LINEAR EQUATIONS OVER CONES,COLLATZ-WIELANDT NUMBERS AND ALTERNATING SEQUENCES

Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this paper:(i)(λⅠ_n-A)x=b,x∈K,and(ii)(A-λⅠ_n)x=b,x∈K.We obtain several equivalent conditions for the solvability of the first equation.

Weighted (<i>λ</i>, <i>μ</i>)-Ideal Statistical Convergence and Strongly Weighted (<i>λ</i>, <i>μ</i>)-Ideal Convergence of Double Sequences of Fuzzy Numbers

The paper aims to investigate different types of weighted ideal statistical convergence and strongly weighted ideal convergence of double sequences of fuzzy numbers. Relations connecting ideal statistical convergence and strongly ideal convergence have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. At the same time, we have examined relevant inclusion relations concerning weighted (λ, μ)-ideal statistical convergence and strongly weighted (λ, μ)-ideal convergence of double sequences of fuzzy numbers. Also, some properties of these new sequence spaces are investigated.

Clinical manifestations and the prenatal diagnosis of trisomy 7 mosaicism:Two case reports

BACKGROUND The clinical manifestations of trisomy 7 mosaicism are diverse and nonspecific,so prenatal diagnosis is very difficult.CASE SUMMARY Two pregnant women with abnormal prenatal screening results were included.One was a 22-year-old woman(G1P0).At 31st week of gestation,ultrasound revealed that the posterior horn of the left lateral ventricle was 10 mm and the right renal pelvis had a separation of 7 mm.The other pregnant woman was 33 years old(G2P1L1A0),and her fetus was found to have a cardiac malformation at the 24th week of gestation.Copy number variation sequencing,whole-exome sequencing and karyotype analysis were carried out after amniocentesis,and both fetuses were diagnosed with trisomy 7 mosaicism.After parental counseling,one woman continued the pregnancy,and the other woman terminated the pregnancy.CONCLUSION In trisomy 7 mosaicism,the low proportion of trisomy does not lead to abortion,but can result in abnormal fetal development,which can be detected via ultrasound.Therefore,clinicians need to pay more attention to various aspects of fetal growth and development,combining with imaging,cellular,molecular genetics and other methods to perform comprehensive evaluations of fetuses to provide more reliable genetic counseling for pregnant women.

The Formula of General Term of the Sum of Equal Powers of Natural Numbers

The formula of general term of the sum of equal powers of natural numbers.

Constructions with Numbers

Pascal Triangle is more of a number construction （body） then an array of the binomial coefficients. It is a mathematical body, like the digital code feeds for computer but with 2 dimensions. And there should be bodies with x-dimensions and even abnormal or irregular appearances.

Confusing finding of quantitative fluorescent polymerase chain reaction analysis in invasive prenatal genetic diagnosis:A case report

BACKGROUND Quantitative fluorescent polymerase chain reaction(QF-PCR)is a rapid prenatal diagnostic method for abnormalities on chromosomes 21,18,and 13 and sex chromosomal aneuploidy.However,the value of QF-PCR in diagnosing chromosomal structural abnormalities is limited.In this article,we report a confusing QF-PCR finding in a pregnant woman who underwent amniocentesis.CASE SUMMARY The short tandem repeat marker AMXY(Xp22.2/Yp11.2)located on the sex chromosome exhibited a trisomic biallelic pattern,indicating that the karyotype of the fetus might be 47,XYY.Chromosome analysis performed on cultured amniocytes showed a normal male karyotype of the fetus.Copy number variation sequencing confirmed a 500 kb duplication at Yp11.2-Yp11.2(chrY:6610001_7110000)and a 250 kb duplication at Yp11.2-Yp11.2(chrY:7110001_7360000).CONCLUSION In conclusion,the comprehensive application of different methods could achieve a higher detection rate and accuracy for the prenatal diagnosis of chromosomal disorders through chromosomal testing.

