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In this paper we consider a family of projective embeddings of the geometry \(A_{n,\{1,n\}}(\mathbb{F})\) of point-hyperplanes flags of \(\mathrm{PG}(n,\mathbb{F})\). The natural embedding \(\varepsilon_{\mathrm{nat}}\) is one of them. It maps every point-hyperplane flag \((p,H)\) onto the vector-line \(\langle {\bf x}\otimes\xi\rangle\), where \({\bf x}\) is a representative vector of \(p\) and \(\xi\) is a linear functional describing \(H\). The other embeddings have been discovered more than two decads ago by Thas and Van Maldeghem for the case \(n = 2\) and recently generalized to any \(n\) by De Schepper, Schillewaert and Van Maldeghem. They are obtained as twistings of \(\varepsilon_{\mathrm{nat}}\) by non-trivial automorphisms of \(\mathbb{F}\). Explicitly, for \(\sigma\in \mathrm{Aut}(\mathbb{F})\setminus\{\mathrm{id}_\mathbb{F}\}\), the twisting \(\varepsilon_\sigma\) of \(\varepsilon_{\mathrm{nat}}\) by \(\sigma\) maps \((p,H)\) onto \(\langle {\bf x}^\sigma\otimes \xi\rangle\). We shall prove that, when \(\mathrm{Aut}(\mathbb{F}) \neq \{\mathrm{id}_\mathbb{F}\}\), a geometric hyperplane \({\cal H}\) of \(A_{n,\{1,n\}}(\mathbb{F})\) arises from \(\varepsilon_{\mathrm{nat}}\) and at least one of its twistings or from at least two distinct twistings of \(\varepsilon_{\mathrm{nat}}\) if and only if \({\cal H} = \{(p,H)\in A_{n,\{1,n\}}(\mathbb{F}) \mid p\in A or a \in H\}\) for a possibly non-incident point-hyperplane pair \((a,A)\) of \(\mathrm{PG}(n,\mathbb{F})\). We call these hyperplanes quasi-singular hyperplanes. With the help of this result we shall prove that if \(|\mathrm{Aut}(\mathbb{F})| › 1\) then \(A_{n,\{1,n\}}(\mathbb{F})\) admits no absolutely universal embedding.Mathematics Subject Classifications: 51A45, 20F40, 15A69Keywords: Lie geometries, Segre varieties, embeddings, hyperplanes

Given a root system Φ of type A n , B n , C n , or D n in Euclidean space E, let W be the associated Weyl group. For a point p ∈ E not orthogonal to any of the roots in Φ , we consider the W-permutohedron P W , which is the convex hull of the W-orbit of p. The representation of W on the rational cohomology ring H ∗ (X Φ) of the toric variety X Φ associated to (the normal fan to) P W has been studied by various authors. Let { s 1 , ... , s n } be a complete set of simple reflections in W. For K ⊆ [ n ] , let W K be the standard parabolic subgroup of W generated by { s k : k ∈ K } . We show that the fixed subring H ∗ (X Φ) W K is isomorphic to the cohomology ring of the toric variety X Φ (K) associated to a polytope obtained by intersecting P W with half-spaces bounded by reflecting hyperplanes for the given generators of W K . We also obtain explicit formulas for h-vectors of these polytopes. By a result of Balibanu–Crooks, the cohomology rings H ∗ (X Φ (K)) are isomorphic with cohomology rings of certain regular Hessenberg varieties. [ABSTRACT FROM AUTHOR]

In this paper, we characterise point sets having the same intersection numbers with respect to hyperplanes as an ovoidal cone. In particular, we show that a set of points of PG(4 , q ) $\text{PG}(4,q)$ which blocks all planes and intersects solids in q + 1 $q+1$, q 2 + 1 ${q}^{2}+1$ or q 2 + q + 1 ${q}^{2}+q+1$ points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted. [ABSTRACT FROM AUTHOR]

