Search Results (14)

We study the Hilbert function of a union of a fixed set of double points and a prescribed number of general double points. Our main results are for double points of Veronese varieties and of Segre–Veronese varieties. Among the Segre–Veronese varieties, we study the ones with all factors of dimension one and the ones with two factors, one of them of dimension one. We give many examples with exceptional or controlled behavior for a small number of double points. Full article

Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. Blomenhofer and Casarotti proved important results on the embeddings of G-varieties, G being an algebraic group, embedded in the projectivations of an irreducible G-representation, proving that no proper secant variety is a cone. In this paper, we give other conditions which assure that no proper secant varieties of X are a cone, e.g., that X is G-homogeneous. We consider the Segre product of two varieties with the product action and the case of toric varieties. We present conceptual tests for it, and discuss the information we obtained from certain linear projections of X. For the Segre–Veronese embeddings of ${\mathbb{P}}^n\times {\mathbb{P}}^n$ with respect to forms of bidegree $(1,d)$, our results are related to the simultaneous rank of degree d forms in $n+1$ variables. Full article

Let $X\subset {\mathbb{P}}^n$, where $3\le n\le 5$, be an irreducible hypersurface of degree $d\ge 2$. Fix an integer $t\ge 3$. If $n=5$, assume $t\ge 4$ and $(t,d)\ne (4,2)$. Using the Differential Horace Lemma, we prove that ${\mathcal{O}}_X\left(t\right)$ is not secant defective. For a fixed X, our proof uses induction on the integer t. The key points of our method are that for a fixed X, we only need the case of general linear hyperplane sections of X in lower-dimension projective spaces and that we do not use induction on d, allowing an interested reader to tackle a specific X for $n>5$. We discuss (as open questions) possible extensions of some weaker forms of the theorem to the case where $n>5$. Full article

For all integers $n\ge 1$ and $k\ge 2$, the Hadamard product $v_1★\cdots ★v_k$ of k elements of $K^{n+1}$ (with K being the complex numbers or real numbers) is the element $v\in K^{n+1}$ which is the coordinate-wise product of $v_1,\dots ,v_k$ (introduced by Cueto, Morton, and Sturmfels for a model in Algebraic Statistics). This product induces a rational map $h:{\mathbb{P}}^n{\left(K\right)}^k⤏{\mathbb{P}}^n\left(K\right)$. When $K=\mathbb{C}$, $k=2$ and $X_i\left(\mathbb{C}\right)\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$, $i=1,2$ are irreducible, we prove four theorems for the case $dim{X}_2\left(\mathbb{C}\right)=1$, three of them with $X_2\left(\mathbb{C}\right)$ as a line. We discuss the existence (non-existence) of a cancellation law for ★-products and use the symmetry group of the Hadamard product. In the second part, we work over $\mathbb{R}$. Under mild assumptions, we prove that by knowing $X_1\left(\mathbb{R}\right)★\cdots ★X_k\left(\mathbb{R}\right)$, we know $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$. The opposite, i.e., taking and multiplying a set of complex entries that are invariant for the complex conjugation and then seeing what appears in the screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$, very often provides real ghosts, i.e., images that do not come from a point of $X_1\left(\mathbb{R}\right)\times \cdots \times X_k\left(\mathbb{R}\right)$. We discuss a case in which we certify the existence of real ghosts as well as a few cases in which we certify the non-existence of these ghosts, and ask several open questions. We also provide a scenario in which ghosts are not a problem, where the Hadamard data are used to test whether the images cover the full screen. Full article

We prove a strong theorem on the partial non-defectivity of secant varieties of embedded homogeneous varieties developing a general set-up for families of subvarieties of Grassmannians. We study this type of problem in the more general set-up of joins of embedded varieties. Joins are defined by taking a closure. We study the set obtained before making the closure (often called the open part of the join) and the set added after making the closure (called the boundary of the join). For a point q of the open part, we give conditions for the uniqueness of the set proving that q is in the open part. Full article

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X, we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions. Full article

In this article, we present a hyperbolic secant-squared distribution via the nonlinear evolution equation. Namely, for this equation, the probability density function of the hyperbolic secant-squared (HSS) distribution has been determined. The density of our model has a variety of shapes, including symmetric, left-skewed, and right-skewed. Eight distinct frequent list estimation methods have been proposed for estimating the parameters of our models. Additionally, these estimation techniques have been used to examine the behavior of the HSS model parameters using data sets that were generated randomly. To demonstrate how the findings may be used to model real data using the HSS distribution, we also use real data. Finally, the proposed justification can be applied to a variety of other complex physical models. Full article

