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 im...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.展开更多
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 di...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展开更多
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 conject...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.展开更多
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 repr...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.展开更多
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...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.展开更多
In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The m...In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> 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.展开更多
In recent years,artificial intelligence(AI)has achieved tremendous development,akin to a significant leap,similar to progressing from 1 to 100.However,a significant gap still exists between current machine intelligenc...In recent years,artificial intelligence(AI)has achieved tremendous development,akin to a significant leap,similar to progressing from 1 to 100.However,a significant gap still exists between current machine intelligence and human wisdom:machine intelligence is constrained to post hoc inference based on existing data,lacking the ability for genuine exploratory innovation and possessing no prospective reasoning inherent to human wisdom.Drawing inspiration from human wisdom,this article presents conjectures for overcoming the four dilemmas faced by machine intelligence:neglect of silicon-based cognition,lack of artistry,pitfall of perfectionism,and obsession with uniformity.These conjectures aim to propel machine intelligence toward machine wisdom,achieving a great leap from 1 to i.展开更多
In this study,we revisit supersymmetric(SUSY)hybrid inflation within the context of considering the latest Cosmic Microwave Background(CMB)observations and swampland conjectures.We demonstrate that SUSY hybrid inflati...In this study,we revisit supersymmetric(SUSY)hybrid inflation within the context of considering the latest Cosmic Microwave Background(CMB)observations and swampland conjectures.We demonstrate that SUSY hybrid inflation remains consistent with Planck 2018 data when radiative,soft mass,and supergravity(SUGRA)corrections are applied to the scalar potential.It is commonly perceived that SUSY hybrid inflation with a minimal Kahler potential results in a gauge symmetry breaking scale M of O(10^(15))GeV,leading to issues with the proton decay rate.In our analysis,we introduce a novel parameter space that alleviates the proton decay issue by achieving M~10^(16) GeV with M_(S)^(2)<0 and am_(3/2)>0.This scenario necessitates a soft SUSY breaking scale|M_(S)|■10^(6) GeV.Further,we find that the tensor-to-scalar ratio r spans from 10^(-16) to 10^(-6),indicating a very small value.This small ratio allows the modified swampland criteria to hold,although satisfying the trans-Planckian censorship conjecture(TCC)remains challenging.To address this,we also explo non-minimal Kahler potentials.By fixing the spectral index at n_(S)=0.9665,consistent with the central value of Planck 2018 data,and setting M=2×10^(16)GeV,we present our calculations.We show that the canonical measure of primordial gravitational waves,r,for M_(S)=1 TeV,m_(3/2)=1 TeV,and k_(S)<0 for N=1 and N=2,ranges from 10^(-5) to 0.01,rendering it detectable by Planck and upcoming experiments such as LiteBIRD,Simons Observatory,PRISM,PIXIE,CORE,CMB-S4,and CMB-HD.Additionally,we outline the parametric space and provide benchmark points for the non-minimal case to ensure compatibility with both the modified swampland conjecture and TCC.展开更多
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...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.展开更多
A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vert...A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vertex.This concept was introduced by Tutte as an extension of face colorings,and Tutte in 1954 conjectured that every bridgeless graph admits a nowhere-zero 5-flow,known as the 5-Flow Conjecture.This conjecture is verified for some graph classes and remains unresolved as of today.In this paper,we show that every bridgeless graph of Euler genus at most 20 admits a nowhere-zero 5-flow,which improves several known results.展开更多
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.
We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through developm...We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.展开更多
This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved...This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved yet.First,we investigate the height functions and their properties of the position vector and normal vector in natural coordinate vectors,and then prove the existence of a Simons-type integral formula on the hypersurface that simultaneously includes the first,second,and third gap point terms of S.These results can provide new avenues of thought and methods for solving Chern's conjecture.展开更多
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 conje...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∧.展开更多
Let f （x） ∈ C [-1, 1], p<sub>n</sub><sup>*</sup> （x） be the best approximation polynomial of degree n tof （x）. G. Iorentz conjectured that if for all n, p<sub>2n</sub><sup...Let f （x） ∈ C [-1, 1], p<sub>n</sub><sup>*</sup> （x） be the best approximation polynomial of degree n tof （x）. G. Iorentz conjectured that if for all n, p<sub>2n</sub><sup>*</sup> （x） = p<sub>2n+1</sub><sup>*</sup> （x）, then f is even; and ifp<sub>2n+1</sub><sup>*</sup> （x） = p<sub>2n+2</sub><sup>*</sup> （x）, p<sub>o</sub><sup>*</sup> （z） = 0, then f is odd. In this paper, it is proved that, under the L<sub>1</sub>-norm, the Lorentz conjecture is validconditionally, i. e. if （i） （1-x<sup>2</sup>） f （x） can be extended to an absolutely convergentTehebyshev sories; （ii） for every n, f （x） - p<sub>2n+1</sub><sup>*</sup> （x） has exactly 2n + 2 zeros (or, in thearcond situation, f （x） - p<sub>2n+2</sub><sup>*</sup> （x） has exaetly 2n+3 zeros), then Lorentz conjecture isvalid.展开更多
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, addres...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.展开更多
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...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.展开更多
基金Supported by National Natural Science Foundation of China under Grant No.11675143the Fundamental Research for the Provincial Universities of Zhejiang in China under Grant No.RF-A2019015
文摘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.
