International Journal of Mathematics

We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg–Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann.311 (1998) 533–547] as a special case.

Let X be a Cantor set, and let A be a unital separable simple amenable -stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|_1⊗A] = [id|_1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and for all u ∈ U(A) (where α^‡ and id^‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊_α ℤ is a unital simple -stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋ⁿ → X × 𝕋ⁿ be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

Recently by Casas et al. a notion of chain of evolution algebras (CEAs) is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. It is known 25 distinct classes of chains of two-dimensional evolution algebras. In our previous paper we gave the classification of two-dimensional real evolution algebras. This classification contains seven (pairwise non-isomorphic) such algebras. For each known CEA we study its dynamics to be an element of a given class.

Let X be a smooth projective curve of genus g over an algebraically closed field k of characteristic p > 2. We prove that any rank 3 nilpotent semistable Higgs bundle (E, θ) on X is a strongly semistable Higgs bundle. This gives a partially affirmative answer to a conjecture of Lan–Sheng–Zuo [Semistable Higgs bundles and representations of algebraic fundamental groups: positive characteristic case, preprint (2012), arXiv:1210.8280][(Very recently, A. Langer [Semistable modules over Lie algebroids in positive characteristic, preprint (2013), arXiv:1311.2794] and independently Lan–Sheng–Yang–Zuo [Semistable Higgs bundles of small ranks are strongly Higgs semistable, preprint (2013), arXiv:1311.2405] have proven the conjecture for ranks less than or equal to p case.)] In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with p satisfying some bounds (Theorem 4.3). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan–Sheng–Zuo conjecture holds (Corollary 4.4).

In this paper, we prove that two meromorphic functions f and g must be linked by a quasi-Möbius transformation if they share a pair of small functions ignoring multiplicities and share other four pairs of small functions with multiplicities truncated by 2. We also show that two meromorphic functions which share q (q ≥ 6) pairs of small functions ignoring multiplicities are linked by a quasi-Möbius transformation. Moreover, in our results, the zeros with multiplicities more than a certain number are not needed to be counted in the condition sharing pairs of small functions of meromorphic functions. These results are generalization and improvements of some recent results.

We find a 2-parameter family of deformations in of the classical Chen–Gackstatter surface explicitly, and show the existence of a larger 4-parameter family of deformations. Each of them still has genus one, a unique end, with total Gaussian curvature - ∫ K = 8π. On the other hand, a uniqueness theorem is obtained when we assume that the surface has more than four symmetries. The problem of embeddedness is also discussed with some partial results.

Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi–Pantev formula holds on maximal rank symplectic leaves of the G-generalized Hitchin system.

We construct and classify (1 2 ⋯ k)-twisted V^⊗k-modules for k even and V a vertex operator superalgebra. In particular, we show that the category of weak (1 2 ⋯ k)-twisted V^⊗k-modules for k even is isomorphic to the category of weak parity-twisted V-modules. This result shows that in the case of a cyclic permutation of even order, the construction and classification of permutation-twisted modules for tensor product vertex operator superalgebras are fundamentally different than in the case of a cyclic permutation of odd order, as previously constructed and classified by the first author. In particular, in the even order case it is the parity-twisted V-modules that play the significant role in place of the untwisted V-modules that play the significant role in the odd order case.

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.