Zur allgemeinen Theorie der halbgeordneten Räume

Foreword by K. Kopotun11Correspondence to: K. Kopotun, Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada ^^IR3T 2N2. Email: [email protected] paper “On the general theory of semi-ordered spaces” (“Zur allgemeinen Theorie der halbgeordneten Räume”) was written by L.V. Kantorovich and G.R. Lorentz22Until 1946, G.G. (Georg Gunter) Lorentz was using the name Geogrij Rudolfovich (G.R.) Lorentz. sometime in 1937–1939, and this is the first time it appears in print.The following is a short history of this manuscript.In his letter to I.P. Natanson written on October 11, 1937, G.G. Lorentz mentioned a talk on joint work with L.V. Kantorovich that he gave at a Session on Functional Analysis in Moscow earlier that year. The records of the Academy of Sciences of USSR indicate that a Session on Functional Analysis took place in Moscow during September 27–29, 1937, and that G.R. Lorentz gave a talk “Topological theory of semi-ordered spaces” there, and that L.V. Kantorovich was speaking on “Theory of linear operations in semi-ordered spaces”.The manuscript “On the general theory of semi-ordered spaces” was found in the archives of L.V. Kantorovich. According to Vsevolod Leonidovich Kantorovich, L.V. Kantorovich’s son, it was submitted to Trudy Tomskogo Gosudarstvennogo Universiteta imeni V. V. Kuibysheva (Proceedings of Tomsk State University). The typed version33See www.math.ohio-state.edu/~nevai/LORENTZ/KANTOROVICH_LORENTZ_typed.pdf/. of the manuscript has a handwritten note by N. Romanov44N.P. Romanov (1907–1972) was a Professor at Tomsk University from 1935 until 1944. After 1944 he worked in Uzbekistan. His main area of research was Number Theory and Theory of Functions of Complex Variables. For more information see “Nikolaĭ Pavlovich Romanov (on the eightieth anniversary of his birth)”, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 1987, no. 3, 92–93, MR0914654 (89b:01069). dated by August 31, 1939 stating that the manuscript is accepted for publication. The manuscript was never published (probably because of the World War II) and around 1945 was returned to L.V. Kantorovich.It has been decided to publish this manuscript in its original language (German), and translate the extended abstract accompanying this manuscript from Russian to English. The manuscript appears here in its original form with only minor editorial corrections.Publication of this historical document would not have been possible without the assistance and effort of many people. In particular, the significant help of C. de Boor, Ya.I. Fet, V.L. Kantorovich, V.N. Konovalov, and S.S. Kutateladze is acknowledged and greatly appreciated.Extended abstract55Translated from Russian by K. Kopotun.The current manuscript is devoted to the investigation of general semi-ordered spaces that are not necessarily linear. Hence, it may be considered a generalization of the work of L.V. Kantorovich [Linear semi-ordered spaces, Mat. Sbornik, 2 (1) 1937, 121–168].We say that a set Y={y} is a semi-ordered space if its elements are partially ordered using a relation “<” so that I.If y1<y2, y2<y3, then y1<y3.II.For any pair y1, y2, there exist elements y3,y4 such that y3⩽y1, y3⩽y2, y1⩽y4, and y2⩽y4.III.Every set E⊂Y bounded above has a least upper bound (supE).IV.For every set E⊂Y, there exists a countable subset E′ that has the same least upper and greatest lower bound as E. The above assumptions allow us to introduce notions of a limit superior, limit inferior, and of a convergent sequence in Y. For example, define lim¯yn=infn(sup(yn,yn+1,…)). It is possible to introduce, e.g., the limit superior differently, for example, by defining lim¯∗yn to be the least element y having the property that, for any subsequence {ynk}, there exists a subsequence {ynki} such that y⩾lim¯i→∞ynki. This type of convergence, ∗-convergence, turns out to be identical with the topological convergence that we arrive at if we turn Y into a topological space using the convergence defined initially. Relationships among various limits which we can define using the above approaches as well as some properties of these limits are studied in § 1 and § 2. In § 3, we study semi-ordered spaces equipped with a nonnegative metric function ρ(y1,y2) defined for all pairs y1, y2 such that y1⩽y2, and satisfying 1∘.ρ(y1,y2)=0 is equivalent to y1=y2.2∘.ρ(y1,y3)⩽ρ(y1,y2)+ρ(y2,y3) (y1⩽y2⩽y3).3∘.ρ(sup(y,y1),sup(y,y2))⩽ρ(y1,y2) (an analogous inequality holds with inf).4∘.If yn→y monotonically, then ρ(yn,y)→0 (or ρ(y,yn)→0).5∘.If yn monotonically tends to infinity, then the condition limn,m→∞ρ(yn,ym)=0 should not hold.Let ρ(y1,y2,…,yn)=ρ(inf(y1,…,yn),sup(y1,…,yn)). Then yn→y turns out to be equivalent to ρ(y,yn,…,yn+p)→0 when n→∞, and yn→y(∗) is equivalent to ρ(y,yn)→0. In addition, Cauchy’s convergence principle holds. Moreover, if Y is distributive, i.e., inf(y,sup(y1,y2))=sup(inf(y,y1),inf(y,y2)), then it is also strongly distributive: inf(y,supnyn)=supn(inf(y,yn)). In § 4, we study similar spaces under weaker assumptions. Particular examples of such spaces are the Hausdorff space of closed sets (see Hausdorff “Set theory”, p. 165) and the space of semicontinuous functions. § 5 is devoted to applications of the general theorems to the theory of semicontinuous functions y=f(x) that map a metric space {x}=X into a semi-ordered space {y}=Y. Under some additional assumptions (Y is regular, distributive, and between any two elements y1 and y2 such that y1<y2 there is a third element y3, y1<y3<y2) it is possible to develop a complete theory of semicontinuous functions including a theorem that every semicontinuous function is a limit of a monotone sequence of continuous functions as well a theorem on separation by a continuous function

