in this paper, we construct two new primitive Evans triangles by using the Pell equation x^2-Dy^2=1, and give the two forms of the Evans triangle and the corresponding Evans ratio, we also get a new conclusion that the ratio is an integer from a few different forms of the bottom and high.
In this paper we study the generalization of the Brocard congurations.
The denition of q-period broken line is given and then the new necessary and
sucient condition about the existence of two conjugate Brocard points of the
qperiod loop broken line is investigated. Based on the past research, a geometric
method for the construction of all possible loop broken lines with conjugate Brocard
points is developed.
Motivated by Gutkin curves arising in capillary floating and billiard ball problem, we
introduce a new projection function for convex plane curves, and study the curves
when the defined projection function is constant. These curves can be regarded as a
generalization of curves with constant width. We give a necessary and sufficient
condition for the existence of such curves, and provide an explicit expression for the
radius of curvature. Properties of our curves and their relation to Gutkin curves are
also discussed. Finally, we construct and exhibit some curves with constant projection
Geometric compression plays a fundamental rolein virtual reality and augmented reality (VR/AR) applications. Dense meshesare re-sampled and re-tessellatedtoreduce thecomplexity. This process is called remeshing. In this work,we propose a novel remeshing algorithm based on both angle-preservingparameterization, and measurecontrollable parameterization. The conformal parameterization iscarried out by discrete surface Ricci flow method, the measurecontrollable parameterization is obtaind by an optimal mass transportation map. The sampling is performed on the measurecontrollable parameterization domain, the triangulation is computed on the conformal parameterization domain using Delaunay refinement algorithm. This method gives the user full control of sampling distribution, and produces mesheswith curvature measure convergence. The meshing resultcan emphsizethe region of interests, is curvature sensitive. Experimental results demonstrate the efficiency and efficacy of the proposed method.
As the barcode becomes more widely used, its applications and data capacity demands grow,
increasing the need for barcodes with greater data density. Utilizing the quick response (QR)
code–one of the many types of barcodes–we developed two algorithms. The first algorithm
creates a color QR code that stores more information than a standard QR code and embeds extra
data with limited access privilege. The second algorithm denoises a noisy color QR code. These
algorithms consist of three techniques: (1) enlarging the data capacity of a compact QR code
image by stacking multiple classical QR codes to form a color barcode, (2) embedding
information into the color QR code using pseudo quantum signals in an M-band wavelet domain
and selecting the discrete 4-band wavelet transforms to compress the QR images, and (3)
applying Discrete M-band Wavelet Transform (DMWT) and Patch Group Prior based Denoising
(PGPD) methods to denoise noisy QR code images. The peak-signal-to-noise-ratio (PSNR)
summary indicates that information in a color QR code can be efficiently stored and retrieved
with these methods. Moreover, it shows that our denoising algorithm effectively removes heavy
noise from the noisy color QR code. Our algorithms are implemented in a flexible framework,
which allows for further modifications to improve both the data capacity of a color QR code and
the effectiveness of signal extraction from noisy data to meet future demands.
In this paper we first introduce a fractional form formula among a number
of Euler’s formulas. We then extend the formula and with mathematical
induction prove the case when the number of terms increases and the
exponent is integer. Afterwards, we study the connection between Euler’s
formula and Lagrange interpolating polynomial and use the latter to prove part
of the extended formula. We then obtain a new formula from this connection.
At last, we derive a set of new equations from the extended formula.
The optimal mass transportation problem, proposed by Monge, is dominated by the
Monge-Ampere equation. In general, as the Monge-Ampere equation is highly non-linear, this type
of partial dierential equations is beyond the solving ability of a conventional nite element method.
This diculty led Gu et al. alternatively to the discrete optimal mass transportation problem. They
developed variational principles for this problem and reported the calculation for the optimal mapping,
based on theorem that among all possible cell decompositions, with constrained measures, the trans-
portation cost of the discrete mapping from cells to the corresponding discrete points is minimized
by the decomposition induced by a power Voronoi diagram. Their research inspired us to consider a
similar discrete optimal mass transportation problem. Here we replace the L2 Euclidean distance by
the length of the shortest path connecting two points on a line network. To solve the optimal trans-
portation problem on a line network, we study a type of Voronoi diagram on undirected and connected
networks and propose an elegant construction algorithm. We further consider weighted distances in a
network and develop a method to compute the centroidal Voronoi tessellation (CVT) for a network.
By using real geographic data, the method proposed is proved ecient and eective in several practical
applications, including charging station distribution in a trac network, and trash can distribution in
In this paper, the differential equations for copolymerization of multiple monomers are derived based on the steady-status assumption of Alfrey-Goldfinger and Valvassori-Sartori respectively, and a numerical method is implemented in C++ language. The differential equations of copolymer composition with wide application presented in this paper provide a theoretical guidance for industrial production and academic research of multiple free radical copolymerization.