ON GROWTH OF MEROMORPHIC SOLUTIONS OF NONLINEAR DIFFERENCE EQUATIONS AND TWO CONJECTURES OF C.C. YANG

In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a difference polynomial of degree at most n - 1 in f with coefficients. Moreover, we give two examples to show that one conjecture and Laine [2] does not hold in general if the hyper-order of f（z） is no less

Bayesian Equilibrium by Iterative Conjectures as a Refinement of Nash Equilibrium

BEIC （Bayesian equilibrium by iterative conjectures） analyzes games with players forming their conjectures about what other players will do through iterative reasoning starting with first order uninformative conjectures and keep updating their conjectures iteratively with game theoretic reasoning until a convergence of conjectures is achieved. In a BEIC, beliefs about the other players＇ strategies are specified and they are consistent with the equilibrium strategies they supported. A BEIC is therefore a perfect Bayesian equilibrium and hence a refinement of Nash equilibrium. Through six examples, the BE1C solutions are compared with those obtained by the other refining criteria of payoff-dominance, risk-dominance, iterated admissibility, subgame perfect equilibrium, Bayesian Nash equilibrium, perfect Bayesian equilibrium and the intuitive criterion. The outstanding results from the comparisons are that the BEIC approach is able to pick the natural focal point of a game when the iterated admissibility criterion fails to, the BEIC approach rules out equilibrium depending upon non credible threat, and that in simultaneous and sequential games of incomplete information, the BEIC approach not only normally narrows down the equilibriums to one but it also picks the most compelling equilibrium compare with Bayesian Nash equilibrium or perfect Bayesian equilibrium or intuitive criterion.

A Comparison of Sufficiency Condtions for the Goldbach and the Twin Primes Conjectures

It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the representation function over the minor arcs. One of the new results here is that the assumption of GRH can be removed. We compare this and other such sufficiency results with similar results for the Goldbach conjecture.

On the Four Conjectures of G G Lorentz Type in L_ρ~2[-a,a]

In the space C[-1,1],G G Lorentz proposed four conjectures on the properties of the polynomials of the best approximation in 1977,1978 and 1980.The present paper transplants the four conjectures in the space Lρ2[-a,a] and proves them being all right in only one theorem under the corresponding conditions,although each of the original conjectures is very difficulty.

The Proofs of Legendre’s Conjecture and Three Related Conjectures

In this paper, we prove Legendre’s conjecture: There is a prime number between n2 and (n +1)2 for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.

Chaplygin gas inspired warm inflation and swampland conjectures through various scalar potentials

In this paper,we analyze inflationary parameters and swampland conjectures in the presence of a scalar field and Chaplygin models.We examine inflationary parameters,such as slow-roll parameters,scalar and tensor power spectra,spectral index,and tensor-to-scalar ratio,in the presence of a scalar field and Chaplygin gas models.We also discuss recently proposed swampland conjectures.We assume that the inflationary expansion is driven by a standard scalar field with a decay ratioΓthat has a generic power-law dependence on the scalar field 0 and that the temperature of the thermal bath T is given byΓ(φ,T)=C_(φ)T^(a)/φ^(a-1),where C_(φ)is a dimensionless parameter and a is the inflation decay rate.In a scenario where our model operates within a robust dissipative environment(R>>1),we analyze both fundamental and perturbative dynamics to extract key inflationary parameters.These include the scalar power spectrum P_(R),dissipative ratio R,scalar spectral index n_(s),tensor-to-scalar ratio r,running of the scalar spectral index dn_s/dlnk,and generalized ratio of the swampland de-Sitter conjecture T'V/V'T for three different potentials.

Remarks on Hard Lefschetz conjectures on Chow groups Dedicated in celebration of SCIENCE CHINA on the occasion of its 60th birthday

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we show that they are equivalent to the well-known conjectures of Beauville and Murre.

Inflationary Cosmology with Quantum Gravitational Effects and Swampland Conjectures

Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which implies a large tensor-to-scalar ratio is strongly in tension with recent observational results. In this paper, we explore the possibility the swampland conjectures could be compatible with single-field inflationary scenarios if the effects due to the quantum theory of gravity are considered. We show that the quantum gravitational effects due to the nonlinear dispersion relation provides significant modifications on the amplitude of both the scalar and tensor perturbation spectra. Such modifications could be either raise or reduce the perturbation spectra depending on the values of the parameters in the nonlinear terms of the dispersion relations. Therefore, these effects can reduce the tensor-to-scalar ratio to a smaller value, which helps to relax the tension between the swampland conjecture and observational data.

