Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

Inverse full state hybrid projective synchronization for chaotic maps with different dimensions

A new synchronization scheme for chaotic（hyperchaotic） maps with different dimensions is presented.Specifically,given a drive system map with dimension n and a response system with dimension m,the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states.The method,based on the Lyapunov stability theory and the pole placement technique,presents some useful features：（i） it enables synchronization to be achieved for both cases of n 〈 m and n 〉 m;（ii） it is rigorous,being based on theorems;（iii） it can be readily applied to any chaotic（hyperchaotic） maps defined to date.Finally,the capability of the approach is illustrated by synchronization examples between the two-dimensional H′enon map（as the drive system） and the three-dimensional hyperchaotic Wang map（as the response system）,and the three-dimensional H′enon-like map（as the drive system） and the two-dimensional Lorenz discrete-time system（as the response system）.

Fixed-Time Adaptive Time-Varying Matrix Projective Synchronization of Time-Delayed Chaotic Systems with Different Dimensions

This paper deals with the fixed-time adaptive time-varying matrix projective synchronization(ATVMPS)of different dimensional chaotic systems(DDCSs)with time delays and unknown parameters.Firstly,to estimate the unknown parameters,adaptive parameter updated laws are designed.Secondly,to realize the fixed-time ATVMPS of the time-delayed DDCSs,an adaptive delay-unrelated controller is designed,where time delays of chaotic systems are known or unknown.Thirdly,some simple fixed-time ATVMPS criteria are deduced,and the rigorous proof is provided by employing the inequality technique and Lyapunov theory.Furthermore,the settling time of fixed-time synchronization(Fix-TS)is obtained,which depends only on controller parameters and system parameters and is independent of the system’s initial states.Finally,simulation examples are presented to validate the theoretical analysis.

Statistics of Projected Motion in One Dimension of a D-Dimensional Random Walker

We are studying the motion of a random walker in generalised d-dimensional continuum with unit step length (up to 10 dimensions) and its projected one dimensional motion numerically. The motion of a random walker in lattice or continuum is well studied in statistical physics but what will be the statistics of projected one dimensional motion of higher dimensional random walker is yet to be explored. Here in this paper, by addressing this particular type of problem, it shows that the projected motion is diffusive irrespective of any dimension;however, the diffusion rate is changing inversely with dimensions. As a consequence, it can be predicted that for the one dimensional projected motion of infinite dimensional random walk, the diffusion rate will be zero. This is an interesting result, at least pedagogically, which implies that though in infinite dimensions there is diffusion, its one dimensional projection is motionless. At the end of the discussion we are able to make a good comparison between projected one dimensional motion of generalised d-dimensional random walk with unit step length and pure one dimensional random walk with random step length varying uniformly between -h to h where h is a “step length renormalizing factor”.

A study of making use of computer-aided projection fringe system to measure breast dimension

To measure breast basic dimension by using computer-aided projection fringe system.Methods A system has been developed for measuring breast basic dimension based on computer-aided projection fringe measurement and programming software.Plastic manikins breast’s SN-N (sternal notch to nipple distance),N-ML (nipple to midline distance),N-N (internipple distance),MBW (base width of breast) and N-IMF (nipple to inframammary fold distance) are measured with this system.At the same time,these items are also measured with routine ruler.Results This study indicate that the system has some merits:① non-touching measurement;② it is very rapid,the patient measured need hold his breath only 0.5 second,and all the time it takes is about 2.5 minutes;③ the measurement’s sensitivity is as high as to 0.6 mm,which meets the clinic requirement entirely;④ the measurement’s accuracy of the system is not significantly when comparing to the routine ruler’s.Conclusion Computer-adided projection fringe system for measuring breast basic dimension is feasible and advanced.14 refs,1 fig.

Possibilities and challenges of expanding dimensions of waterway downstream of Three Gorges Dam

The waterway in the middle and lower reaches of the Yangtze River has long been known as the Golden Waterway and has served as an important link in the construction of the Yangtze River Economic Belt.Therefore,expanding its dimensions is a significant goal,particularly given the long-range cumulative erosion occurring downstream of the Three Gorges Dam (TGD),which has been concentrated in the dry river channel.With the regulation of the volume from upstream reservoirs and the TGD,the minimum discharge and water level of the river downstream are increasing,and creating favorable conditions for the increase of the depth of the waterway.The discharge compensation effect during the dry season offsets the decline in the water level of the river channel caused by the down-cutting of part of the riverbed,but the minimum navigable water level of the segment near the dam still shows a declining trend.In recent years,several waterway remediation projects have been implemented in the downstream reaches of the TGD and although the waterway depth and width have been increased,the channel dimensions are still insufficient in the Yichang-Anqing reach (with a total length of 1026 km),as compared to the upstream reservoir area and the deep water channel in the downstream tidal reaches.A comprehensive analysis of the water depth and the number and length of shoals in the waterway indicates that its dimensions can be increased to 4.5 m ×200 m and 6.0 m×200 m in the Yichang-Wuhan and Wuhan-Anqing reaches,respectively.This is also feasible given the remediation technologies currently available,but remediation projects need to be coordinated with those for flood prevention and ecological protection.

On Ding homological dimensions

In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a complete hereditary cotorsion pair such that a module M belongs to DD∩W1 if and only if M is projective, moreover, 1421 = （M[pd（M） 〈 ∞} = {MIfd（M） ≤ n} = {MIpd（M） ≤ n}. Then we introduce and inves- tigate Ding derived functor Dext^i（-, -）, and use it to characterize global Ding dimension. We show that if R is a Ding-Chen ring, or if R is a ring with rDPD（R） 〈≤ and rDID（R） 〈 ≤, then rDPD（R） 〈 n if and only if rDID（R） 〈 n if and only if Dext^n＋i（M,N） = 0 for all modules M and N and all integer i ≥ 1.

