Corresponding Prime Number Distribution Equation

The conjecture of twin prime numbers is a mathematical problem. Proving the twin prime conjecture using traditional modern number theory is extremely profound and complex. We propose an elementary research method for corresponding prime number, proved that the conjecture of twin prime numbers and obtain the corresponding prime distribution equation. According to the distribution rate of corresponding prime numbers, the distribution pattern of twin prime numbers was proved the distribution rate theorem. This is the distribution rate of prime numbers corresponding to composite numbers, which approaches the distribution rate of prime numbers corresponding to integers. Based on the corresponding prime distribution equation, obtain the twin prime inequality function. Then, the formula for calculating twin prime numbers was discussed. There is also the Hardy Littlewood conjecture. This provides a practical and feasible approach for studying the distribution of twin prime numbers.

Topological phases and edge modes of an uneven ladder

We investigate the topological properties of a two-chain quantum ladder with uneven legs,i.e.,the two chains differ in their periods by a factor of 2.Such an uneven ladder presents rich band structures classified by the closure of either direct or indirect bandgaps.It also provides opportunities to explore fundamental concepts concerning band topology and edge modes,including the difference of intracellular and intercellular Zak phases,and the role of the inversion symmetry(IS).We calculate the Zak phases of the two kinds and find excellent agreement with the dipole moment and extra charge accumulation.We also find that configurations with IS feature a pair of degenerate two-side edge modes emerging as the closure of the direct bandgap,while configurations without IS feature one-side edge modes emerging as not only the closure of both direct and indirect bandgaps but also within the band continuum.Furthermore,by projecting to the two sublattices,we find that the effective Bloch Hamiltonian corresponds to that of a generalized Su–Schrieffer–Heeger model or the Rice–Mele model whose hopping amplitudes depend on the quasimomentum.In this way,the topological phases can be efficiently extracted through winding numbers.We propose that uneven ladders can be realized by spin-dependent optical lattices and their rich topological characteristics can be examined by near future experiments.

On calculating glacial isostatic adjustment

Modeling the earth's fluid and elastic response to the melting of the glaciers of the last ice age is the most direct way to infer the earth's radial viscosity profile.Here,we compare two methods for calculating the viscoelastic response to surface loading.In one,the elastic equation of motion is converted to a viscoelastic equation using the Correspondence Principle.In the other,elastic deformation is added to the viscous flow as isostatic adjustment proceeds.The two modeling methods predict adjustment histories that are different enough to potentially impact the interpretation of the observed glacial isostatic adjustment(GIA).The differences arise from buoyancy and whether fluid displacements are subjected to hydrostatic pre-stress.The methods agree if they use the same equations and boundary conditions.The origin of the differences is determined by varying the boundary conditions and pre-stress application.

Selective sampling with Gromov–Hausdorff metric:Efficient dense-shape correspondence via Confidence-based sample consensus

Background Functional mapping, despite its proven efficiency, suffers from a “chicken or egg” scenario, in that, poor spatial features lead to inadequate spectral alignment and vice versa during training, often resulting in slow convergence, high computational costs, and learning failures, particularly when small datasets are used. Methods A novel method is presented for dense-shape correspondence, whereby the spatial information transformed by neural networks is combined with the projections onto spectral maps to overcome the “chicken or egg” challenge by selectively sampling only points with high confidence in their alignment. These points then contribute to the alignment and spectral loss terms, boosting training, and accelerating convergence by a factor of five. To ensure full unsupervised learning, the Gromov–Hausdorff distance metric was used to select the points with the maximal alignment score displaying most confidence. Results The effectiveness of the proposed approach was demonstrated on several benchmark datasets, whereby results were reported as superior to those of spectral and spatial-based methods. Conclusions The proposed method provides a promising new approach to dense-shape correspondence, addressing the key challenges in the field and offering significant advantages over the current methods, including faster convergence, improved accuracy, and reduced computational costs.