Given positive integers k ≤ d and a finite field F , a set S ⊂ F d is (k, c)-subspace evasive if every k-dimensional affine subspace contains at most c elements of S. By a simple averaging argument, the maximum size of a (k, c)-subspace evasive set is at most c | F | d - k . When k and d are fixed, and c is sufficiently large, the matching lower bound Ω (| F | d - k) is proved by Dvir and Lovett. We provide an alternative proof of this result using the random algebraic method. We also prove sharp upper bounds on the size of (k, c)-evasive sets in case d is large, extending results of Ben-Aroya and Shinkar. The existence of optimal evasive sets has several interesting consequences in combinatorial geometry. We show that the minimum number of k-dimensional linear hyperplanes needed to cover the grid [ n ] d ⊂ R d is Ω d (n d (d - k) d - 1 ) , which matches the upper bound proved by Balko et al., and settles a problem proposed by Brass et al. Furthermore, we improve the best known lower bound on the maximum number of incidences between points and hyperplanes in R d assuming their incidence graph avoids the complete bipartite graph K c , c for some large constant c = c (d) . [ABSTRACT FROM AUTHOR]

Too much reliance has been placed on calculating single images meeting possibly arbitrary optimization criteria. By generating a dispersion of multiple images consistent with the data, we may be able to learn how wrong we could be. Our problem is to generate a way of seeing a representative sample of all of the solutions in the hyperplane of solutions, which would be an array of images, each of which is a solution to the equations. To accomplish this, we suggest that our sampling is in a space of image basis functions, rather than directly in the hyperplane. As the number of basis functions is large, we design a selection criterion for choosing a subset that reasonably spans the space of images. First, we try a random sampling, which gives high frequency or sequency samples. Then we turn to more systematic sampling, based on the methods developed for one-pixel imaging. Some numerical experiments demonstrate that the use of basis functions as starting images for ART-like iterative algorithms may suffice to span the hyperplane of solutions, allowing choices between solutions other than simple optimization of arbitrary criteria such as minimum norm or maximum entropy, or deconvolution of the point spread function of the algorithm. [ABSTRACT FROM AUTHOR]

Let V be a vector space over the binary field F 2 = { 0 , 1 } with dim V ≥ 2 . If W = { W 1 , ... , W n } is a family of hyperplanes of V, one says that W is an irredundant covering for V provided that W covers V, that is, ∪ i = 1 n W i = V and no proper subfamily of W covers V. We show that W is an irredundant covering for V if and only if the following three conditions are satisfied: (i) n is odd, (ii) dim V / ∩ i = 1 n W i = n − 1 , and (iii) any (n − 1) -fold intersection of the elements of W has codimension n – 1. Moreover, we show that for any odd integer n with 3 ≤ n ≤ dim V + 1 there is an irredundant hyperplane covering for V with n elements. [ABSTRACT FROM AUTHOR]

Twin support vector machine (TWSVM) and twin support vector regression (TSVR) are newly emerging efficient machine learning techniques which offer promising solutions for classification and regression challenges respectively. TWSVM is based upon the idea to identify two nonparallel hyperplanes which classify the data points to their respective classes. It requires to solve two small sized quadratic programming problems (QPPs) in lieu of solving single large size QPP in support vector machine (SVM) while TSVR is formulated on the lines of TWSVM and requires to solve two SVM kind problems. Although there has been good research progress on these techniques; there is limited literature on the comparison of different variants of TSVR. Thus, this review presents a rigorous analysis of recent research in TWSVM and TSVR simultaneously mentioning their limitations and advantages. To begin with, we first introduce the basic theory of support vector machine, TWSVM and then focus on the various improvements and applications of TWSVM, and then we introduce TSVR and its various enhancements. Finally, we suggest future research and development prospects. [ABSTRACT FROM AUTHOR]

The purpose of representation learning is to encode the entities and relations in a knowledge graph as low-dimensional and real-valued vectors through machine learning technology. Traditional representation learning methods like TransE, a method which models relationships by interpreting them as translations operating on the low-dimensional embeddings of a graph's entities, are effective for learning the embeddings of knowledge bases, but struggle to effectively model complex relations like one-to-many, many-to-one, and many-to-many. To overcome the above issues, we introduce a new method for knowledge representation, reasoning, and completion based on multi-translation principles and TransE (TransE-MTP). By defining multiple translation principles (MTPs) for different relation types, such as one-to-one and complex relations like one-to-many, many-to-one, and many-to-many, and combining MTPs with a typical translating-based model for modeling multi-relational data (TransE), the proposed method, TransE-MTP, ensures that multiple optimization objectives can be targeted and optimized during training on complex relations, thereby providing superior prediction performance. We implement a prototype of TransE-MTP to demonstrate its effectiveness at link prediction and triplet classification on two prominent knowledge graph datasets: Freebase and Wordnet. Our experimental results show that the proposed method enhanced the performance of both TransE and knowledge graph embedding by translating on hyperplanes (TransH), which confirms its effectiveness and competitiveness. [ABSTRACT FROM AUTHOR]

