On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces

We are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial’s condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points.

Strange Duality on Rational Surfaces II: Higher-Rank Cases

We study Le Potier’s strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ that involves the moduli space of rank $r$ sheaves with trivial 1st Chern class and 2nd Chern class $n$, and the moduli space of one-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show there is an exact sequence relating the map $SD_{c_r^r,L}$ to $SD_{c^{r-1}_{r},L}$ and $SD_{c_r^r,L\otimes K_X}$ for all $r\geq 1$ under some conditions on $X$ and $L$ that applies to a large number of cases on $\mathbb{P}^2$ or Hirzebruch surfaces. Also on $\mathbb{P}^2$ we show that for any $r>0$, $SD_{c^r_r,dH}$ is an isomorphism for $d=1,2$, injective for $d=3,$ and moreover $SD_{c_3^3,rH}$ and $SD_{c_3^2,rH}$ are injective. At the end we prove that the map $SD_{c_n^2,L}$ ($n\geq 2$) is an isomorphism for $X=\mathbb{P}^2$ or Fano rational-ruled surfaces and $g_L=3$, and hence so is $SD_{c_3^3,L}$ as a corollary of our main result.

Dark solitons, D-branes and noncommutative tachyon field theory

In this paper we discuss the boson/vortex duality by mapping the (3[Formula: see text]+[Formula: see text]1)D Gross–Pitaevskii theory into an effective string theory in the presence of boundaries. Via the effective string theory, we find the Seiberg–Witten map between the commutative and the noncommutative tachyon field theories, and consequently identify their soliton solutions with D-branes in the effective string theory. We perform various checks of the duality map and the identification of soliton solutions. This new insight between the Gross–Pitaevskii theory and the effective string theory explains the similarity of these two systems at quantitative level.

Being versus becoming ontology of paradox management

Purpose The purpose of this paper is to discuss the theoretical contribution of Li’s (2016) “Yin-Yang balancing” approach of paradox management, as well as its future development to guide paradox management research across the east and west contexts. Design/methodology/approach It begins by recognizing the importance of paradox management research, especially the indigenous epistemological approach as Li (2016) has followed. The authors take “being” and “becoming” ontology toward social reality as the basic premise in this commentary, and summarize the knowledge that the study has contributed to existing literature. Findings The “Yin-Yang balancing” approach can extend the knowledge about paradox management phenomena at least from four aspects: the “either/and” frame to view a paradox system, the importance of “seed” or “threshold” in defining moderate rather than extreme groups, duality map as a novel tool for paradox management, and comparison of being and becoming ontology. Originality/value Based on the comparison of “being” and “becoming” ontological view, the authors suggest to further develop this “Yin-Yang balancing” approach by emphasizing the following issues: eastern culture does not have exclusive ownership of the “becoming” ontology toward the world, elaboration of alternative theoretical explanation to win out the identity approach about organizational existence, the linkage between the “Yin-Yang balancing” epistemological system and process research method, and boundary condition of the “Yin-Yang balancing” approach.

Global implications of the indigenous epistemological system from the east

Purpose – The author argues and explains that the indigenous Eastern epistemological system of Yin-Yang balancing should be taken as a novel system or frame of thinking, which is deeply rooted in the indigenous Eastern culture traditions, but it has significant global implications, especially in the domain of paradox management. The purpose of this paper is twofold: first, to provide a detailed elaboration of the indigenous Eastern epistemological system of Yin-Yang balancing in contrast to the Western logic systems; and second, to provide a roadmap for applying the system of Yin-Yang balancing to complex issues in the area of management, in general, and paradoxical issues, in particular. Design/methodology/approach – This is a conceptual paper with a focus on theory-building. Findings – The author elaborates on the indigenous features of Yin-Yang balancing, in contrast to Aristotle’s formal logic and Hegel’s dialectical logic in the West, to further explore the former’s global implications for the increased attention to research on paradox management. In particular, the author posits that Yin-Yang balancing appears to be better suited for paradox management than the more commonly used logics available in the Western literature. Built upon the Yin-Yang balancing, a practical tool of Duality Map for paradox management is proposed. Research limitations/implications – The system of Yin-Yang balancing proposed in this paper has the potential to embrace logical systems available in the West into a geocentric (East-meeting-West) meta-system. This paper further shows how to apply Yin-Yang balancing with the tool of Duality Map to the most salient paradoxes in the domain of management, including value-profit balance (triple bottom lines), exploration-exploitation balance (ambidexterity), cooperation-competition balance (co-opetition), globalization-localization balance (glocalization), institution-agency balance (institutional entrepreneurship), simultaneously positive and negative attitudes toward an entity (ambivalence), and etic-emic balance (geocentric) across all domains of management research. Originality/value – The primary challenge for management researchers is to find a way to achieve a geocentric integration between the West and the East at the fundamental level of philosophy. The hope is that the philosophical traditions in the East will facilitate such integration. In particular, the Eastern philosophy of wisdom has a unique capacity to reframe paradox from a negative problem (i.e. a problem of inconsistency to be resolved by dualism in terms of separating opposite elements) to a positive solution (i.e. a solution of completeness or holism to be achieved by duality in terms of partially separating and partially integrating opposite elements).

A path convergence theorem and construction of fixed points for nonexpansive mappings in certain Banach spaces

In this paper, we introduce a new iterative process to approximate fixed points of nonexpansive maps in real Banach spaces having weakly continuous duality map and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption on K or on T. Our results improve important recent results.

Genus one partitions

International audience We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides another way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations. Nous démontrons la conjecture de M. Yip affirmant que compter les partitions de genre un par le nombre de leurs éléments et leurs parties donne, à un décalage d’indices près, la même gamme de nombres que celle qui résulte en comptant les hypermonopoles de genre un. Notre preuve utilise une représentation de chaque permutation de genre un par une partition non-croisée quatricolorée. Cette représentation peut être choisi d’une manière unique pour les permutations qui ne contiennent pas des cycles triviaux. La conclusion suit d’une formule des fonctions génératrices générale qui vaut pour toute classe de permutations qui est fermé sous l’enlèvement et la réinsertion des cycles triviaux. Notre méthode offre une autre manière de compter les hypermonopoles enracinés de genre un, et dirige l’attention sur une extension évidente de la dualité de Kreweras sur les permutations de genre un.

NONLINEAR MARKOV SEMIGROUPS ON C*-ALGEBRAS

A notion of a nonlinear quantum dynamical semigroup is introduced and discussed. Some sufficient conditions, expressed solely in terms of the duality map, in order that a multivalued mapping on a C*-algebra generates the nonlinear Markov semigroup are proposed.