基金supported by the NNSF of China(11171013,11371225,11201014)the YWF-14-SXXY-008 of Beihang Universitythe Fundamental Research Funds for the Central University
文摘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
文摘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.
文摘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.
文摘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.
文摘In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> 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.
基金supported by the National Natural Science Foundation of China(NSFC72421002,72025405,and 62206303)+2 种基金the Science and Technology Innovation Program of Hunan Province(2023RC3009)the Science and Technology Project for Young andMiddle-aged Talents of Hunan(2023TJ-Z03)the Scientific Project ofNUDT(23-ZZCX-JDZ-28 and JS24-05).
文摘In recent years,artificial intelligence(AI)has achieved tremendous development,akin to a significant leap,similar to progressing from 1 to 100.However,a significant gap still exists between current machine intelligence and human wisdom:machine intelligence is constrained to post hoc inference based on existing data,lacking the ability for genuine exploratory innovation and possessing no prospective reasoning inherent to human wisdom.Drawing inspiration from human wisdom,this article presents conjectures for overcoming the four dilemmas faced by machine intelligence:neglect of silicon-based cognition,lack of artistry,pitfall of perfectionism,and obsession with uniformity.These conjectures aim to propel machine intelligence toward machine wisdom,achieving a great leap from 1 to i.
文摘In this study,we revisit supersymmetric(SUSY)hybrid inflation within the context of considering the latest Cosmic Microwave Background(CMB)observations and swampland conjectures.We demonstrate that SUSY hybrid inflation remains consistent with Planck 2018 data when radiative,soft mass,and supergravity(SUGRA)corrections are applied to the scalar potential.It is commonly perceived that SUSY hybrid inflation with a minimal Kahler potential results in a gauge symmetry breaking scale M of O(10^(15))GeV,leading to issues with the proton decay rate.In our analysis,we introduce a novel parameter space that alleviates the proton decay issue by achieving M~10^(16) GeV with M_(S)^(2)<0 and am_(3/2)>0.This scenario necessitates a soft SUSY breaking scale|M_(S)|■10^(6) GeV.Further,we find that the tensor-to-scalar ratio r spans from 10^(-16) to 10^(-6),indicating a very small value.This small ratio allows the modified swampland criteria to hold,although satisfying the trans-Planckian censorship conjecture(TCC)remains challenging.To address this,we also explo non-minimal Kahler potentials.By fixing the spectral index at n_(S)=0.9665,consistent with the central value of Planck 2018 data,and setting M=2×10^(16)GeV,we present our calculations.We show that the canonical measure of primordial gravitational waves,r,for M_(S)=1 TeV,m_(3/2)=1 TeV,and k_(S)<0 for N=1 and N=2,ranges from 10^(-5) to 0.01,rendering it detectable by Planck and upcoming experiments such as LiteBIRD,Simons Observatory,PRISM,PIXIE,CORE,CMB-S4,and CMB-HD.Additionally,we outline the parametric space and provide benchmark points for the non-minimal case to ensure compatibility with both the modified swampland conjecture and TCC.
文摘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.
文摘A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vertex.This concept was introduced by Tutte as an extension of face colorings,and Tutte in 1954 conjectured that every bridgeless graph admits a nowhere-zero 5-flow,known as the 5-Flow Conjecture.This conjecture is verified for some graph classes and remains unresolved as of today.In this paper,we show that every bridgeless graph of Euler genus at most 20 admits a nowhere-zero 5-flow,which improves several known results.
文摘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.
文摘We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.
文摘This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved yet.First,we investigate the height functions and their properties of the position vector and normal vector in natural coordinate vectors,and then prove the existence of a Simons-type integral formula on the hypersurface that simultaneously includes the first,second,and third gap point terms of S.These results can provide new avenues of thought and methods for solving Chern's conjecture.
文摘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∧.
文摘Let f （x） ∈ C [-1, 1], p<sub>n</sub><sup>*</sup> （x） be the best approximation polynomial of degree n tof （x）. G. Iorentz conjectured that if for all n, p<sub>2n</sub><sup>*</sup> （x） = p<sub>2n+1</sub><sup>*</sup> （x）, then f is even; and ifp<sub>2n+1</sub><sup>*</sup> （x） = p<sub>2n+2</sub><sup>*</sup> （x）, p<sub>o</sub><sup>*</sup> （z） = 0, then f is odd. In this paper, it is proved that, under the L<sub>1</sub>-norm, the Lorentz conjecture is validconditionally, i. e. if （i） （1-x<sup>2</sup>） f （x） can be extended to an absolutely convergentTehebyshev sories; （ii） for every n, f （x） - p<sub>2n+1</sub><sup>*</sup> （x） has exactly 2n + 2 zeros (or, in thearcond situation, f （x） - p<sub>2n+2</sub><sup>*</sup> （x） has exaetly 2n+3 zeros), then Lorentz conjecture isvalid.
文摘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.
基金supported by the National Natural Science Foundation of China(Grant Nos.12261064 and 11861048)the Natural Science Foundation of Inner Mongolia,China(Grant No.2021MS01004)the Innovation Program for Graduate Education of Inner Mongolia University(Grant No.11200-5223737).
文摘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.