Let X be a smooth projective variety and $f:X\to {\mathbb{P}}^r$ a morphism birational onto its image. We define the Terracini loci of the map f. Most results are only for the case $dimX=1$. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set $B⊊X$ or solutions containing a prescribed $p\in X$). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools. Full article

Stainless-steel has proven to be a first-class material with unique mechanical properties for a variety of applications in the building and construction industry. High ductility, strain hardening, durability and aesthetic appeal are only a few of them. From a specific point of view, its nonlinear stress–strain behaviour appears capable of providing a significant increase in the rotational capacity of stainless-steel connections. This, in turn, may provide significant benefits for the overall response of a structure in terms of capacity and ductility. However, the bulk of the research on stainless-steel that has been published so far has mostly ignored the analysis of the deformation capabilities of the stainless-steel connections and has mostly focused on the structural response of individual members, such as beams or columns. For such a reason, the present study aims to contribute to the general understanding of the behaviour of stainless-steel connections from a conceptual, numerical and design standpoint. After a brief review of the available literature, the influence of the use of stainless-steel for column–beam connections is discussed from a theoretical standpoint. As a novel contribution, a different approach to compute the pseudo-plastic moment resistance that takes into account the post-elastic secant stiffness of the stainless-steel is proposed. Successively, a refined finite element model is employed to study the failure of stainless-steel column–beam connections. Finally, a critical assessment of the employment of carbon-steel-based design guidelines for stainless-steel connections provided by the Eurocode 3 design (EN 1993-1-8) is performed. The findings prove the need for the development of novel design approaches and more precise capacity models capable of capturing the actual stainless-steel joint response and their impact on the overall ductility and capacity of the whole structure. Full article

Fix a format $(n_1+1)\times \cdots \times (n_k+1)$, $k>1$, for real or complex tensors and the associated multiprojective space Y. Let V be the vector space of all tensors of the prescribed format. Let $S(Y,x)$ denote the set of all subsets of Y with cardinality x. Elements of $S(Y,x)$ are associated to rank 1 decompositions of tensors $T\in V$. We study the dimension $\delta (2S,Y)$ of the kernel at S of the differential of the associated algebraic map $S(Y,x)\stackrel{}{\to }\mathbb{P}V$. The set ${\mathbb{T}}_1(Y,x)$ of all $S\in S(Y,x)$ such that $\delta (2S,Y)>0$ is the largest and less interesting x-Terracini locus for tensors $T\in V$. Moreover, we consider the one (minimally Terracini) such that $\delta (2A,Y)=0$ for all $A⊈S$. We define and study two different types of subsets of ${\mathbb{T}}_1(Y,x)$ (primitive Terracini and solution sets). A previous work (Ballico, Bernardi, and Santarsiero) provided a complete classification for the cases $x=2,3$. We consider the case $x=4$ and several extremal cases for arbitrary x. Full article

This paper introduces and further applies an approach to support the decision makers in construction projects differentiating among a variety of deep excavation supporting systems (DESSs). These kinds of problems include dealing with uncertainty in data, multi-criteria affecting the decision, and multi-alternatives to select one from them. The proposed approach combines the analytic hierarchy process (AHP) with the fuzzy technique for order of preference by similarity to ideal solution (fuzzy TOPSIS) in a multicriteria decision-making (MCDM) model. The MCDM model emphasize the ability to combine expert knowledge, cost calculations, and laboratory test results for soil properties to achieve the scope. The model proved it had a superior ability to deal with the complexity and vague data that are related to construction projects. Furthermore, it was applied to a real case study for a governmental housing project in Egypt. Secant pile walls, sheet pile walls, and soldier piles and lagging are selected and studied as being the most common DESSs and as they satisfy the project requirements. The model utilized four criteria and fourteen comparing factors, including site characteristics, safety, cost, and environmental impacts. Based on the results of the model application on the investigated case study, a decision was reached that using secant piles as a supporting system in this project is mostly preferred. Furthermore, sheet pile wall, and soldier piles and lagging, come next in the ranking order. A sensitivity analysis is carried out to investigate how sensitive the results are to the criteria weights. In addition, the paper discusses in detail the reasons and factors which affect and control the decision-making process. Full article

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector. Full article

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject. Full article

Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties. Full article