Evaluation of Soil Fertility Quality and Environmental Driving Factors in Different Soil Types of Artificial Forests

The aim was to clarify the environmental driving factors of soil fertility indicators in artificial forests of Guangxi and comprehensively evaluate the soil fertility level.By collecting data on the current status of soil in artificial forests,the spatial distribution of major soil fertility indicators was analyzed,and the distribution map of the fertility index of artificial forests in the entire region and the comprehensive fertility index of artificial forests of different soil types were obtained.Canonical correspondence analysis method was used to analyze soil fertility indicators and environmental factors,and the environmental driving factors of soil fertility indicators for artificial forests of the main soil types in Guangxi were obtained.The results showed that over 90%of the soil fertility index of artificial forests in the entire region was between 0.20 and 0.50.The order of soil fertility index of different soil types of artificial forests from high to low was yellow brown soil>yellow red soil>yellow soil>red soil>limestone soil>latosolic red soil>laterite.In artificial forests of latosolic red soil,the correlation between soil alkaline nitrogen and organic matter,annual average temperature was high,while the correlation between soil available phosphorus and organic matter,pH was high,and the correlation between soil available potassium and environmental factors such as slope,altitude,rainfall,accumulated temperature,and slope aspect was high.In artificial forests of red soil,the correlation between soil alkaline nitrogen and slope,altitude was high,while the correlation between soil available phosphorus and accumulated temperature,rainfall was high,and the correlation between soil available potassium and pH was high.In artificial forests of limestone soil,there was a high correlation between soil alkaline nitrogen and slope,organic matter,a high correlation between soil available phosphorus and accumulated temperature,rainfall,and a high correlation between soil available potassium and pH.

The Deconvolution Method for Obtaining Correspondence in Data-Independent Acquisition Mass Spectrometry Data

Although data-independent acquisition (DIA) shows powerful potential in achieving comprehensive peptide information acquisition, the difficulty in determining the precursor m/z and distinguishing fragment ions has posed challenges in DIA data analysis. To address this challenge, a common approach is to recover the correspondence between precursor ions and fragment ions, followed by peptide identification using traditional data-dependent acquisition (DDA) database searching. In this study, we propose a cosine similarity-based deconvolution method that rapidly establishes the correspondence between chromatographic profiles of precursor ions and fragment ions through matrix calculations. Experimental results demonstrate that our method, referred to as CosDIA, yields a peptide identification count close to that of DIA-umpire. However, compared to DIA-umpire, we can establish the correspondence between original MS/MS spectra and pseudo-MS/MS spectra. Furthermore, compared to the CorrDIA method, our approach achieves higher efficiency in terms of time, reducing the time cost of the analysis process. These results highlight the potential advantages of the CosDIA method in DIA data analysis, providing a powerful tool and method for large-scale proteomics research.

Special Relativity’s “Newtonization” in Complex “Para-Space”: The Two Theories Equivalence Question

Complex model, say C3, of “para-space” as alternative to the real M4 Minkowski space-time for both relativistic and classical mechanics was shortly introduced as reference to our previous works on that subject. The actual aim, however, is an additional analysis of the physical and para-physical phenomena’ behavior as we formally transport observable mechanical phenomena [motion] to non-real interior of the complex domain. As it turns out, such procedure, when properly set, corresponds to transition from relativistic to more classic (or, possibly, just classic) kind of the motion. This procedure, we call the “Newtonization of relativistic physical quantities and phenomena”, first of all, includes the mechanical motion’s characteristics in the C3. The algebraic structure of vector spaces was imposed and analyzed on both: the set of all relativistic velocities and on the set of the corresponding to them “Galilean” velocities. The key point of the analysis is realization that, as a matter of fact, the relativistic theory and the classical are equivalent at least as for the kinematics. This conclusion follows the fact that the two defined structures of topological vector spaces i.e., the structure imposed on sets of all relativistic velocities and the structure on set of all “Galilean” velocities, are both diffeomorphic in their topological parts and are isomorphic as the vector spaces. As for the relativistic theory, the two approaches: the hyperbolic (“classical” SR) with its four-vector formalism and Euclidean, where SR is modeled by the complex para-space C3, were analyzed and compared.

