing‚ and one thing well‚ it has been political revolution. Whether it was the movement from Conservatism to Liberalism‚ or Democracy to Dictatorship; revolution was the push that made these changes happen. In the Post-World War II era‚ one particular political movement rose to prominent status: Communism. China‚ Cuba‚ North Korea and many Eastern Bloc countries made this shift‚ all through the help of revolution. This is understandable given that
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