On Nontrivial Zeros of Riemann Zeta Function

Let {\Xi} be a function relating to the Riemann zeta function with . In this paper, we construct a function containing and {\Xi} , and prove that satisfies a nonadjoint boundary value problem to a nonsingular differential equation if is any nontrivial zero of {\Xi} . Inspecting properties of and using known results of nontrivial zeros of , we derive that nontrivial zeros of all have real part equal to , which concludes that Riemann Hypothesis is true.Comment: 14Page

On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

In this paper, we are interested in developing polynomial decomposition techniques to reformulate real valued multivariate polynomials into difference-of-sums-of-squares (namely, D-SOS) and difference-of-convex-sums-of-squares (namely, DC-SOS). Firstly, we prove that the set of D-SOS and DC-SOS polynomials are vector spaces and equivalent to the set of real valued polynomials. Moreover, the problem of finding D-SOS and DC-SOS decompositions are equivalent to semidefinite programs (SDP) which can be solved to any desired precision in polynomial time. Some important algebraic properties and the relationships among the set of sums-of-squares (SOS) polynomials, positive semidefinite (PSD) polynomials, convex-sums-of-squares (CSOS) polynomials, SOS-convex polynomials, D-SOS and DC-SOS polynomials are discussed. Secondly, we focus on establishing several practical algorithms for constructing D-SOS and DC-SOS decompositions for any polynomial without solving SDP. Using DC-SOS decomposition, we can reformulate polynomial optimization problems in the realm of difference-of-convex (DC) programming, which can be handled by efficient DC programming approaches. Some examples illustrate how to use our methods for constructing D-SOS and DC-SOS decompositions. Numerical performance of D-SOS and DC-SOS decomposition algorithms and their parallelized methods are tested on a synthetic dataset with 1750 randomly generated large and small sized sparse and dense polynomials. Some real-world applications in higher order moment portfolio optimization problems, eigenvalue complementarity problems, Euclidean distance matrix completion problems, and Boolean polynomial programs are also presented.Comment: 47 pages, 19 figure

On a Conjecture of Cai-Zhang-Shen for Figurate Primes

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer n > 1 is the sum of two figurate primes. In this paper we give respectively equivalent propositions to the conjecture in the cases of even and odd integers and then confirm the conjecture by considering functions with several variables.Comment: 12page

Retail Bottle Pricing at the Border: Evidence of Cross-Border Shopping, Fraudulent Redemptions, and Use Tax Evasion

This paper examines the pattern of retail prices for deposit eligible goods near Michigan’s borders. Michigan’s unique bottle redemption system and lower sales tax generate incentives for various potentially illegal household responses. Such incentives and behavior should be capitalized in the prices of affected goods. I empirically quantify the spatial price effects and find patterns consistent with theoretical predictions. Michigan’s border prices are higher (lower) for goods with higher (lower) per unit costs by up to 38%. Price-distance trends reflect the waning of these effects away from the border