Phase Transitions Governed by the Fifth Power of the Golden Mean and Beyond

In this contribution results from different disciplines of science were compared to show their intimate interweaving with each other having in common the golden ratio φ respectively its fifth power φ5. The research fields cover model calculations of statistical physics associated with phase transitions, the quantum probability of two particles, new physics of everything suggested by the information relativity theory (IRT) including explanations of cosmological relevance, the ε-infinity theory, superconductivity, and the Tammes problem of the largest diameter of N non-overlapping circles on the surface of a sphere with its connection to viral morphology and crystallography. Finally, Fibonacci anyons proposed for topological quantum computation (TQC) were briefly described in comparison to the recently formulated reverse Fibonacci approach using the Janičko number sequence. An architecture applicable for a quantum computer is proposed consisting of 13-step twisted microtubules similar to tubulin microtubules of living matter. Most topics point to the omnipresence of the golden mean as the numerical dominator of our world.

On the Space bv_p(F) of Sequences of p-bounded Variation of Fuzzy Numbers

Recently, the space bvp of real or complex numbers consisting of all sequences whose differences are in the space lp has been studied by Basar, Altay [Ukrainian Math. J. 55（1）（2003）, 136-147], where 1 ≤ p ≤ ∞. The main purpose of the present paper is to introduce the space bvp（F） of sequences of p-bounded variation of fuzzy numbers. Moreover, it is proved that the space bvp（F） includes the space lp（F） and also shown that the spaces bvp（F） and lp（F） axe isomorphic for 1 ≤ p ≤∞. Furthermore, some inclusion relations have been given.

Number Sequencing and Architecture

In Chinese, Mathematics is “数学”, i.e. Numbers （数） Knowledge （学）. Human identifies us ourselves from the whole one of the Universe by means of numbers, and numbers are equal object status identities, and numbers counting means repetition, i.e. the process or time dimension. Given a series of n numbers, a1 , a2 , … , an , n∈N, i.e. the n objects of status, and a general formula a（i） for ordering all of the number terms by i （i=1 to n） can be generated, which shows a process of counting, and from sequencing to progression, some properties are extracted.

The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable-With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence

The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes—that is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and so on. The main conclusion revolves entirely around two points. First, on the one hand, it is shown that the prime sequence exhibits an extremely high level of organization. But second, on the other hand, it is also shown that the clearly detectable organization of the primes is ultimately beyond human comprehension. This conclusion runs radically counter and opposite—in regard to both points—to what may well be the default view held widely, if not universally, in current theoretical mathematics about the prime sequence, namely the following. First, on the one hand, the prime sequence is deemed by all appearance to be entirely random, not organized at all. Second, on the other hand, all hope has not been abandoned that the sequence may perhaps at some point be grasped by human cognition, even if no progress at all has been made in this regard. Current mathematical research seems to be entirely predicated on keeping this hope alive. In the present paper, it is proposed that there is no reason to hope, as it were. According to this point of view, theoretical mathematics needs to take a drastic 180-degree turn. The manner of demonstration that will be used is direct and empirical. Two key observations are adduced showing, 1), how the prime sequence is highly organized and, 2), how this organization transcends human intelligence because it plays out in the dimension of infinity and in relation to π. The present paper is part of a larger project whose design it is to present a complete and final mathematical and physical theory of rational human intelligence. Nothing seems more self-evident than that rational human intelligence is subject to absolute limitations. The brain is a material and physically finite tool. Everyone will therefore readily agree that, as far as reasoning is concerned, there are things that the brain can do and things that it cannot do. The search is therefore for the line that separates the two, or the limits beyond which rational human intelligence cannot go. It is proposed that the structure of the prime sequence lies beyond those limits. The contemplation of the prime sequence teaches us something deeply fundamental about the human condition. It is part of the quest to Know Thyself.

Validation of the diagnostic potential of mtDNA copy number derived from whole genome sequencing

Diagnosis of mitochondrial DNA（mt DNA）disorders has traditionally been focused on the presence of point mutations and large deletions.However,deviations in mitochondrial abundance or mt DNA copy number can also be associated with many physiological and pathological conditions（Bai and Wong,2005）.