The article discusses a correction to a previous construction of the Salvetti complex of a real hyperplane arrangement. It is noted that the chosen linear extension in the previous construction cannot be arbitrary, and the results remain valid with the use of an appropriate total ordering of the chambers. The article also presents counterexamples to a claim made in the previous construction and discusses open questions regarding nonrealizable oriented matroids and Euclidean orderings. The implications of the correction on further results are also discussed. [Extracted from the article]

The Euler characteristic transform (ECT) is an integral transform used widely in topological data analysis. Previous efforts by Curry et al. and Ghrist et al. have independently shown that the ECT is injective on all compact definable sets. In this work, we first study the injectivity of the ECT on definable sets that are not necessarily compact and prove a complete classification of constructible functions that the Euler characteristic transform is not injective on. We then introduce the quadric Euler characteristic transform (QECT) as a natural generalization of the ECT by detecting definable shapes with quadric hypersurfaces rather than hyperplanes. We also discuss some criteria for the injectivity of QECT. [ABSTRACT FROM AUTHOR]

The amount of hyperplanes that are needed in order to cover the Boolean cube has been studied in various contexts. Linial and Radhakrishnan introduced the notion of essential covers. An essential cover is a collection of hyperplanes that form a minimal cover of the vertices of the hypercube, and every coordinate is influential in at least one of the hyperplanes. Linial and Radhakrishnan proved using algebraic tools that every essential cover of the n-cube must be of size at least Ω (n) . We devise a stronger lower bound method, and show that the size of every essential cover is at least Ω (n 0.52) . This result has implications in proof complexity, because essential covers have been used to prove lower bounds for several proof systems. [ABSTRACT FROM AUTHOR]

In a discrete group generated by hyperplane reflections in the 푛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the 푛-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the 푛-dimensional hyperbolic space without common boundary points have a unique common perpendicular. [ABSTRACT FROM AUTHOR]

We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus 3, 4 or 5. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the 2-torsion subgroup. Using p-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton–Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the 2-torsion points. We demonstrate the practicality of our method by computing the 2-torsion of the modular Jacobians J0(N) for N = 42, 55, 63, 72, 75. As a result of this we are able to verify the generalized Ogg conjecture for these values. [ABSTRACT FROM AUTHOR]

In this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc. [ABSTRACT FROM AUTHOR]

A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e., is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e., asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary. [ABSTRACT FROM AUTHOR]

Sea surface target detection is a key stage in a typical target detection system and directly influences the performance of the whole system. As an effective discriminator, the one-class support vector machine (OCSVM) has been widely used in target detection. In OCSCM, training samples are first mapped to the hypersphere in the kernel space with the Gaussian kernel function, and then, a linear classification hyperplane is constructed in each cluster to separate target samples from other classes of samples. However, when the distribution of the original data is complex, the transformed data in the kernel space may be nonlinearly separable. In this situation, OCSVM cannot classify the data correctly, because only a linear hyperplane is constructed in the kernel space. To solve this problem, a novel one-class classification algorithm, referred to as ensemble one-class support vector machine (En-OCSVM), is proposed in this paper. En-OCSVM is a hybrid model based on k -means clustering and OCSVM. In En-OCSVM, training samples are clustered in the kernel space with the k -means clustering algorithm, while a linear decision hyperplane is constructed in each cluster. With the combination of multiple linear classification hyperplanes, a complex nonlinear classification boundary can be achieved in the kernel space. Moreover, the joint optimization of the k -means clustering model and OCSVM model is realized in the proposed method, which ensures the linear separability of each cluster. The experimental results based on the synthetic dataset, benchmark datasets, IPIX datasets, and SAR real data demonstrate the better performance of our method over other related methods. [ABSTRACT FROM AUTHOR]