Fully Dynamic Matching in Bipartite Graphs

Maximum cardinality matching in bipartite graphs is an important and well-studied problem. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for dynamic matching (in general graphs) seem to fall into two groups: there are fast (mostly randomized) algorithms that do not achieve a better than 2-approximation, and there slow algorithms with \O(\sqrt{m}) update time that achieve a better-than-2 approximation. Thus the obvious question is whether we can design an algorithm -- deterministic or randomized -- that achieves a tradeoff between these two: a $o(\sqrt{m})$ approximation and a better-than-2 approximation simultaneously. We answer this question in the affirmative for bipartite graphs. Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give stronger results for graphs whose arboricity is at most \al, achieving a (1+ \eps) approximation in worst-case time O(\al (\al + \log n)) for constant \eps. When the arboricity is constant, this bound is $O(\log n)$ and when the arboricity is polylogarithmic the update time is also polylogarithmic. The most important technical developement is the use of an intermediate graph we call an edge degree constrained subgraph (EDCS). This graph places constraints on the sum of the degrees of the endpoints of each edge: upper bounds for matched edges and lower bounds for unmatched edges. The main technical content of our paper involves showing both how to maintain an EDCS dynamically and that and EDCS always contains a sufficiently large matching. We also make use of graph orientations to help bound the amount of work done during each update.Comment: Longer version of paper that appears in ICALP 201

Latest Developments on the IEEE 1788 Effort for the Standardization of Interval Arithmetic

(Standardization effort supported by the INRIA D2T.)International audienceInterval arithmetic undergoes a standardization effort started in 2008 by the IEEE P1788 working group. The structure of the proposed standard is presented: the mathematical level is distinguished from both the implementation and representation levels. The main definitions are introduced: interval, mathematical functions, either arithmetic operations or trigonometric functions, comparison relations, set operations. While developing this standard, some topics led to hot debate. Such a hot topic is the handling of exceptions. Eventually, the system of decorations has been adopted. A decoration is a piece of information that is attached to each interval. Rules for the propagation of decorations have also been defined. Another hot topic is the mathematical model used for interval arithmetic. Historically, the model introduced by R. Moore in the 60s covered only non-empty and bounded intervals. The set-based model includes the empty set and unbounded intervals as well. Tenants of Kaucher arithmetic also insisted on offering "reverse" intervals. It has eventually been decided that an implementation must provide at least one of these flavors of interval arithmetic. The standard provides hooks for these different flavors. As the preparation of the draft should end in December 2013, no chapter is missing. However, a reference implementation would be welcome

