61 research outputs found
Non-polynomial Worst-Case Analysis of Recursive Programs
We study the problem of developing efficient approaches for proving
worst-case bounds of non-deterministic recursive programs. Ranking functions
are sound and complete for proving termination and worst-case bounds of
nonrecursive programs. First, we apply ranking functions to recursion,
resulting in measure functions. We show that measure functions provide a sound
and complete approach to prove worst-case bounds of non-deterministic recursive
programs. Our second contribution is the synthesis of measure functions in
nonpolynomial forms. We show that non-polynomial measure functions with
logarithm and exponentiation can be synthesized through abstraction of
logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem
using linear programming. While previous methods obtain worst-case polynomial
bounds, our approach can synthesize bounds of the form
as well as where is not an integer. We present
experimental results to demonstrate that our approach can obtain efficiently
worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the
divide-and-conquer algorithm for the Closest-Pair problem, where we obtain
worst-case bound, and (ii) Karatsuba's algorithm for
polynomial multiplication and Strassen's algorithm for matrix multiplication,
where we obtain bound such that is not an integer and
close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201
Kothe dual of Banach lattices generated by vector measures
We study the Kothe dual spaces of Banach function lattices generated by abstract methods having roots in the theory of interpolation spaces. We apply these results to Banach spaces of integrable functions with respect to Banach space valued countably additive vector measures. As an application we derive a description of the Banach dual of a large class of these spaces, including Orlicz spaces of integrable functions with respect to vector measuresThe first author was supported by the Foundation for Polish Science (FNP). The second author was supported by the Ministerio de Economia y Competitividad (Spain) under Grant #MTM2012-36740-C02-02.Mastylo, M.; Sánchez Pérez, EA. (2014). Kothe dual of Banach lattices generated by vector measures. Monatshefte fur Mathematik. 173(4):541-557. https://doi.org/10.1007/s00605-013-0560-8S5415571734Aronszajn, N., Gagliardo, E.: Interpolation spaces and interpolation methods. Ann. Mat. Pura. Appl. 68, 51–118 (1965)Bartle, R.G., Dunford, N., Schwartz, J.: Weak compactness and vector measures. Canad. J. Math. 7, 289–305 (1955)Brudnyi, Yu.A., Krugljak, N.Ya.: Interpolation functors and interpolation spaces I . North-Holland, Amsterdam (1991)Curbera, G.P.: Operators into L 1 of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)Curbera, G.P., Ricker, W.J.: The Fatou property in p -convex Banach lattices. J. Math. Anal. Appl. 328, 287–294 (2007)Delgado, O.: Banach function subspaces of L 1 of a vector measure and related Orlicz spaces. Indag. Math. 15(4), 485–495 (2004)Diestel, J., Jr., Uhl, J.J.: Vector measures, Amer. Math. Soc. Surveys 15, Providence, R.I. (1977)Fernández, A., Mayoral, F., Naranjo, F., Sánchez-Pérez, E.A.: Spaces of p -integrable functions with respect to a vector measure. Positivity 10, 1–16 (2006)Ferrando, I., Rodríguez, J.: The weak topology on L p of a vector measure. Topol. Appl. 155, 1439–1444 (2008)Ferrando, I., Sánchez Pérez, E.A.: Tensor product representation of the (pre)dual of the L p -space of a vector measure. J. Aust. Math. Soc. 87, 211–225 (2009)Galaz-Fontes, F.: The dual space of L p of a vector measure. Positivity 14(4), 715–729 (2010)Kamińska, A.: Indices, convexity and concavity in Musielak-Orlicz spaces, dedicated to Julian Musielak. Funct. Approx. Comment. Math. 26, 67–84 (1998)Kantorovich, L.V., Akilov, G.P.: Functional analysis, 2nd edn. Pergamon Press, New York (1982)Krein, S.G., Petunin, Yu.I., Semenov, E.M.: Interpolation of linear operators. In: Translations of mathematical monographs, 54. American Mathematical Society, Providence, R.I., (1982)Lewis, D.R.: Integration with respect to vector measures. Pacific. J. Math. 33, 157–165 (1970)Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 583–599 (1973)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices, (Russian). Sibirsk. Mat. Z. 10, 419–430 (1969)Musielak, J.: Orlicz spaces and modular spaces. In: Lecture Notes in Math. 1034, Springer-Verlag, Berlin (1983)Okada, S.: The dual space of L 1 ( μ ) of a vector measure μ . J. Math. Anal. Appl. 177, 583–599 (1993)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces, operator theory. Adv. Appl., vol. 180, Birkhäuser, Basel (2008)Rao, M.M., Zen, Z.D.: Applications of Orlicz spaces. Marcel Dekker, Inc., New York (2002)Rivera, M.J.: Orlicz spaces of integrable functions with respect to vector-valued measures. Rocky Mt. J. Math. 38(2), 619–637 (2008)Sánchez Pérez, E.A.: Compactness arguments for spaces of p -integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Ill. J. Math. 45(3), 907–923 (2001)Sánchez Pérez, E.A.: Vector measure duality and tensor product representation of L p spaces of vector measures. Proc. Amer. Math. Soc. 132, 3319–3326 (2004)Zaanen, A.C.: Integration. North Holland, Amsterdam (1967
Boundary value problems for nonlinear fractional integro-differential equations: theoretical and numerical results
Some Newton-like methods with sharper error estimates for solving operator equations in Banach spaces
Output feedback stabilisation for uncertain nonlinear time-delay systems subject to input constraints
Object classification in analytical chemistry via data‐driven discovery of partial differential equations
Magic Square of Real Spectral and Time Series Analysis with an Application to Moving Average Processes
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