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon.65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the 1 -dimensional heat equation u t = u x x on ℝ 2 with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the n -dimensional heat equation u t = △ u on ℝ n + 1 with zero initial data (i.e. vanishing on the hyperplane P : { t = 0 }) can be easily obtained as a consequence of the examples on ℝ 2. Finally, given an arbitrary hyperplane P ⊂ ℝ n + 1 , we can construct a family of nonzero solutions which vanish on P ⊂ ℝ n + 1. [ABSTRACT FROM AUTHOR]

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let r, n be positive integers with n ≥ 6r. Let Pn ∩ Mr be the space of polynomials of degree at most n on [0, 1] with integer coefficients such that P(k)(0)/k! and P(k)(1)/k! are integers for k = 0,. . ., r - 1 and let PZn ∩ Mr be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of PZn ∩ Mr from Pn ∩ Mr in L2(0, 1). We give rather precise quantitative estimations for successive minima of PZn in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in Pn ∩ Mr. [ABSTRACT FROM AUTHOR]

For any Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \}, we establish a structure theory for the class \mathcal {S}_Q of stable codimension 1 stationary integral varifolds admitting no classical singularities of density

We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated with root lattices. We give a uniform construction of 147 hyperplane arrangements on type IV symmetric domains for which the algebras of modular forms with constrained poles are free and therefore the Looijenga compactifications of the arrangement complements are weighted projective spaces. We also construct eight free algebras of modular forms on complex balls with poles on hyperplane arrangements. The most striking example is the discriminant kernel of the 2U ⊕D11 lattice, which admits a free algebra on 14 meromorphic generators. Along the way, we determine minimal systems of generators for non-free algebras of orthogonal modular forms for 26 reducible root lattices and prove the modularity of formal Fourier-Jacobi series associated with them. By exploiting an identity between weight 1 singular additive and multiplicative lifts on 2U ⊕ D11, we prove that the additive lift of any (possibly weak) theta block of positive weight and q-order 1 is a Borcherds product. The special case of holomorphic theta blocks of one elliptic variable is the theta block conjecture of Gritsenko, Poor and Yuen. [ABSTRACT FROM AUTHOR]

In this paper, we establish some modified defect relations for the Gauss map g$g$ of a complete minimal surface S⊂Rm$S\subset \mathbb {R}^m$ into a k$k$‐dimension projective subvariety V⊂Pn(C)(n=m−1)$V\subset \mathbb {P}^n(\mathbb {C})\ (n=m-1)$ with hypersurfaces Q1,…,Qq$Q_1,\ldots,Q_q$ of Pn(C)$\mathbb {P}^n(\mathbb {C})$ in N$N$‐subgeneral position with respect to V(N≥k)$V\ (N\ge k)$. In particular, we give the upper bound for the number q$q$ if the image g(S)$g(S)$ intersects each hypersurface Q1,…,Qq$Q_1,\ldots,Q_q$ a finite number of times and g$g$ is nondegenerate over Id(V)$I_d(V)$, where d=lcm(degQ1,…,degQq)$d=\text{lcm}(\deg Q_1,\ldots,\deg Q_q)$, that is, the image of g$g$ is not contained in any hypersurface Q$Q$ of degree d$d$ with V⊄Q$V\not\subset Q$. Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces. [ABSTRACT FROM AUTHOR]

Let |$\mathcal{X}$| be a skew-symmetrizable cluster Poisson variety. The cluster complex |$\Delta ^{+}(\mathcal{X})$| was introduced in [ 12 ]. It codifies the theta functions on |$\mathcal{X}$| that restrict to a character of a seed torus. Every seed |$\textbf{s}$| for |$\mathcal{X}$| determines a fan realization |$\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$| of |$\Delta ^{+}(\mathcal{X})$|⁠. For every |$\textbf{s}$| we provide a simple and explicit description of the cones of |$\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$| and their facets using c -vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of |$\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$| in terms of |$F$| -polynomials. In case |$\mathcal{X}$| is skew-symmetric and the quiver |$Q$| associated to |$\textbf{s}$| is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of |$\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$| using g -vectors of (non-necessarily rigid) objects in |$\textsf{K}^{\textrm{b}}(\textrm{proj} \; kQ)$|⁠. [ABSTRACT FROM AUTHOR]