Comprehensive precursor anomaly research based on earthquake corresponding relevancy spectrum

With the aim to the quantification of anomaly identification and extraction for observed or analyzed records, we present two statistical methods of earthquake corresponding relevancy spectrum （ECRS） and sliding mean relevancy （SMR）. With ECRS method, we can obtain the abnormal confidence attribute of data in different value ranges. Based on the relevancy spectrum in different studied time-intervals, we convert the original time sequence into relevancy time sequence, and can obtain the SMR time series by using the multi-point cumulative sliding mean method. Then we can identify the seismic precursor anomaly. We test this method by taking the time sequence of r/-value in the northern Tianshan region as original data. The result shows that when the studied time-interval is 18 months, the precursor anomaly can be identified bet- ter from sliding mean relevancy. The anomaly corresponding rate is 83 percent, the earthquake corresponding rate is 86 per- cent, and the anomaly is characteristic of the middle term. To try the research on multi-parameter comprehensive application, we take the Kalpin tectonic block in Xinjiang as our studied region, and analyze the spatial and temporal abnormal characters of multi-parameter sliding extreme-value relevancy （MSER） before mid-strong earthquakes in the Kalpin block. The result indicates that ECRS and SMR sequence in different time-intervals can not only be used to identify the precursor anomaly of single-item data, but also offer the data of quantitative single-item anomaly for comprehensive earthquake analysis and prediction.

Estimation of Acentric Factor of Organic Compounds with Corresponding States Group Contribution Method

A new estimation method was proposed by combining the corresponding state principle with the group contribution method through introducing the concept of assumed-critical properties. Combining it with the Reidel equation, a new acentric factor correlation equation (CSGC-Reidel) was developed. Contribution values of 70 groups were obtained by correlating acentric factor data of 228 organic compounds of 14 type substances including saturated hydrocarbons, unsaturated hydrocarbons, cyclanes, aromatics, oxygen compounds, nitrogen compounds,halohydrocarbons, etc. The average error of acentric factor is 3.52% between the literature data and the predicated with the new estimation method.

NEW SHAPE FACTOR FOR CORRESPONDING STATES PRINCIPLE

1 INTRODUCTIONThe CSP(Corresponding States Principle)has been considered to be a useful methodfor the prediction of fluid properties from a minimum amount of information.One ofits form,the shape factor methods is developed through relating the pVT of any interestedpure fluid"a"to that of another"0"by

ESTIMATION OF THEMAL PRESSURE COEFFICIENTS FOR LIQUIDS BASED ON THE LAW OF CORRESPONDING STATE

1 INTRODUCTIONThe p-V-T relation of liquids can be represented not only by equations of state,but alsoby the following three partial differentials: α=(V/T)_p/V (1) β=-(V/p)_T/V (2) γ=(p/T)_V (3)Where p,V and T denote the pressure,volume and temperature of liquids;α,βand γ are thethermal expansion,isothermal compressibility and thermal pressure coefficient,respectively. Since the densities of liquids at various temperature and constant pressure can easily bemeasured by experiments,values of α can be obtained easily.If the values of.γ can beestimated,then values of β can be acquired from the following equation:

THE CORRESPONDING ANALYSIS OF HEAVY-METAL POLLUTION OF SOIL IN ZHUZHOU CITY

ＴＨＥ ＣＯＲＲＥＳＰＯＮＤＩＮＧ ＡＮＡＬＹＳＩＳ ＯＦ ＨＥＡＶＹ－ＭＥＴＡＬ ＰＯＬＬＵＴＩＯＮ ＯＦ ＳＯＩＬ ＩＮ ＺＨＵＺＨＯＵ ＣＩＴＹ ＳｈａｎｇＪｉｎｃｈｅｎｇ（尚金城）；ＬｏｎｇＡｉｍｉｎ（龙爱民）ＬｉＢｉｎ（李斌）；ＪｉａｎｇＪ...

