Logic and Paradox

This article discusses the logical errors in the liar paradox, G\"odel's incompleteness theorems, Russell's paradox, and the halting problem. In order to avoid these errors, a redefinition of logic has been presented, which is concluded as four principles in set-theoretic language, including 1. Don't talk about empty; 2. Elements of set should have identity; 3. Sets should have definitions; and 4. A total set should be defined. The new definition of logic can eliminate all paradoxes and invalid statements, thus becoming a solid foundation for human language and knowledge

An upper bound for the crossing number of augmented cubes

A {\it good drawing} of a graph $G$ is a drawing where the edges are non-self-intersecting and each two edges have at most one point in common, which is either a common end vertex or a crossing. The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a good drawing of $G$ in the plane. The {\it $n$-dimensional augmented cube} $AQ_n$, proposed by S.A. Choudum and V. Sunitha, is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of $AQ_n$ less than $26/324^{n}-(2n^2+7/2n-6)2^{n-2}$.Comment: 39 page

Dense Granular Flow as Heat Transfer Media: A New Type of High Power Target Design

High power target systems require reliable high-temperature heat transfer media. Dense granular flow materials have potential to work as heat transfer media, especially at high temperature. In this chapter, a brief review of dense granular and their heat transfer properties is introduced, including basic concepts of heat transfer, thermal behavior of these materials, and factors that affect this behavior. The implementation of these materials as targets is addressed, where two targets were designed, constructed, and tested based on the concept of dense granular flow. The results of the application of a hopper flow-type target in 2.5 MW accelerator-driven system (ADS) and a chute flow-type target in the material irradiation facility will be presented

CDO pricing using single factor MG-NIG copula model with stochastic correlation and random factor loading

AbstractWe consider the valuation of CDO tranches with single factor MG-NIG copula model, where the involved distributions are mixtures of Gaussian distribution and NIG distribution. In addition, we consider two cases for stochastic correlation and random factor loadings instead of constant factor loadings. We analyze the unconditional characteristic function of accumulated loss of the reference portfolio, and derive the loss distribution through the fast Fourier transform. Moreover, using the loss distribution and semi-analytic approach, we can get the CDO tranches spreads

Recitation-Augmented Language Models

We propose a new paradigm to help Large Language Models (LLMs) generate more accurate factual knowledge without retrieving from an external corpus, called RECITation-augmented gEneration (RECITE). Different from retrieval-augmented language models that retrieve relevant documents before generating the outputs, given an input, RECITE first recites one or several relevant passages from LLMs' own memory via sampling, and then produces the final answers. We show that RECITE is a powerful paradigm for knowledge-intensive NLP tasks. Specifically, we show that by utilizing recitation as the intermediate step, a recite-and-answer scheme can achieve new state-of-the-art performance in various closed-book question answering (CBQA) tasks. In experiments, we verify the effectiveness of RECITE on three pre-trained models (PaLM, UL2, and OPT) and three CBQA tasks (Natural Questions, TriviaQA, and HotpotQA)