A Family of Integer-Point Ternary Parametric Subdivision Schemes

New subdivision schemes are always required for the generation of smooth curves and surfaces. The purpose of this paper is to present a general formula for family of parametric ternary subdivision schemes based on the Laurent polynomial method. The different complexity subdivision schemes are obtained by substituting the different values of the parameter. The important properties of the proposed family of subdivision schemes are also presented. The continuity of the proposed family is C 2 m . Comparison shows that the proposed family of subdivision schemes has higher degree of polynomial generation, degree of polynomial reproduction, and continuity compared with the exiting subdivision schemes. Maple software is used for mathematical calculations and plotting of graphs.

Complexity of linear relaxations in integer programming

For a setXof integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with Xis called the relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$rc(X). This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of Xwithout using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding$${{\,\mathrm{rc}\,}}(X)$$rc(X)and its variant$${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rcQ(X), restricting the descriptions of Xto rational polyhedra. As our main results we show that$${{\,\mathrm{rc}\,}}(X) = {{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rc(X)=rcQ(X)when: (a)Xis at most four-dimensional, (b)Xrepresents every residue class in$$(\mathbb {Z}/2\mathbb {Z})^d$$(Z/2Z)d, (c) the convex hull of Xcontains an interior integer point, or (d) the lattice-width of Xis above a certain threshold. Additionally,$${{\,\mathrm{rc}\,}}(X)$$rc(X)can be algorithmically computed when Xis at most three-dimensional, orXsatisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on$${{\,\mathrm{rc}\,}}(X)$$rc(X)in terms of the dimension of X.

Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

Zero-one Schubert polynomials

We prove that if $$\sigma \in S_m$$ σ ∈ S m is a pattern of $$w \in S_n$$ w ∈ S n , then we can express the Schubert polynomial $$\mathfrak {S}_w$$ S w as a monomial times $$\mathfrak {S}_\sigma $$ S σ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar’s orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on $$\mathfrak {S}_w$$ S w being zero-one. In this case, the Schubert polynomial $$\mathfrak {S}_w$$ S w is equal to the integer point transform of a generalized permutahedron.

The positive integral points on the elliptic curve 𝒚𝟐=𝟕𝒑𝒙(𝒙𝟐+𝟖)

The integral point of elliptic curve is a very important problem in both elementary number theory and analytic number theory. In recent years, scholars have paid great attention to solving the problem of positive integer points on elliptic curve 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), where 𝑘,𝑎,𝑏,𝑐 are integers. As a special case of 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), when 𝑎 = 1,𝑏 = 0,𝑐 = 22𝑡−1, it turns into 𝑦2 = 𝑘𝑥(𝑥2+22𝑡−1), which is a very important case. However ,at present, there are only a few conclusions on it, and the conclusions mainly concentrated on the case of 𝑡 = 1,2,3,4. The case of 𝑡 = 1, main conclusions reference [1] to [7]. The case of 𝑡 = 2, main conclusions reference [8]. The case of 𝑡 = 3, main conclusions reference [9] to [11]. The case of 𝑡 = 4, main conclusions reference [12] and [13]. Up to now, there is no relevant result on the case of 𝑘 = 7𝑝 when 𝑡 = 2, here the elliptic curve is 𝑦2 = 7𝑝(𝑥2 + 8), this paper mainly discusses the positive integral points of it. And we obtained the conclusion of the positive integral points on the elliptic curve 𝑦2 = 7𝑝(𝑥2 + 8). By using congruence, Legendre symbol and other elementary methods, it is proved that the elliptic curve in the title has at most one integer point when 𝑝 ≡ 5,7(𝑚𝑜𝑑8).

Implementation of Numerical Mesostructure Concrete Material Models: A Dot Matrix Method

We develop a dot matrix method (DMM) using the principles of computational geometry to place aggregates into matrices for the construction of mesolevel concrete models efficiently and rapidly. The basic idea of the approach is to transform overlap detection between polygons (or polyhedrons) into checking the possibility of any intersection between the point sets within a trial placement aggregate and the already placed ones in mortar. Through the arithmetic operation of integer point sets, the efficiency of the underlying algorithm in the dot matrix method is higher. Our parking algorithm holds several advantages comparing with the conventional placement issues. First, it is suitable for arbitrary-shape aggregate particles. Second, it only needs two sets for examining if the overlap between a trial placement aggregate and the already placed ones. Third, it accurately places aggregates according to aggregate grading curves, by order of reduction, led to more efficiently reducing aggregate placement time. The present method is independent of the size of aggregate particles. Combing with 3D laser scanning technology, the present method can also be used to create mesostructure concrete models conveniently and flexibly. Several examples show that DDM is a robust and valid method to construct mesostructure concrete models.

A A Novel Risk Score to Predict New Onset Atrial Fibrillation in Patients Undergoing Isolated Coronary Artery Bypass Grafting

Background: Atrial fibrillation (AF) is common after cardiac surgery and contributes to increased morbidity and mortality. Our objective was to derive and validate a predictive model for AF after CABG in patients, incorporating novel echocardiographic and laboratory values. Methods: We retrospectively reviewed patients at our institution without preexisting dysrhythmia who underwent on-pump, isolated CABG from 2011-2015. The primary outcome was new onset AF lasting >1 hour on continuous telemetry or requiring medical treatment. Patients with a preoperative echocardiographic measurement of left atrial diameter were included in a risk model, and were randomly divided into derivation (80%) and validation (20%) cohorts. The predictors of AF after CABG (PAFAC) score was derived from a multivariable logistic regression model by multiplying the adjusted odds ratios of significant risk factors (P < .05) by a factor of 4 to derive an integer point system. Results: 1307 patients underwent isolated CABG, including 762/1307 patients with a preoperative left atrial diameter measurement. 209/762 patients (27%) developed new onset AF including 165/611 (27%) in the derivation cohort. We identified four risk factors independently associated with postoperative AF which comprised the PAFAC score: age > 60 years (5 points), White race (5 points), baseline GFR < 90 mL/min (4 points) and left atrial diameter > 4.5 cm (4 points). Scores ranged from 0-18. The PAFAC score was then applied to the validation cohort and predicted incidence of AF strongly correlated with observed incidence (r = 0.92). Conclusion: The PAFAC score is easy to calculate and can be used upon ICU admission to reliably identify patients at high risk of developing AF after isolated CABG.

Farey Sequence for Error Correcting Codes and Medical Images

A new scheme for Error Correcting Codes and the coding of medical images through Farey Sequence is brought in the paper. It brings out the reduced real point representations and more of integer point representations. The test case and outcomes are explained in the section 4 and 5.