Some ideas about quantitative convergence of collision models to their mean field limit

We consider a stochastic $N$-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when $N\to \infty$. For any time $T>0$ we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in some homogeneous negative Sobolev space. The control we get is Gaussian, i.e. we prove that the distance is bigger than $x N^{-1/2}$ with a probability of type $O(e^{-x^2})$. The two main ingredients are first a control of fluctuations due to the discrete nature of collisions, secondly a Lipschitz continuity for the Boltzmann collision kernel. The latter condition, in our present setting, is only satisfied for Maxwellian models. Numerical computations tend to show that our results are useful in practice.Comment: 27 pages, references added and style improve

Maharam-type kernel representation for operators with a trigonometric domination

[EN] Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in L2[0, 1] by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.This research has been supported by MTM2016-77054-C2-1-P (Ministerio de Economia, Industria y Competitividad, Spain).Sánchez Pérez, EA. (2017). Maharam-type kernel representation for operators with a trigonometric domination. Aequationes Mathematicae. 91(6):1073-1091. https://doi.org/10.1007/s00010-017-0507-6S10731091916Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Generalized perfect spaces. Indag. Math. 19(3), 359–378 (2008)Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Delgado, O., Sánchez Pérez, E.A.: Strong factorizations between couples of operators on Banach function spaces. J. Convex Anal. 20(3), 599–616 (2013)Dodds, P.G., Huijsmans, C.B., de Pagter, B.: Characterizations of conditional expectation type operators. Pacific J. Math. 141(1), 55–77 (1990)Flores, J., Hernández, F.L., Tradacete, P.: Domination problems for strictly singular operators and other related classes. Positivity 15(4), 595–616 (2011). 2011Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)Hu, G.: Weighted norm inequalities for bilinear Fourier multiplier operators. Math. Ineq. Appl. 18(4), 1409–1425 (2015)Halmos, P., Sunder, V.: Bounded Integral Operators on $$L^2$$ L 2 Spaces. Springer, Berlin (1978)Kantorovitch, L., Vulich, B.: Sur la représentation des opérations linéaires. Compositio Math. 5, 119–165 (1938)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise multipliers of Calderón- Lozanovskii spaces. Math. Nachr. 286, 876–907 (2013)Kolwicz, P., Leśnik, K., Maligranda, L.: Pointwise products of some Banach function spaces and factorization. J. Funct. Anal. 266(2), 616–659 (2014)Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Conditional expectations on Riesz spaces. J. Math. Anal. Appl. 303, 509–521 (2005)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Maharam, D.: The representation of abstract integrals. Trans. Am. Math. Soc. 75, 154–184 (1953)Maharam, D.: On kernel representation of linear operators. Trans. Am. Math. Soc. 79, 229–255 (1955)Maligranda, L., Persson, L.E.: Generalized duality of some Banach function spaces. Indag. Math. 51, 323–338 (1989)Neugebauer, C.J.: Weighted norm inequalities for averaging operators of monotone functions. Publ. Mat. 35, 429–447 (1991)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)Rota, G.C.: On the representation of averaging operators. Rend. Sem. Mat. Univ. Padova. 30, 52–64 (1960)Sánchez Pérez, E.A.: Factorization theorems for multiplication operators on Banach function spaces. Integr. Equ. Oper. Theory 80(1), 117–135 (2014)Schep, A.R.: Factorization of positive multilinear maps. Ill. J. Math. 28(4), 579–591 (1984)Schep, A.R.: Products and factors of Banach function spaces. Positivity 14(2), 301–319 (2010