Competition through capacity investment under asymmetric existing capacities and costs

This paper discusses the way that different operational characteristics including existing capacity, scale economies, and production policy have an important influence on the capacity outcomes when firms compete in the market place. We formulate a game-theoretical model where each firm has an existing capacity and faces both fixed and variable costs in purchasing additional capacity. Specifically, the firms simultaneously (or sequentially) make their expansion decisions, and then simultaneously decide their production decisions with these outputs being capacity constrained. We also compare our results with cases where production has to match capacity. By characterizing the firms’ capacity and production choices in equilibrium, our analysis shows that the operational factors play a crucial role in determining what happens. The modeling and analysis in the paper gives insight into the way that the ability to use less production capacity than has been built will undermine the commitment value of existing capacity. If a commitment to full production is not possible, sinking operational costs can enable a firm to keep some preemptive advantage. We also show that the existence of fixed costs can introduce cases where there are either no pure strategy equilibrium or multiple equilibria. The managerial implications of our analysis are noted in the discussion. Our central contribution in this paper is the innovative integration of the strategic analysis of capacity expansion and well-known (s,S)$(s\text{,}S)$ policy in operations and supply chain theory.     

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobi?s triple product identity and the t  -coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewell?s and Chen–Chen–Huang?s octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system _A2$A_2$ as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.     

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.     

Harnack inequalities for graphs with non-negative Ricci curvature

We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.     

用高中数学求解异面直线公垂线方程

用高中向量基础知识和待定法可非常方便地得出n（n≥3）维线性空间中异面直线的公垂线方程.若在大学数学教学中挖掘与中学数学的衔接和联系,则可在很大程度上降低大学数学的教学难度.