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The paper mainly discusses, based on the properties and related knowledge of continuous functions, the conditions that the graphs of separated binary polynomials are closed curves by the method of separation of variables. It also studies the properties of closed curves in the ways of analytic geometry, and draws a sucient and necessary condition to determine whether the graph of a separated binary polynomial is closed.

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This paper explores the Nash equilibria of a variant of the Colonel Blotto game, which we call the Asymmetric Colonel Blotto game. In the Colonel Blotto game, two players simultaneously distribute forces across n battlefields. Within each battlefield, the player that allocates the higher level of force wins. The payo↵ of the game is the proportion of wins on the individual battlefields. In the asymmetric version, the levels of force distributed to the battlefields must be nondecreasing. In this paper, we find a family of Nash equilibria for the case with three battlefields and equal levels of force and prove the uniqueness of the marginal distributions. We also find the unique equilibrium payo↵ for all possible levels of force in the case with two battlefields, and obtain partial results for the unique equilibrium payo↵ for asymmetric levels of force in the case with three battlefields.

A physical method is developed to analyze the Riemann zeta function and the L-function. First, a physical model of an elastic bar fixed at both ends and subjected to symmetric axial loads with respect to the mid-span is constructed. The axial force of the bar at the left end is calculated with two methods: the equilibrium of forces and the principle of minimum potential energy. With the help of the orthogonality of {sinix} (i 1, 2, 3, ) , the Parseval identity and the Bessel inequality of the physical model are obtained. Further, it is proven that, in all the possible displacements which satisfy the boundary conditions, the real one minimizes the total potential energy of the bar. With the equivalence of the two methods, an identity of a type of infinite series is derived. Based on the identity, L(1) , L(3) , a recurrence formula of L(2k 1) (k  3) ,  (2) ,  (4) , and a relationship between  (2k) (k  2) and L(2i 1) (i  1, 2, ..., k) are then deduced with power axial load functions. The upper and lower bounds for the Riemann zeta function  (2k -1) and the L-function L(2k) are given. The numerical results show that the upper and lower bounds are in excellent agreement with the accurate value. Finally, two improvement methods are proposed for estimating the circumference ratio. The numerical results show that the two methods can estimate the circumference ratio with high accuracy and that the L -function is more efficient than the Riemann zeta function in estimating the circumference ratio.