In this note, we are mainly interested in orthogonality, including Birkhoff orthogonality, isosceles orthogonality, Pythagorean orthogonality and approximate Birkhoff orthogonality ⊥ B ϵ . First, we show the relation between linear functionals and hyperplanes by means of approximate Birkhoff orthogonality. Second, we establish the sufficient conditions for Birkhoff orthogonality and approximate Birkhoff orthogonality. Then some sufficient and necessary conditions for isosceles orthogonality and Pythagorean orthogonality of bounded operators are shown. In addition, we characterize the inner product space in terms of functional orthogonality. [ABSTRACT FROM AUTHOR]

Classical jittered sampling partitions [ 0 , 1 ] d into m d cubes for a positive integer m and randomly places a point inside each of them, providing a point set of size N = m d with small discrepancy. The aim of this note is to provide a construction of partitions that works for arbitrary N and improves straight-forward constructions. We show how to construct equivolume partitions of the d-dimensional unit cube with hyperplanes that are orthogonal to the main diagonal of the cube. We investigate the discrepancy of such point sets and optimise the expected discrepancy numerically by relaxing the equivolume constraint using different black-box optimisation techniques. [ABSTRACT FROM AUTHOR]

We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ for the following fractional Hardy–Hénon system: \[ \left\{\begin{array}{@{}ll} (-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x), & x\in\mathbb{R}_+^n, \\ (-\Delta)^{s_2}v(x)=|x|^\beta u^q(x), & x\in\mathbb{R}_+^n, \\ u(x)\geq 0, & v(x)\geq 0,\ x\in\mathbb{R}_+^n, \end{array}\right. \] where $0 , $n>2\max \{s_1,s_2\}$. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of $(p,q)$ under some corresponding assumptions of $\alpha,\beta$ via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$ , one of our results shows that one domain of $(p,q)$ , where nonexistence of anti-symmetric solutions with appropriate decay conditions at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropriate condition of $\alpha, \beta$. [ABSTRACT FROM AUTHOR]

We study the global approximate controllability of the reaction-diffusion process in a parallelepiped $ \Omega = (a_1, b_1) \times \ldots (a_n, b_n) \subset R^n $, controlled via a multiplicative factor in a reaction term. It is assumed that the initial state $ u_0 $ admits zeros only on the intersections of $ \Omega $ with finitely many hyperplanes, parallel to the sides of $ \Omega $, and that $ u_0 $ changes its sign after crossing such hyperplanes (we further refer to them as the 'hyperplanes of change of sign' or simply the 'zero hyperplanes'). This paper can be viewed as a continuation of research presented in [2, 3] for the controllability of the one dimensional reaction-diffusion equation with solutions admitting finitely many zeros. However, the methods of [2, 3] are intrinsically one dimensional. In this paper we introduce a new approach designed to deal with the case of multiple spatial variables. [ABSTRACT FROM AUTHOR]

Processes of random tessellations of the Euclidean space $\mathbb{R}^d$ , $d\geq 1$ , are considered that are generated by subsequent division of their cells. Such processes are characterized by the laws of the life times of the cells until their division and by the laws for the random hyperplanes that divide the cells at the end of their life times. The STIT (STable with respect to ITerations) tessellation processes are a reference model. In the present paper a generalization concerning the life time distributions is introduced, a sufficient condition for the existence of such cell division tessellation processes is provided, and a construction is described. In particular, for the case that the random dividing hyperplanes have a Mondrian distribution—which means that all cells of the tessellations are cuboids—it is shown that the intrinsic volumes, except the Euler characteristic, can be used as the parameter for the exponential life time distribution of the cells. [ABSTRACT FROM AUTHOR]

We establish a new Harnack inequality for nonnegative solutions to the p(x)-Laplace equation with two-phase exponent p(x) taking two constant values p and q in the case where the phase interface is a hyperplane. [ABSTRACT FROM AUTHOR]

