Criteria for strict monotonicity of the mixed volume of convex polytopes

Let P1, …, Pn and Q1, …, Qn be convex polytopes in ℝn with Pi ⊆ Qi. It is well-known that the mixed volume is monotone: V(P1, …, Pn) ≤ V(Q1, …, Qn). We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1, …, Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 ∪ … ∪ Pn. In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.

Index of Grassmann manifolds and orthogonal shadows

In this paper, we study the {\mathbb{Z}/2} action on the real Grassmann manifolds {G_{n}(\mathbb{R}^{2n})} and {\widetilde{G}_{n}(\mathbb{R}^{2n})} given by taking the (appropriately oriented) orthogonal complement. We completely evaluate the related {\mathbb{Z}/2} Fadell–Husseini index utilizing a novel computation of the Stiefel–Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For {n=2^{a}(2b+1)} , {k=2^{a+1}-1} , a convex body C in {\mathbb{R}^{2n}} , and k real-valued functions {\alpha_{1},\ldots,\alpha_{k}} continuous on convex bodies in {\mathbb{R}^{2n}} with respect to the Hausdorff metric, there exists a subspace {V\subseteq\mathbb{R}^{2n}} such that projections of C to V and its orthogonal complement {V^{\perp}} have the same value with respect to each function {\alpha_{i}} , that is, {\alpha_{i}(p_{V}(C))=\alpha_{i}(p_{V^{\perp}}(C))} for all {1\leq i\leq k} .

GEOMETRIA E TRIGONOMETRIA: POSSIBILIDADES DE UM VÍNCULO VANTAJOSO

http://dx.doi.org/10.5902/2179460X14639The relationship between geometry and trigonometry can go far beyond the classic problem involving the two areas in basic education, which is the resolution of triangles. Possibilities of using the geometry in the service of Trigonometry, or the Trigonometry in the service of Geometry, are many. There is a tendency to separate them by rigid boundaries, affecting cooperation among the methods and techniques of both areas in solving certain mathematical problems, not necessarily limited to one area, for example, the problems posed in the Olympics. In this paper we show some of these possibilities, working with examples where from a geometric result we obtain certain trigonometric results, or, from a trigonometric result we obtain some geometric result. We also present the solution of some geometric problems using both methods.

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

A Dual of the Rectangle-Segmentation Problem for Binary Matrices

We consider the problem to decompose a binary matrix into a small number of binary matrices whose 1-entries form a rectangle. We show that the linear relaxation of this problem has an optimal integral solution corresponding to a well known geometric result on the decomposition of rectilinear polygons.

The Role of Objective Plane Angulation on the Mandibular Image Using Cross-Sectional Tomography

Cross-sectional jaw images in the buccolingual direction obtained by conventional or computerized tomography are used in the image diagnosis of dental implant treatment. This study was performed to clarify the subjective image quality of the mandibular depiction by shifting the angles of the tomographic objective plane. A panoramic machine with a linear tomographic function was used to obtain cross-sectional tomographic images on bilateral first molar regions of 10 dried human mandibles. The angles of tomographic objective planes were shifted horizontally within a range of ±20° at intervals of 5° from the tomographic objective plane, which was automatically determined. The image qualities of 4 anatomical structures—alveolar crest, buccal and lingual cortical bone, and mandibular canal—were subjectively scored on a 5-point scale method. As a result, the permitted tomographic objective angles were from −1.7° to 2.5°, a range of 4.2° for all 4 anatomical structures. When this result was compared with a previous geometric result, the permitted range of the angles was quite narrow. The tomographic objective angles should be manually set in accordance with an optimal tomographic plane for individual patients by using the positioning technique in linear tomography.

Multiplicity and t-isomultiple ideals

Let V be an irreducible non degenerate variety in Pn; a classical geometric result says that degree (V) ≥ codim V + 1 and, if equality holds, V is said to be of minimal degree. Varieties of minimal degree has been classified by Del Pezzo and Bertini and they all are intersections of quadrics. The local version of this result is due to J. Sally who proved that if is a regular local ring and is a Cohen-Macaulay local ring of minimal multiplicity, according to the bound e(R) ≥ height (I) + 1 given by Abhyankar, then the tangent cone of R is intersection of quadrics and it is Cohen-Macaulay.