Affine function valued valuations

A classification of SL$(n)$ contravariant, continuous function valued valuations on convex bodies is established. Such valuations are natural extensions of SL$(n)$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0< p < 1$ and in the Orlicz Brunn-Minkowski theory.Comment: accepted by Int. Math. Res. No

Directional dark matter by polar angle direct detection and application of columnar recombination

We report a systematic study on the directional sensitivity of a direct dark matter detector that detects the polar angle of a recoiling nucleus. A weakly interacting massive particle (WIMP)-mass independent method is used to obtain the sensitivity of a general detector in an isothermal galactic dark matter halo. By using two-dimensional distributions of energy and polar angle, a detector without head-tail information with 6.3 times the statistics is found to achieve the same performance level as a full three-dimensional tracking dark matter detector. Optimum operation orientations are obtained for various experimental configurations, with detectors that are space- or Earth-fixed, have head-tail capability or not, and use energy information or not. Earth-fixed detectors are found to have best sensitivity when the polar axis is oriented at a 45 degree angle from the Earth's pole. With background contamination that mimics the WIMP signal's energy distribution, the performance is found to decrease at a rate less than the decrease of signal purity. The WIMP-mass dependence of the performance of a detector with various energy thresholds that uses gaseous xenon as target material is reported. We find that with a $5\times 10^{-46} \mathrm{cm}^2$ spin-independent WIMP-nucleon cross-section and a 30 GeV WIMP, a $770$ kg$\cdot$year's exposure with a polar detector of 10 keV threshold can make a three sigma discovery of directional WIMPs in the isothermal galactic dark matter halo. For a columnar recombination detector, experimental considerations are discussed.Comment: 10 pages, 6 figures, 1 tabl

On the trace of Hecke operators for Maass forms for congruence subgroups II

Let E_lambda be the Hilbert space spanned by the eigenfunctions of the non-Euclidean Laplacian associated with a positive discrete eigenvalue lambda. In this paper, the trace of Hecke operators T_n acting on the space E_lambda is computed for Hecke congruence subgroups Gamma_0(N) of non-square free level. This extends the computation of Conrey-Li [2], where only Hecke congruence subgroups Gamma_0(N) of square free level N were considered.Comment: 32 page

A note on zeros of LL-series of elliptic curves

In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an affirmative answer would imply the analogue of the Riemann hypothesis for elliptic curves over the rational number field

A residue scalar product for algebraic function fields over a number field

In 1952 Peter Roquette gave an arithmetic proof of the Riemann hypothesis for algebraic function fields of a finite constants field, which was proved by Andr\'e Weil in 1940. The construction of Weil's scalar product is essential in Roquette's proof. In this paper a scalar product for algebraic function fields over a number field is constructed which is the analogue of Weil's scalar product

An explicit formula for Hecke LL-functions

In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials, which are associated with the space of cusp forms of weight $k$ for Hecke congruence subgroups, lie on the critical line

Zero-determinant strategies in iterated multi-strategy games

Self-serving, rational agents sometimes cooperate to their mutual benefit. The two-player iterated prisoner's dilemma game is a model for including the emergence of cooperation. It is generally believed that there is no simple ultimatum strategy which a player can control the return of the other participants. The recent discovery of the powerful class of zero-determinant strategies in the iterated prisoner's dilemma dramatically expands our understanding of the classic game by uncovering strategies that provide a unilateral advantage to sentient players pitted against unwitting opponents. However, strategies in the prisoner's dilemma game are only two strategies. Are there these results for general multi-strategy games? To address this question, the paper develops a theory for zero-determinant strategies for multi-strategy games, with any number of strategies. The analytical results exhibit a similar yet different scenario to the case of two-strategy games. Zero-determinant strategies in iterated prisoner's dilemma can be seen as degenerate case of our results. The results are also applied to the snowdrift game, the hawk-dove game and the chicken game

Estimates and Existence Results for a Fully Nonlinear Yamabe Problem on Manifolds with Boundary

This paper concerns a fully nonlinear version of the Yamabe problem on manifolds with boundary. We establish some existence results and estimates of solutions

Partitioning complete graphs by heterochromatic trees

A {\it heterochromatic tree} is an edge-colored tree in which any two edges have different colors. The {\it heterochromatic tree partition number} of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum positive integer $p$ such that whenever the edges of the graph $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $p$ vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of an $r$-edge-colored complete graph.Comment: 7 page

A Hopf's Lemma and the Boundary Regularity for the Fractional P-Laplacian

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions to a fractional p-Laplacian equation possess certain Holder continuity up to the boundary