Generic character sheaves on reductive groups over a finite ring

In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne--Lusztig characters when the parameters are generic. We formulate a conjecture on the simple perversity of these complexes, and we prove it in the level two case (thus generalised a result of Lusztig from the function field case). We then discuss the induction and restriction functors, as well as the Frobenius reciprocity, based on the perversity.Comment: Add a new result and some minor correction

Machine Ruling

Emerging technologies, such as big data, Internet of things, cloud computing, mobile Internet, and robotics, breed and expedite new applications and fields. In the mean while, the long-term prosperity and happiness of human race demands advanced technologies. In this paper, the aforementioned emerging technologies are applied to management and governance for the long-term prosperity and happiness of human race. The term "machine ruling" is coined, introduced, and justified. Moreover, the framework and architecture of machine ruling are proposed. Enabling technologies and challenges are discussed

A note on cusp forms and representations of SL2(Fp)\mathrm{SL}_2(\mathbb{F}_p)

Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup $\Gamma(p)$, $p$ a prime, is acted by $\mathrm{SL}_2(\mathbb{F}_p)$. Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne--Lusztig (or Drinfeld) curve, whose cohomology space is also acted by $\mathrm{SL}_2(\mathbb{F}_p)$. In this note we compute the relation between these two spaces in the weight $2$ case.Comment: shortened to 6 pages, and Lem~2.2 is upgrade

On the Generative Power of Omega-Grammars and Omega-Automata

An \omega-grammar is a formal grammar used to generate \omega-words (i.e. infinite length words), while an \omega-automaton is an automaton used to recognize \omega-words. This paper gives clean and uniform definitions for \omega-grammars and \omega-automata, provides a systematic study of the generative power of \omega-grammars with respect to \omega-automata, and presents a complete set of results for various types of \omega-grammars and acceptance modes. We use the tuple (\sigma,\rho,\pi) to denote various acceptance modes, where \sigma denotes that some designated elements should appear at least once or infinitely often, \rho denotes some binary relation between two sets, and \pi denotes normal or leftmost derivations. Technically, we propose (\sigma,\rho,\pi)-accepting \omega-grammars, and systematically study their relative generative power with respect to (\sigma,\rho)-accepting \omega-automata. We show how to construct some special forms of \omega-grammars, such as \epsilon-production-free \omega-grammars. We study the equivalence or inclusion relations between \omega$-grammars and \omega-automata by establishing the translation techniques. In particular, we show that, for some acceptance modes, the generative power of \omega-CFG is strictly weaker than \omega-PDA, and the generative power of \omega-CSG is equal to \omega-TM (rather than linear-bounded \omega-automata-like devices). Furthermore, we raise some remaining open problems for two of the acceptance modes

On the inner products of some Deligne--Lusztig type representations

In this paper we introduce a family of Deligne--Lusztig type varieties attached to connected reductive groups over quotients of discrete valuation rings, naturally generalising the higher Deligne--Lusztig varieties and some constructions related to the algebraisation problem raised by Lusztig. We establish the inner product formula between the representations associated to these varieties and the higher Deligne--Lusztig representations.Comment: 14 page

A Classification of 6R Manipulators

This paper presents a classification of generic 6-revolute jointed (6R) manipulators using homotopy class of their critical point manifold. A part of classification is listed in this paper because of the complexity of homotopy class of 4-torus. The results of this classification will serve future research of the classification and topological properties of maniplators joint space and workspace.Comment: 4 page

Characterization of Pentagons Determined by Two X-rays

This paper contains some results of pentagons which can be determined by two X-rays. The results reveal this problem is more complicated.Comment: 4 pages, 2 figure

On the Monotonicity of Work Function in k-Server Conjecture

This paper presents a mistake in work function algorithm of k-server conjecture. That is, the monotonicity of the work function is not always true.Comment: 3 pages, corrected typo

Reconstruction of Missing Big Sensor Data

With ubiquitous sensors continuously monitoring and collecting large amounts of information, there is no doubt that this is an era of big data. One of the important sources for scientific big data is the datasets collected by Internet of things (IoT). It's considered that these datesets contain highly useful and valuable information. For an IoT application to analyze big sensor data, it is necessary that the data are clean and lossless. However, due to unreliable wireless link or hardware failure in the nodes, data loss in IoT is very common. To reconstruct the missing big sensor data, firstly, we propose an algorithm based on matrix rank-minimization method. Then, we consider IoT with multiple types of sensor in each node. Accounting for possible correlations among multiple-attribute sensor data, we propose tensor-based methods to estimate missing values. Moreover, effective solutions are proposed using the alternating direction method of multipliers. Finally, we evaluate the approaches using two real sensor datasets with two missing data-patterns, i.e., random missing pattern and consecutive missing pattern. The experiments with real-world sensor data show the effectiveness of the proposed methods

Family income affects children's altruistic behavior in the dictator game.

This study aimed to examine how family income and social distance influence young rural Chinese children's altruistic behavior in the dictator game (DG). A total of 469 four-year-old children from eight rural areas in China, including many children left behind by parents who had migrated to urban areas for work, played the DG. Stickers comprised the resource, while recipients in the game were assumed to be either their friends or strangers, with the social distance (i.e., strangers compared to friends) as a between-subjects variable. Children donated significantly more stickers to their friends than to strangers. Moreover, children from lower income families donated more stickers than children from higher income families. However, no gender and parental migrant status differences in children's prosocial behaviors were evident in this sample. Findings of this study suggest that children's altruistic behaviours to peers are influenced by family characteristics since preschool age. The probable influence of local socialization practices on development and the possible adaptive significance were discussed