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Short Proof (short + proof)
Selected AbstractsLandau's inequalities for tournament scores and a short proof of a theorem on transitive sub-tournamentsJOURNAL OF GRAPH THEORY, Issue 4 2001Richard A. Brualdi Abstract Ao and Hanson, and Guiduli, Gyárfás, Thomassé and Weidl independently, proved the following result: For any tournament score sequence S,=,(s1, s2,,,,,sn) with s1,s2,,,,,,,sn, there exists a tournament T on vertex set {1,2,,,,,n} such that the score of each vertex i is si and the sub-tournaments of T on both the even and the odd indexed vertices are transitive in the given order; that is, i dominates j whenever i,>,j and i,,,j (mod 2). In this note, we give a much shorter proof of the result. In the course of doing so, we show that the score sequence of a tournament satisfies a set of inequalities which are individually stronger than the well-known set of inequalities of Landau, but collectively the two sets of inequalities are equivalent. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 244,254, 2001 [source] On the Existence of Finite-Dimensional Realizations for Nonlinear Forward Rate ModelsMATHEMATICAL FINANCE, Issue 2 2001Tomas Björk We consider interest rate models of the Heath,Jarrow,Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite-dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite-dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view. [source] The play operator on the rectifiable curves in a Hilbert spaceMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2008Vincenzo Recupero Abstract The vector play operator is the solution operator of a class of evolution variational inequalities arising in continuum mechanics. For regular data, the existence of solutions is easily obtained from general results on maximal monotone operators. If the datum is a continuous function of bounded variation, then the existence of a weak solution is usually proved by means of a time discretization procedure. In this paper we give a short proof of the existence of the play operator on rectifiable curves making use of basic facts of measure theory. We also drop the separability assumptions usually made by other authors. Copyright © 2007 John Wiley & Sons, Ltd. [source] A short proof of the preservation of the ,, -bounding propertyMLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 1 2004Chaz Schlindwein Abstract There are two versions of the Proper Iteration Lemma. The stronger (but less well-known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ,, -bounding property. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A comparison of abstract versions of deflation, balancing and additive coarse grid correction preconditionersNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2008R. Nabben Abstract In this paper we consider various preconditioners for the conjugate gradient (CG) method to solve large linear systems of equations with symmetric positive definite system matrix. We continue the comparison between abstract versions of the deflation, balancing and additive coarse grid correction preconditioning techniques started in (SIAM J. Numer. Anal. 2004; 42:1631,1647; SIAM J. Sci. Comput. 2006; 27:1742,1759). There the deflation method is compared with the abstract additive coarse grid correction preconditioner and the abstract balancing preconditioner. Here, we close the triangle between these three methods. First of all, we show that a theoretical comparison of the condition numbers of the abstract additive coarse grid correction and the condition number of the system preconditioned by the abstract balancing preconditioner is not possible. We present a counter example, for which the condition number of the abstract additive coarse grid correction preconditioned system is below the condition number of the system preconditioned with the abstract balancing preconditioner. However, if the CG method is preconditioned by the abstract balancing preconditioner and is started with a special starting vector, the asymptotic convergence behavior of the CG method can be described by the so-called effective condition number with respect to the starting vector. We prove that this effective condition number of the system preconditioned by the abstract balancing preconditioner is less than or equal to the condition number of the system preconditioned by the abstract additive coarse grid correction method. We also provide a short proof of the relationship between the effective condition number and the convergence of CG. Moreover, we compare the A -norm of the errors of the iterates given by the different preconditioners and establish the orthogonal invariants of all three types of preconditioners. Copyright © 2008 John Wiley & Sons, Ltd. [source] A remark on the coercivity for a first-order least-squares methodNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2007Jaeun Ku Abstract This study present a short proof concerning the coercivity of a first-order least-squares finite element method for general second-order elliptic problems proposed by Cai, Lazarov, Manteuffel and McCormick (Cai et al. J Numer Anal 31 (1994), 1785,1799). Our proof is based on a priori estimate and the technique can be applied to prove L2 -norm error estimate for the primary function u. After establishing the coercivity bound from the assumed a priori estimate, we observe that the coercivity bound is actually equivalent to the a priori estimate. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] Greedy colorings of uniform hypergraphsRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2009András Pluhár Abstract We give a very short proof of an Erd,s conjecture that the number of edges in a non-2-colorable n -uniform hypergraph is at least f(n)2n, where f(n) goes to infinity. Originally it was solved by József Beck in 1977, showing that f(n) at least clog n. With an ingenious recoloring idea he later proved that f(n) , cn1/3+o(1). Here we prove a weaker bound on f(n), namely f(n) , cn1/4. Instead of recoloring a random coloring, we take the ground set in random order and use a greedy algorithm to color. The same technique works for getting bounds on k -colorability. It is also possible to combine this idea with the Lovász Local Lemma, reproving some known results for sparse hypergraphs (e.g., the n -uniform, n -regular hypergraphs are 2-colorable if n , 8). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source] Negative correlation and log-concavityRANDOM STRUCTURES AND ALGORITHMS, Issue 3 2010J. Kahn Abstract We give counterexamples and a few positive results related to several conjectures of R. Pemantle (Pemantle, J Math Phys 41 (2000), 1371,1390) and D. Wagner (Wagner, Ann Combin 12 (2008), 211,239) concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. (Borcea et al., J Am Math Soc 22 (2009), 521,567). We also give short proofs of a pair of results from (Pemantle, J Math Phys 41 (2000), 1371,1390) and (Borcea et al., J Am Math Soc 22 (2009), 521,567); prove that "almost exchangeable" measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious" example of a class of measures for which this important property can be shown to hold; and mention some further questions. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 [source] | ||||||||||