Large-scale replication of the natural process of photosynthesis is a crucial subject of storing solar energy and saving our environment. Here, we report femtosecond laser induced self-assembled metal nanostructure arrays, which are easily mass producible on earthabundant metals, can directly synthesize liquid and solid hydrocarbon compounds from carbon dioxide, water, and sunlight at a production rate of more than 1 × 10 5 µL/(gh) that is significantly (10 3 -10 6 times) higher than those in previous studies. 1,2 The efficiency for storing solar energy of the photosynthesis is about 10% in the present simple experimental setup which can be further improved. Moreover, different from previous artificial photosynthesis works, this phenomenon presents a new mechanism that, through a surface-enhanced photodissociation process, nature-like photosynthesis can be performed artificially.

The “Schur-Siegel-Smyth trace problem” is a famous open problem that has existed for nearly 100 years. To study this problem with the known methods, we need to find all totally positive algebraic integers with small trace. In this work, on the basis of the classical algorithm, we construct a new type of explicit auxiliary functions related to Chebyshev polynomials to give better bounds for S k S_k , and reduce sharply the computing time. We are then able to push the computation to degree 15 15 and prove that there is no such totally positive algebraic integer with absolute trace 1.8 1.8 . As an application, we improve the lower bound for the absolute trace of totally positive algebraic integers to 1.793145 ⋯ 1.793145\cdots .

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Let $\alpha $ be a totally positive algebraic integer of degree d, with conjugates $\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$ . The absolute $S_k$ -measure of $\alpha $ is defined by $s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$ . We compute the lower bounds $\upsilon _k$ of $s_k(\alpha )$ for each integer in the range $2\leq k \leq 15$ and give a conjecture on the results for integers $k>15$ . Then we derive the lower bounds of $s_k(\alpha )$ for all real numbers $k>2$ . Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.