Natural Numbers and the Strong Goldbach Conjecture

This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.

Identities on q-Harmonic Numbers

With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.

On Prime Numbers between kn and (k + 1) n

In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

A General Representation of Hankel Matrix about Bell Numbers

The purpose of this note is to establish a general representation of Hankel matrices of Bell numbers and the convoluted Bell numbers. As a special case, the results of Aigner are extended.

Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory.

On Polynomials Rn(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the p-Adic Integral on

In this paper, one introduces the polynomials Rn(x) and numbers Rn and derives some interesting identities related to the numbers and polynomials: Rn and Rn(x). We also give relation between the Stirling numbers, the Bell numbers, the Rn and Rn(x).

A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable

In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions are given.By using these generating functions and some identities,relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials,Stirling numbers are presented.Computational formulae for these polynomials are obtained.Applying a partial derivative operator to these generating functions,some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained.In addition,some remarks and observations on these polynomials are given.

Strong Law of Large Numbers and Complete Convergence for Sequences of -Mixing Random Variables

We give some theorems of strong law of large numbers and complete convergence for sequences of φ-mixing random variables. In particular, Wittmann＇s strong law of large numbers and Teicher＇s strong law of large nnumbers for independent random variables are generalized to the case of φ -minxing random variables.

Recovery from Bell’s palsy after treatment using uncultured umbilical cord-derived mesenchymal stem cells:A case report

BACKGROUND Bell’s palsy is an idiopathic facial palsy with an unknown cause,and 75%of patients heal spontaneously.However,the other 25%of patients continue experiencing mild or severe disabilities,resulting in a reduced quality of life.Currently,various treatment methods have been developed to treat this disease.However,there is controversy regarding their effectiveness,and new alternative treatments are needed.CASE SUMMARY The patient suffered from left-sided facial paralysis due to Bell’s palsy for 7 years.The patient received an uncultured umbilical cord-derived mesenchymal stem cell transplant eight times for treatment.After follow-up for 32 mo,the paralysis was cured,and there was no recurrence.CONCLUSION Uncultured umbilical cord-derived mesenchymal stem cell transplantation may be a potential treatment for patients with Bell’s palsy who do not spontaneously recover.

Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

Bell’s Palsy Post COVID-19 Vaccination: A Retrospective Cross-Sectional Study in a Single Tertiary Center

Introduction: Bell’s palsy is an uncommon adverse effect of the COVID-19 vaccine that has been reported in clinical trials. Even though a few studies have linked the vaccination to Bell’s palsy, the actual mechanism is uncertain. Objectives: To describe the demographic data and COVID-19 vaccines-related data with Bell’s palsy in a tertiary centre of Malaysia, Hospital Kuala Lumpur. Methods: A retrospective cross-sectional study was observed among vaccinated recipients who developed Bell’s palsy within 60 days and sought treatment in the Otorhinolaryngology Department Hospital Kuala Lumpur, Malaysia between 1st May 2021 and 30th November 2021. The demographic data, clinical history, and vaccination history were collected from clinical records. The facial paralysis was graded according to the House-Brackmann grading system. Results: A total of 26 patients with a mean age was 38.5 years;higher incidence in younger age, below 60 years old (n = 24), specifically 18 - 30 years old (n = 11). We observed an equal number in relation to gender and onset (after the first or second dose) of facial palsy. Predominantly were Malay (n = 21) and only 6 patients had comorbidities. We found there was no difference in regard to the type of vaccine among Bell’s palsy patients;Pfizer (n = 9), followed by Sinovac (n = 9) and AstraZeneca (n = 8). Conclusion: Bell’s palsy was found to be a possible adverse event of the COVID-19 vaccine. Younger groups were noted as susceptible to this rare adverse effect. However, the benefits of vaccination outweigh the risk of Bell’s palsy, which has a good prognosis. More research with larger samples is needed to determine the true relationship between vaccination and Bell’s palsy.

Einstein-Local Counter-Arguments and Counter-Examples to Bell-Type Proofs

J. S. Bell’s well-known proofs of inequalities (and related work) are shown to be invalidated by two counter-arguments (-examples) that are based on Einstein-local propositions: Bell-type inequalities of Einstein-Podolsky-Rosen experiments must include, as virtually all physical theories do, elements of physical reality and their mathematical representations that relate to continua as opposed to exclusively finite numbers. Furthermore, Bell-type inequalities must be valid for all possible experimental geometries that lead to the quantum result. Based on these propositions, violations of Bell-type inequalities are demonstrated without violating Einstein locality, without conspiracy type theories and even for the case that all known “loopholes” are closed.

