Proof of Collatz Conjecture Using Division Sequence

The purpose of this study is to prove the Collatz conjecture using a theorem proving system. First, the division sequence is defined as an alignment of the number of times division by 2 is performed in the Collatz operation. Then, the star conversion is defined, which is a mapping from a specific division sequence to a division sequence. Here it is important to map to some division sequence, not which division sequence. The important point is that the finite length of the division sequence does not change before and after the star conversion. In theorem proving system, we considered two parallel methods: main-proof is a claim to a computer proposition that has the same meaning as the Collatz conjecture. Theorem proving support system “Idris” was used. Moreover, we sub-proved that the 12 “extended star conversion” are closed to the “Collatz operation”. Egison’s computer algebra system is used for proof. The results of the two methods achieved the goal of proving the Collatz conjecture using a theorem proving system.

STUDY ON EFFICIENT COMPUTATION AND PERFORMANCE OF AV-BASED REDUCED-RANK FILTERING

In this paper, the complexity and performance of the Auxiliary Vector (AV) based reduced-rank filtering are addressed. The AV filters presented in the previous papers have the general form of the sum of the signature vector of the desired signal and a set of weighted AVs,which can be classified as three categories according to the orthogonality of their AVs and the optimality of the weight coefficients of the AVs. The AV filter with orthogonal AVs and optimal weight coefficients has the best performance, but requires considerable computational complexity and suffers from the numerical unstable operation. In order to reduce its computational load while keeping the superior performance, several low complexity algorithms are proposed to efficiently calculate the AVs and their weight coefficients. The diagonal loading technique is also introduced to solve the numerical unstability problem without complexity increase. The performance of the three types of AV filters is also compared through their application to Direct Sequence Code Division Multiple Access (DS-CDM A) systems for interference suppression.

A CLASS OF BAND-LIMITED CHIP WAVEFORMS FOR DS-CDMA SYSTEMS

In Direct Sequence Code Division Multiple Access (DS-CDMA) systems,the chip wave-form affects the implementation,system bandwidth,envelope uniformity,eye pattern and Multiple user Access Interference (MAI). In this paper,based on an elementary density function of a second order polynomial,a class of second order continuity pulses is proposed. From this class of pulses,we can find some members having faster decaying rate,bigger eye opening,more uniform envelope and stronger anti-MAI capability than the Nyquist waveform. The normalized-bandwidth-pulse-shape-factor product,the decaying rate of the tail of the time waveform,the opening of the eye diagram,and the envelope uniformity of the second order continuity pulses are addressed in the paper that provide the basic information for the selection of the chip pulse for CDMA systems.

Characteristics of the Cretaceous Magmatism in Huanghua Depression and Their Relationships with Hydrocarbon Enrichment

The Huanghua depression located in the hinterland of the Bohai Bay Basin in eastern China is a typical area for the research of multistage magmatic activities with hydrocarbon enrichment,many high-yield wells related to igneous rocks were discovered within the Cretaceous strata in recent years.However,the spatial and temporal distribution of Cretaceous igneous rocks remains unclear,and the relationships among magmatic sequence,lithology,and hydrocarbon enrichment have been poorly studied.In order to solve these problems,core observation,logging analysis,major element analysis,zircon U-Pb chronology,oil-bearing grade statistics and reservoir spaces analysis were used to subdivide the magmatism cycles and to investigate the characteristics of igneous reservoirs.Our results show that the Mesozoic magmatism in Huanghua depression started in 140.1±1.4 Ma and could be divided into two stages including the Early Cretaceous stage and Late Cretaceous stage.The Early Cretaceous magmatism principally developed basic-intermediate rocks in the north zone,and could be subdivided into 3 cycles with their forming ages of 140,125–119,and 118–111 Ma,respectively.By contrast,the late stage mainly developed intermediate-acid rocks centralized in the south zone and formed at 75–70 Ma.The GR and SP curves are good indicators for the contrast of different lithologies,lithofacies and for magmatic sequences division.Intensive magmatism may have an advantage to form reservoirs,since basalt in cycle III in the Early Cretaceous and dacite porphyrite in the Late Cretaceous have great exploration potential.Lithology and tectonic fractures have an important influence on the formation of reservoir spaces and hydrocarbon enrichment.The characteristics of Cretaceous magmatism and igneous reservoirs in Huanghua depression and adjacent areas are summarized,providing important information for igneous reservoirs research and oil-gas exploration in the Cretaceous in related areas.