In the field of machine learning, clustering has become an increasingly popular research topic due to its critical importance. Many clustering algorithms have been proposed utilizing a variety of approaches. This study focuses on clustering of high-dimensional data using the maximum margin clustering approach. In this paper, two methods are introduced: The first method employs the classical maximum margin clustering approach, which separates data into two clusters with the greatest margin between them. The second method takes cluster compactness into account and searches for two parallel hyperplanes that best fit to the cluster samples while also being as far apart from each other as possible. Additionally, robust variants of these clustering methods are introduced to handle outliers and noise within the data samples. The stochastic gradient algorithm is used to solve the resulting optimization problems, enabling all proposed clustering methods to scale well with large-scale data. Experimental results demonstrate that the proposed methods are more effective than existing maximum margin clustering methods, particularly in high-dimensional clustering problems, highlighting the efficacy of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that if |$E\subset {{\mathbb {R}}}^{n+1}$| is an |$s$| -AD regular compact set, with |$s\in [n-\frac 12,n)$|⁠ , and |$E$| is contained in a hyperplane or, more generally, in an |$n$| -dimensional |$C^{1}$| manifold, then the Hausdorff dimension of the harmonic measure for the domain |${{\mathbb {R}}}^{n+1}\setminus E$| is strictly smaller than |$s$|⁠ , that is, than the Hausdorff dimension of |$E$|⁠. [ABSTRACT FROM AUTHOR]

For any genuinely ramified morphism f : Y ⟶ X between irreducible smooth projective curves we prove that (Y × X Y) ∖ Δ ‾ is connected, where Δ ⊂ Y × X Y is the diagonal. Using this result the following are proved: (1) If f is further Morse then the Galois closure is the symmetric group S d , where d = degree (f). (2) The Galois group of the general projection, to a line, of any smooth curve X ⊂ P n of degree d , which is not contained in a hyperplane and contains a non-flex point, is S d. [ABSTRACT FROM AUTHOR]

Suppose A = { a 1 , ... , a n + 2 } ⊂ Z n has cardinality n + 2 , with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = (f 1 , ... , f n) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixedn, counts exactly the number of real roots of F in time polynomial in log (d H) . We also discuss a number-theoretic hypothesis that would imply a further speed-up to time polynomial in n as well. [ABSTRACT FROM AUTHOR]

Some questions concerning the inversion of the classical and generalized integral Radon transforms are discussed. The main issue is to determine information about the integrand if the values of some integrals are known. A feature of this work is that a function is integrated over hyperplanes in a finite-dimensional Euclidean space and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. The independent variables describing the known integrals are fewer than those in the unknown integrand. We consider discontinuous integrands defined on specifically introduced pseudoconvex sets. A Stefan-type problem of finding discontinuity surfaces of the integrand is posed. Formulas for solving the problem under study are derived by applying special integro-differential operators to known data. [ABSTRACT FROM AUTHOR]

Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some $k\geq 2$ , and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J -groups of Achar and Aubert ['On rank 2 complex reflection groups', Comm. Algebra 36 (6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of 'toric reflection groups' that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter's truncations of the $3$ -strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest. [ABSTRACT FROM AUTHOR]

The traditional NSGA- II algorithm uses the crowding degree as the second index of elite selection, which is difficult to maintain the balance of population convergence and diversity when dealing with high-dimensional multi-objective optimization problems due to insufficient selection pressure and intensified optimization conflicts between different objectives. To solve these problems, an improved NSGA- II algorithm based on external archive updating and truncation mechanism was proposed: NSGA- II -UTEA. The algorithm firstly introduces the external archiving mechanism based on decision variable decomposition into elite selection, then updates the external archiving according to the sum of the weight vector and hyperplane distance between individuals, and realizes the truncation of external archiving based on the Angle calculation between individuals, which further improves the convergence and diversity of the algorithm in the high-dimensional multi-objective optimization problem. Compared with the five classical evolutionary algorithms, NSGA-II, NSGA-III, MOEA/D, NSGA-II-ARSBX and RPD-NSGA-E. The experimental results show that NSGA- II -UTEA algorithm is superior to other algorithms in the performance indexes of high-dimensional DTLZ and WFG series test functions with more than 10 targets, and has significant advantages in the distribution and diversity of solution sets. In particular, the best performance index values can be obtained for most high-dimensional WFG4-7 concave problems. Compared with the traditional NSGA- II algorithm, the IGD performance of NSGA- II -UTEA algorithm is improved by 50.6% on average on the high-dimensional DTLZ series test functions with more than 10 targets. In the high-dimensional WFG series test functions with 15 targets and above, the HV performance is improved by 60.7% on average. Experimental results verify the effectiveness of the improved NSGA- II -UTEA algorithm. [ABSTRACT FROM AUTHOR]

We consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$ , denoted by $\gamma_f$ , in terms of the weights and degrees of f. As a consequence, a necessary condition for a corank 1 finitely determined map germ $g\,{:}\,(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $\gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular. [ABSTRACT FROM AUTHOR]

Given a one-parameter family of continuous linear operators , with , we consider the optimal recovery of the values of on the whole space by approximate information on the values of , where runs over a compact set and . We find a family of optimal methods for recovering the values of . Each of these methods uses approximate measurements at no more than two points in and depends linearly on these measurements. As a corollary, we provide some families of optimal methods for recovering the solution of the heat equation at a given moment of time from inaccurate measurements on other time intervals and for solving the Dirichlet problem for a half-space on a hyperplane by inaccurate measurements on other hyperplanes. The optimal recovery of the values of from the indicated information reduces to finding the value of an extremal problem for the maximum with continuum many inequality-type constraints, i.e., to finding the exact upper bound of the maximized functional under these constraints. This rather complicated task reduces to the infinite-dimensional problem of linear programming on the vector space of all finite real measures on the -algebra of Lebesgue measurable sets in . This problem can be solved by some generalization of the Karush–Kuhn–Tucker theorem, and its significance coincides with the significance of the original problem. [ABSTRACT FROM AUTHOR]

Let F (z 1 , ⋯ , z d) be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and provide the first software package to verify and compute asymptotics in non-smooth cases of analytic combinatorics in several variables. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists. [ABSTRACT FROM AUTHOR]

The celebrated Blahut–Arimoto algorithm computes the capacity of a discrete memoryless point-to-point channel by alternately maximizing the objective function of a maximization problem. This algorithm has been applied to degraded broadcast channels, in which the supporting hyperplanes of the capacity region are again cast as maximization problems. In this work, we consider general broadcast channels and extend this algorithm to compute inner and outer bounds on the capacity regions. Our main contributions are as follows: first, we show that the optimization problems are max–min problems and that the exchange of minimum and maximum holds; second, we design Blahut–Arimoto algorithms for the maximization part and gradient descent algorithms for the minimization part; third, we provide convergence analysis for both parts. Numerical experiments validate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

We provide a concrete isometric description of all the preduals of ⃵1 for which the standard basis in ⃵1 has a finite number of w*-limit points. Then, we apply this result to give an example of an ⃵1-predual X such that its dual X* lacks the weak* fixed point property for nonexpansive mappings (briey, wα-FPP), but X does not contain an isometric copy of any hyperplane Wα of the space c of convergent sequences such that Wα is a predual of ⃵1 and W*α ff lacks the w*-FPP. This answers a question left open in the 2017 paper of the present authors. [ABSTRACT FROM AUTHOR]

This paper introduces a control strategy to recover from various spin motions to level trim flight conditions, utilizing a fixed-time nonsingular terminal sliding mode approach to minimize altitude loss. Firstly, an integral terminal sliding mode surface is constructed to guarantee the tracking errors of the spin recovery system converge to the origin within a fixed time and avoid the singularity problem. Next, a fixed-time type reaching law is proposed to weaken the chattering phenomenon and guarantee terminal sliding hyperplanes converge to zero in a fixed time. Furthermore, the stability analysis indicates the fixed-time convergence stability of the spin recovery control system and provides an accurate calculation of the settling time. Finally, comparative simulations are performed to validate the effectiveness and superiority of the developed spin recovery control scheme. [ABSTRACT FROM AUTHOR]

