Abstract. In this paper, the symplectic genus for any 2−dimensional class in a 4−manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least −1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least −1 in any 4−manifold admitting a symplectic structure. §1 Introduction Let M be a smooth, closed oriented 4−manifold. An orientation-compatible symplectic form on M is a closed two−form ω such that ω ∧ ω is nowhere vanishing and agrees with the orientation. For any oriented 4−manifold M , its symplectic cone C M is defined as the set of cohomology classes which are represented by orientationcompatible symplectic forms. For any class e ∈ H 2 (M ; Z), its minimal genus m(e) is the minimal genus of a smoothly embedded connected surface representing the Poincaré dual PD(e). The problem of determining the minimal genus has involved many of the important techniques in 4−manifold topology, and it bears its origin in the older problem of representing the Poincaré dual to a class by an embedded sphere (See the excellent survey papers [La1-2] and [Kr1] on these two problems).We are here interested in studying both these problems for 4−manifolds with non-empty symplectic cone. We will introduce the notion of symplectic genus η(e) for 4−manifolds with non-empty symplectic cone. Recall that any symplectic structure ω determines a homotopy class of compatible almost complex structures on the cotangent bundle, whose first Chern class is called the canonical class of ω. Roughly, the symplectic genus η(e) of a class e is given by the formula [e 2 +K · e]/2 +1, where K has largest pairing against e amongst canonical classes of symplectic structures for which the symplectic area of e is positive.η(e) has many nice properties, among which are (i) invariance under the action 1

To understand how enzymes work is essential for understanding life processes. And, in enzyme kinetics, a fundamental assumption is the so-called Quasi-Steady-State Assumption, which has the history of more than 80 years and has been proven very fruitful in analyzing the equations of enzyme kinetics. Many experimental results and numerical results have shown the validity of the assumption. So, an important problem is if it is always true. If it is always true, then it should be a law, not only an assumption. In this paper, we prove mathematically rigorously that it is indeed always true. Hence, it is a law, and we name it the Quasi-Steady-State Law. Actually, more precisely, we have two Quasi-Steady-State Laws. In one of them quasi-steady state means that the concentration of the enzyme-substrate complex remains approximately constant, and in the other it means that the change rate of the concentration of enzyme-substrate complex is extremely tiny.

Abstract. For any nonnegative class ξ in H 2 (CP 2 #nCP 2 , Z), the minimal genus of smoothly embedded surfaces which represent ξ is given for n ≤ 9, and in some cases with n ≥ 10, the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus g, we prove that it is true for n ≤ 8 with g ≥ 1 and for n ≤ 9 with g ≥ 2.

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