ON A PROBLEM RELATED TO GOLOMB'S CONJECTURES

The main purpose of this paper is to study a problem related to Golomb's conjectures,and give an interesting asymptotic formula.

Some unsolvable conjectures in finite p-groups

We survey some unsolvable conjectures in finite p-groups and their research progress.

Excellent Extensions and Homological Conjectures

In this paper,we introduce the notion of excellent extensions of rings.Let F be an excellent extension of an Artin algebra∧,we prove that∧satisfies the Gorenstein symmetry conjecture(resp.,finitistic dimension conjecture,Auslander-Gorenstein conjecture,Nakayama conjecture)if and only if so does F.As a special case of excellent extensions,when G is a finite group whose order is invertible in∧acting on∧and∧is G-stable,we prove that if the skew group algebra∧G satisfies strong Nakayama conjecture(resp.,generalized Nakayama conjecture),then so does∧.

ON THE LORENTZ CONJECTURES UNDER THE L_1-NORM

Let f （x） ∈ C [-1, 1], pn* （x） be the best approximation polynomial of degree n tof （x）. G. Iorentz conjectured that if for all n, p2n* （x） = p2n+1* （x）, then f is even; and ifp2n+1* （x） = p2n+2* （x）, po* （z） = 0, then f is odd. In this paper, it is proved that, under the L1-norm, the Lorentz conjecture is validconditionally, i. e. if （i） （1-x2） f （x） can be extended to an absolutely convergentTehebyshev sories; （ii） for every n, f （x） - p2n+1* （x） has exactly 2n + 2 zeros (or, in thearcond situation, f （x） - p2n+2* （x） has exaetly 2n+3 zeros), then Lorentz conjecture isvalid.

ON A PROBLEM RELATED TO GOLOMB'S CONJECTURES

The main purpose of this paper is to study a problem related to Golomb’s conjectures, and give an interesting asymptotic formula.

Presentation of the Berry-Tabor conjecture in Levy plates

The Berry-Tabor(BT)conjecture is a famous statistical inference in quantum chaos,which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used to describe other wave phenomena.In this paper,the BT conjecture has been extended to Lévy plates.As predicted by the BT conjecture,level clustering is present in the spectra of Lévy plates.The consequence of level clustering is studied by introducing the distribution of nearest neighbor frequency level spacing ratios P(r),which is calculated through the analytical solution obtained by the Hamiltonian approach.Our work investigates the impact of varying foundation parameters,rotary inertia,and boundary conditions on the frequency spectra,and we find that P(r)conforms to a Poisson distribution in all cases.The reason for the occurrence of the Poisson distribution in the Lévy plates is the independence between modal frequencies,which can be understood through mode functions.

Uniform isochronous center of a class of higher degree polynomial differential systems

In this paper,we give the necessary and sufficient conditions for a class of higher degree polynomial systems to have a uniform isochronous center.At the same time,we prove that for this system the composition conjecture is correct.

How to Prove Riemann Conjecture by Riemann’s Four Theorems

Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).

A New Proof for Congruent Number’s Problem via Pythagorician Divisors

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .

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.

SQL or NoSQL—Practical Aspect and Rational behind Choosing Data Stores

Data storage solutions are a crucial aspect of any application, significantly impacting data management and system performance. This article explores the rationale behind utilizing both SQL and NoSQL databases, addressing key questions about when each type is preferable. The background emphasizes the importance of selecting the appropriate database technology to meet specific application requirements. The purpose of this research is to provide a comprehensive guide for choosing between SQL and NoSQL databases based on various factors, including workload characteristics, scalability needs, and consistency requirements. To achieve this, we examine different strategies for implementing SQL and NoSQL databases in large-scale distributed applications and systems. The research method involves a comparative analysis of the features, advantages, and limitations of both database types. We specifically focus on scenarios involving read-heavy versus write-heavy systems and the trade-offs between availability and consistency. The results of this research indicate that SQL databases, with their relational structure and ACID compliance, are ideal for applications requiring complex queries and data integrity. In contrast, NoSQL databases, offering schema flexibility and horizontal scalability, are better suited for managing extensive datasets and high-velocity data ingestion. In conclusion, the selection of a database depends on the specific needs of the application. SQL databases are preferred for transactional systems with complex relationships, while NoSQL databases excel in scenarios demanding flexibility and scalability. The study provides insights into hybrid approaches, leveraging both database types to optimize system performance.

The Erdös-Faber-Lovász Conjecture for Gap-Restricted Hypergraphs

An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.