In this paper, we find surprisingly small Oka domains in Euclidean spaces |${\mathbb {C}}^n$| of dimension |$n>1$| at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set |$E$| in |${\mathbb {C}}^n$|⁠ , we show that |${\mathbb {C}}^n\setminus E$| is an Oka domain. In particular, there are Oka domains only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives smooth families of real hypersurfaces |$\Sigma _t\subset {\mathbb {C}}^n$| for |$t\in {\mathbb {R}}$| dividing |${\mathbb {C}}^n$| in an unbounded hyperbolic domain and an Oka domain such that at |$t=0$|⁠ , |$\Sigma _0$| is a hyperplane and the character of the two sides gets reversed. More generally, we show that if |$E$| is a closed set in |${\mathbb {C}}^n$| for |$n>1$| whose projective closure |$\overline E\subset \mathbb {C}\mathbb {P}^n$| avoids a hyperplane |$\Lambda \subset \mathbb {C}\mathbb {P}^n$| and is polynomially convex in |$\mathbb {C}\mathbb {P}^n\setminus \Lambda \cong {\mathbb {C}}^n$|⁠ , then |${\mathbb {C}}^n\setminus E$| is an Oka domain. [ABSTRACT FROM AUTHOR]

Let P λ Σ n be the Ehrhart polynomial associated to an integral multiple λ of the standard simplex Σ n ⊂ ℝ n . In this paper, we prove that if (M , L) is an n -dimensional polarized toric manifold with associated Delzant polytope Δ and Ehrhart polynomial P Δ such that P Δ = P λ Σ n , for some λ ∈ ℤ + , then (M , L) ≅ (ℂ P n , O (λ)) (where O (1) is the hyperplane bundle on ℂ P n ) in the following three cases: (1) arbitrary n and λ = 1 , (2) n = 2 and λ = 3 and (3) λ = n + 1 under the assumption that the polarization L is asymptotically Chow semistable. [ABSTRACT FROM AUTHOR]

In this paper, we consider the matter of on-line clustering of hyperplanes within the presence of bounded noise. This is often a challenging problem that involves the segmentation of the data and the estimation of the hyperplanes parameters. The proposed algorithm consists in two successive steps. At whenever, the first step allows to assign the current data point to the most appropriate hyperplane and therefore the second step realizes an update of the parameters of the hyperplane that contains the data point. The second step springs from an Outer Bounding Ellipsoid type algorithm suitable for on-line parameters estimation within the presence of bounded noise. The performance of our algorithm is proven using synthetic and real data. [ABSTRACT FROM AUTHOR]

A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Sweeps of a point configuration are in bijection with faces of an associated sweep polytope. Mimicking the fact that sweep polytopes are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. We show strong ties between sweep oriented matroids and both modular hyperplanes and Dilworth truncations from (unoriented) matroid theory. Pseudo-sweeps are a generalization of sweeps in which the sweeping hyperplane is allowed to slightly change direction, and that can be extended to arbitrary oriented matroids in terms of cellular strings. We prove that for sweepable oriented matroids, sweep oriented matroids provide a sphere that is a deformation retract of the poset of pseudo-sweeps. This generalizes a property of sweep polytopes (which can be interpreted as monotone path polytopes of zonotopes), and solves a special case of the strong Generalized Baues Problem for cellular strings. A second generalization are allowable graphs of permutations: symmetric sets of permutations pairwise connected by allowable sequences. They have the structure of acycloids and include sweep oriented matroids. [ABSTRACT FROM AUTHOR]

For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When n=3, we recover the recursive description of characters from recent work of Linyuan Liu, while for general n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters. [ABSTRACT FROM AUTHOR]

We classify globally generated vector bundles on a general hyperplane section of P² × P² embedded by the Segre embedding, considering small first Chern classes c1 = (1, 1) and c1 = (2, 1). [ABSTRACT FROM AUTHOR]

Let V $V$ be an n $n$‐dimensional vector space over Fq ${{\mathbb{F}}}_{q}$ and H ${\rm{ {\mathcal H} }}$ is any set of k $k$‐dimensional subspaces of V $V$. We construct two incidence structures Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ and Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$ using subspaces from H ${\rm{ {\mathcal H} }}$. The points are subspaces from H ${\rm{ {\mathcal H} }}$. The blocks of Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ are indexed by all hyperplanes of V $V$, while the blocks of Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$ are indexed by all subspaces of dimension 1. We show that Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ and Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$ are dual in the sense that their incidence matrices are dependent, one can be calculated from the other. Additionally, if H ${\rm{ {\mathcal H} }}$ is a t−(n,k,λ)q $t-{(n,k,\lambda)}_{q}$‐design we prove new matrix equations for incidence matrices of Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ and Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$. [ABSTRACT FROM AUTHOR]