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy pro jection technique

This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method （FEM） by recursive application of the one-dimensional （1D） element energy projection （EEP） technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element （FE） model is reached. This conceptual dimension-by- dimension （D-by-D） discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional （2D） and three-dimensional （3D） problems can be obtained over the domain. Numerical examples of 3D Poisson＇s equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.

Cotorsion Dimension of Weak Crossed Products

Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#;H in terms of the corresponding data for H and A.

On Rings with Finite Global Gorenstein Dimensions

As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou's equality of global Gorenstein dimension is given.

Relative Ding Projective Modules over Formal Triangular Matrix Rings

Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.

Rainfall Assimilation Using a New Four-Dimensional Variational Method:A Single-Point Observation Experiment

Accurate forecast of rainstorms associated with the mei-yu front has been an important issue for the Chinese economy and society. In July 1998 a heavy rainstorm hit the Yangzi River valley and received widespread attention from the public because it caused catastrophic damage in China. Several numerical studies have shown that many forecast models, including Pennsylvania State University National Center for Atmospheric Research’s fifth-generation mesoscale model （MM5）, failed to simulate the heavy precipitation over the Yangzi River valley. This study demonstrates that with the optimal initial conditions from the dimension-reduced projection four-dimensional variational data assimilation （DRP-4DVar） system, MM5 can successfully reproduce these observed rainfall amounts and can capture many important mesoscale features, including the southwestward shear line and the low-level jet stream. The study also indicates that the failure of previous forecasts can be mainly attributed to the lack of mesoscale details in the initial conditions of the models.

New Exact Solutions for the Generalized （2 ＋ 1）-dimensional Nonlinear Schroedinger Equation with Variable Coefficients

With the help of the variable-coefficient generalized projected Ricatti equation expansion method, we present exact solutions for the generalized （2＋1）-dimensional nonlinear SchrSdinger equation with variable coefficients. These solutions include solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time.

基于BIM技术的数维生产工程项目应用研究

为了促进房建工程变革,提高建筑施工效率和施工质量,研究基于BIM技术实现项目数维生产,结合BIM技术,从建模、模型检查、算量一体化、轻量化交付5个方面阐述数维生产工程项目BIM实施路径,并落实到具体案例,观察实践案例在提高项目管理和施工效率等方面的应用效果。通过BIM建模,可以实现对建筑结构的详细模拟,帮助设计师和工程师更好地理解项目需求,减少设计错误和变更。模型检查功能可以确保建筑设计的合规性和安全性,降低施工风险。算量一体化功能则可以实现建筑材料的精确计算和成本控制,提高投资效益。最后,轻量化交付可以加速信息传递和协作效率,使项目团队更加高效地协同工作。

Gorenstein Projective Dimensions of Complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C^m is Gorenstein projective in R-Mod for all m E Z. Basing on this we show that if R is a right coherent left perfect ring then Gpd（C） = sup{Gpd（C^m）｜m ∈ Z} where Gpd（-） denotes Gorenstein projective dimension.

Gorenstein Projective Dimensions of Modules over Minimal Auslander-Gorenstein Algebras

In this article we investigate the relations between the Gorenstein projective dimensions of Λ-modules and their socles for re-minimal Auslander-Gorenstein algebras Λ.First we give a description of projective-injective Λ-modules in terms of their socles.Then we prove that a Λ-module N has Gorenstein projective dimension at most n if and only if its socle has Gorenstein projective dimension at most n if and only if N is cogenerated by a projective Λ-module.Furthermore,we show that n-minimal Auslander-Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.

Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal

Let S = K[x1, x2,..., xn] be the polynomial ring in n variables over a field K, and let I be a squarefree monomial ideal minimally generated by the monomials ul,u2,...,Um. Let w be the smallest number t with the property that for all integers 1 ≤ i1 〈 i2 〈 … 〈 it ≤ m such that lcm（uil, ui2,..., uii） = lcm（ul,u2,...,Um）. We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I. As a corollary, the projective dimension of I is bounded by the number w.

On Gorenstein Projective Dimensions of Unbounded Complexes

Let R→S be a ring homomorphism and X be a complex of R-modules.Then the complex of S-modules S L RX in the derived category D(S)is constructed in the natural way.This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X(possibly unbounded)with those of the S-complex S■R^L X.It is shown that if R is a Noetherian ring of finite Krull dimension and :R→S is a faithfully flat ring homomorphism,then for any homologically degree-wise finite complex X,there is an equality GpdRX=GpdS(S■R^LX).Similar result is obtained for Ding projective dimension of the S-complex S■R^L X.

Tensor Train Random Projection

This work proposes a Tensor Train Random Projection(TTRP)method for dimension reduction,where pairwise distances can be approximately preserved.Our TTRP is systematically constructed through a Tensor Train(TT)representation with TT-ranks equal to one.Based on the tensor train format,this random projection method can speed up the dimension reduction procedure for high-dimensional datasets and requires fewer storage costs with little loss in accuracy,comparedwith existingmethods.We provide a theoretical analysis of the bias and the variance of TTRP,which shows that this approach is an expected isometric projectionwith bounded variance,and we show that the scaling Rademacher variable is an optimal choice for generating the corresponding TT-cores.Detailed numerical experiments with synthetic datasets and theMNIST dataset are conducted to demonstrate the efficiency of TTRP.

R & D project's investment evaluation based on real option and its value at risk

The authors looked upon it as real options and applied the VaR(Value at Risk) method to the evaluation of its risk value based on the analysis of R & D project investment characteristics,and advanced the evaluation model of the project’s return and risk according to financial theories.This paper expounded the two dimension evaluation model of project,and divided it into five decision making regions.