A Perfect Shape Factor Corresponding States Principle for Pure Non-Polar and Polar Fluids

In the two-parameter corresponding states principle(CSP), the critical compressibility factors of the fluid under study(called 'a' fluid) and the reference fluid(called 'o' fluid) must be identical. This is not generally observed in nature. To overcome this limitation, a perfect shape factor CSP is proposed in which the compressibility factors of 'a' and 'o' fluids are corresponded perfectly by introducing a new pressure shape factor 8. Using methane as the 'o' fluid, the shape factors of many fluids are calculated from PVT properties at saturation state and the second virial coefficients. Models are also formulated for the shape factors with the assumption of is a function of temperature and volume while 6 and 5 are temperature dependent only. The models described the shape factors satisfactorily in whole region including vapor, liquid and their co-existing phases. The perfect shape factor CSP could be applied for both polar and non-polar fluids.

Bekenstein-Hawking Cosmological Entropy and Correction Term Corresponding Cosmological Horizon of Rotating and Charged Black String

Utilizing the quantum statistical method and applying the new state density equation motivated by generalized uncertainty principle in quantum gravitaty, we avoid the difficulty in solving wave equation and directly calculate the partition function of bosonic and fermionic field on the background of rotating and charged black string. Then near the cosmological horizon, entropies of bosonic and fermionic field are calculated on the background of black string. When constant A introduced in generalized uncertainty principle takes a proper value, we derive Bekenstein- Hawking entropy and the correction value corresponding cosmologicaJ horizon on the background of rotating and charged black string. Because we use the new state density equation, in our calculation there are not divergent term and small mass approximation in the original brick-wall method. From the view of quantum statistic mechanics, the correction value to Bekenstein-Hawking entropy of the black string is derived. It makes people deeply understand the correction value to the entropy of the black string cosmological horizon in non-spherical coordinate spacetime.

A NEW ROUTE FROM HETEROAROMATIC BORON COMPOUNDS TO CORRESPONDING KETONES VIA DICHLOROCARBENE INSERTION UNDER PHASE TRANSFER CONDITIONS

The reactions of tri-2-benzofuranylborane or aminoethyl di-2-thienylborinate and 2-thienylphenylborinate with dichlorocarbene generated in situ from chloroform under phase transfer conditions were explored as a new route for the preparation of heteroaromatic ketones. It was found that a good yield of the ketones can be obtained by solid-liquid system.

DETERMINATION OF STABILITY CONSTANTS OF THE COORDINATION COMPOUNDS OF Au(Ⅲ)-2-MERCAPTOPYRIDINE-1-OXIDE AND METHYL DERIVATIVES——METHOD OF DUAL-WAVELENGTH CORRESPONDING SOLUTIONS

In this paper,we report the deductive formula used for the method of dual-wavelength corresponding solutions under condition of having ligand interference and the stability constants of three new coordination compounds [AuL_2]^+determined with this method.The stability of the three compounds,the necessity of controlling pH in experimental systems and the advantage of this method are discussed in detail.

CHARACTERIZATION OF HLDER SPACES CORRESPONDING TO DITZIAN-TOTIK MODULUS OF SMOOTHNESS

For the step-weight function , we prove that the Holder spaces ∧a,p on the interval [-1,1], defined in terms of moduli of smoothness with the step-weight function ,are linearly isomorphic to some sequence spaces, and the isomorphism is given by the cofficients of function with respect to a system of orthonormal splines with knots uniformly distributed according to the measure with density . In case ∧a,p is contained in the space of continuous functions, we give a discrete characterization of this space, using only values of function at the appropriate knots. Application of these results to characterize the order of polynomial approximation is presented.

Synthesis of(±)-Deoxyschisandrin and the Corresponding Trans-lsomer

Deoxyschisandrin 1 and the corresponding trans - isomer 2 were readily prepared by intramolecular oxidative coupling of 1.4 -diaryl-2.3-dimethylbutanes 7 and 6 with 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) in trifluoroacetic acid(TFA).