The Theoretical Significance and Reality of Imaginary Number

It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary numbers have been applied in many theoretical theories. One of the biggest functions of imaginary numbers is to represent changes in phase, which is indispensable in signal analysis theory. The imaginary numbers in quantum mechanics pose a greater mystery: do the imaginary numbers really exist? This question still needs further scientific development to be answered.

Mass Constituents of a Flat Lattice Multiverse: Conclusion from Similarity between Two Universal Numbers, the Rocksalt-Type 2<i>D</i>Madelung Constant and the Golden Mean

In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean &phi;=0.6180339887, yielding for dark energy to matter mass fractions .?Assuming the baryonic matter to be only 4.432%, the ratio of matter to baryonic matter would be , and further the ratio of dark matter to baryonic one . If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajdukovic, the remaining mass ratio would yield . Replacing the “Madelung” constant α of Villata’s “lattice universe” by &phi;, one reaches again 1 + &phi;as the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant and φ&minus;1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the square of the most irrational universal number &phi;may indicate that the chaotic cosmological processes have reached a quite stable equilibrium. This may be confirmed by another, but similar representation of the mass constituents by the Archimedes’ constant &pi;, giving for respectively for the dark components . However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.

Misdiagnosis of Facial Nerve Palsy: A Case Report of Sporadic Vestibular Schwannomas (Non-NFM2) Causing Facial Nerve Palsy Be Misled by Bell’s Palsy in a Child

We report a case of a rare sporadic Vestibular Schwannoma of a 9-month-old girl who had a right-sided lower motor type facial nerve palsy. The patient was initially diagnosed with Bell’s palsy and received steroid treatment accordingly, two months later the patient’s condition deteriorated, and further evaluation of CT and MRI brain was conducted that showed a mass lesion in the posterior fossa causing compression on the facial nerve. Misdiagnosis of facial nerve paralysis is common among children due to multiple related etiologies and varying rates of incidence in comparison to adults. The authors hope to address this issue in this report. Background: Facial nerve paralysis has been a matter of concern for many researchers to understand its nature, causes and presentation according to different age groups. In adults, Bell’s palsy (BP), the idiopathic form of facial nerve paralysis, is more common compared to children where most cases are due to secondary etiologies. Therefore, pediatricians are in an important position to identify these patients early in order to launch the most effective diagnostic and treatment approaches.

Biased Random Number Generator Based on Bell's Theorem

We propose a biased random number generation protocol whose randomness is based on the violation of the Clauser Home inequality. Non-maximally entangled state is used to maximize the Bell violation. Due to the rotational asymmetry of the quantum state, the ratio of Os to ls varies with the measurement bases. The experimental partners can then use their measurement outcomes to generate the biased random bit string. The bias of their bit string can be adjusted by altering their choices of measurement bases. When this protocol is implemented in a device-independent way, we show that the bias of the bit string can still be ensured under the collective attack.

Logical Conflict between Bell’s Locality and Probability Theory

Based on “locality” considerations, John Stuart Bell and his followers have derived inequalities and theorems that, when taken together with actual experiments that have been performed by Aspect and others, appear to contradict physical reality as defined by Einstein. However, their specifically applied concept of locality is in conflict with the Fundamental Model of probability theory and the set theoretic definition of conditional probabilities. Bell-type inequalities are, therefore, not adequate to decide ponderous questions regarding physical reality.

Why Bell’s EPRBA Experiment Is Meaningless

We demonstrate that a Bell type of experiment asks the impossible of a Kolmogorovian correlation. An Einstein locality explanation in Bell’s format is therefore excluded beforehand by way of the experimental and statistical method followed.

Bell’s Theorem and Einstein’s Worry about Quantum Mechanics

With the use of a local dependency on instrument setting parameters of the probability density of local hidden variables, it is demonstrated that a Kolmogorov formulation reproduces the quantum correlation. This is the novelty of the work. In a Bell experiment, one cannot distinguish between Bell’s formula and the here presented local Kolmogorov formula. With the presented formula, no CHSH can be obtained. Therefore, the famous CHSH inequality has no excluding power concerning local extra Einstein parameter models. This result concurs with other previous research concerning difficulties with Bell’s formula.