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In this paper, we deal with a particular class of rank two vector bundles (instanton bundles) on the Fano threefold of index one F:=F1×P1. We show that every instanton bundle on F can be described ...

A new Caristi-type inequality is considered and Caristi’s fixed point theorem for mappings of complete metric spaces is developed (in both the single- and set-valued cases). On the basis of this development mappings of complete metric spaces which are contractions with respect to a function of two vector arguments are studied. This function is not required to be a metric or even a continuous function. The proved theorems are generalizations of the Banach contraction principle and Nadler’s theorem.

In this paper, we study Cayley-transform-based gradient and conjugate gradient algorithms for optimization on Grassmann manifolds. We revisit the Cayley transform on Grassmann manifolds as a retraction in the framework of quotient manifolds constructed by Lie group actions and obtain an efficient formula for this retraction in low-rank cases. We also prove that this retraction is the restriction of the Cayley transform on Stiefel manifolds to horizontal spaces. To develop vector transports on Grassmann manifolds, we introduce a concept called induced vector transports on quotient manifolds. Based on this concept, three vector transports associated with the Cayley transform are obtained. The first vector transport is the traditional orthogonal projection onto horizontal spaces, whereas the other two vector transports are newly proposed herein. We show that one of the new vector transports satisfies the Ring–Wirth non-expansion condition and that the other is isometric. We also simplify the formulae of the new vector transports in low-rank cases. Riemannian gradient and conjugate gradient algorithms are established via the Cayley transform and the three abovementioned vector transports. Numerical experiments on two mean-of-subspaces problems demonstrate the effectiveness of the proposed algorithms.

The Canonical Correlation Analysis (CCA) estimates the correlation between two vector variables by maximizing the correlation of linear combinations of their respective components. Here, the CCA is used to find correlation patterns in the last five successive, per pairs, earthquakes ( [Image: see text] ) preceding 271 main shocks ( [Image: see text] ) that occurred in the Greek territory during 1964–2018. The vector variables have two components, the earthquake magnitude and interevent time. The statistical significance of CCA is determined by the standard parametric test along with two proposed randomization tests, one using random shuffling of each paired dataset and one using randomly selected pairs of successive earthquakes. Simulations were designed on synthetic data from vector variables having the statistical characteristics of the real observations. The results on uncorrelated variables showed the correct size for the two randomization tests but larger type I error for the parametric significance test for small sample size. For correlated variables, the test power was equally high for both test types. The application of CCA and the significance tests to the Greek seismicity evidence the significant correlation among the last five successive preshocks, proving to be a promising tool in an a posteriori short-term earthquake forecasting.

In linear algebra, the tensor product construction allows us to take two vector spaces and “multiply” them together to get a new vector space with a universal property regarding bilinear maps.

In this paper we introduce the modified Dirac operators $$\mathcal {M}_{\mathbf {x}}^\kappa f{:}= \,\partial _{\mathbf {x}}f-\frac{\kappa }{x_m}e_m\cdot f$$ and $$\mathcal {M}_{\mathbf {y}}^\tau f{:}= \,\partial _{\mathbf {y}}f-\frac{\tau }{y_m}e_m \cdot f$$ , where $$f:\Omega \subset \mathbb {R}_+^{m}\times \mathbb {R}_+^{m}\rightarrow {{\mathrm{\mathcal {C}\ell }}}_{0,m}$$ is differentiable function, and $${{\mathrm{\mathcal {C}\ell }}}_{0,m}$$ is the Clifford algebra generated by the basis vectors of $$\mathbb {R}^m$$ . We look for solutions $$f(\mathbf {x},\mathbf {y}) =f(\underline{x},x_m,\underline{y},y_m)$$ of the system $$\mathcal {M}_{\mathbf {x}}^\kappa f(\mathbf {x,y})= \mathcal {M}_{\mathbf {y}}^\tau f(\mathbf {x,y})=0$$ , where the first and third variables are invariant under rotations. These functions are called $$(\kappa ,\tau )$$ -hypermonogenic functions. We discuss about axially symmetric functions with respect to the symmetric group $$SO(m-1)$$ . Some examples of axially symmetric $$(\kappa ,\tau )$$ -hypermonogenic functions generated by homogeneous functions and hypergeometric functions are presented.

Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P \in \psi^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i \to M$, $i = 0,1$, and let $\alpha$ be an irreducible representation of the group $G$. Then $P$ induces a map $\pi_\alpha(P) : H^s(M; E_0)_\alpha \to H^{s-m}(M; E_1)_\alpha$ between the $\alpha$-isotypical components. We prove that the map $\pi_\alpha(P)$ is Fredholm if, and only if, $P$ is {\em transversally $\alpha$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$., Comment: eight pages, it explains the main differences in the discrete and non-discrete cases

In this paper, the attitude determination problem from two vector observations is revisited, incorporating the redundant equality constraint obtained by the dot product of vector observations. Analytical solutions to this constrained attitude determination problem are derived. It is found out that the studied two-vector attitude determination problem by Davenport q-method under the dot product constraint has deterministic maximum eigenvalue, which leads to its advantage in error/perturbation analysis and covariance determination. The proposed dot product constrained two-vector attitude solution is applied then to solve several engineering problems. Detailed simulations on spacecrafts attitude determination indicate the efficiency of the proposed theory.

Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and $E_1$, $E_2$ are vector bundles over $C$.In this paper we compute the pseudo effective cones of higher codimension cycles on $X$., Comment: Comments are welcome

We classify pairs of linear mappings (U -> V, U/U' -> V') in which U, V are finite dimensional vector spaces over a field IF, and U', are their subspaces. (C) 2016 Elsevier Inc. All rights re ...

Extended Kalman Filter (EKF) and Complementary Filter (CF) are the two most commonly-used attitude determination algorithms in inertial and magnetic measurement units. It is known that the only difference between the a posterior attitude estimates provided by EKF and CF respectively is the gain matrix (GM) assigned to innovation for attitude update. Through mathematical derivation, it is concluded that the GM of EKF can be simplified to the one of CF, which means that CF is an approximation to EKF. Monte Carlo simulations were done to validate what influence of these simplifications is put on the performance of EKF. The main finding is EKF is more stable than CF in condition of low sampling rate, and hence CF must rely more on the measurements of gyroscope for attitude determination to improve its stability.

The paper contains two main results that are obtained by Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices to the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of $A\!L^p$ and $c_0$ for the class of cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices., Comment: This a preprint of the work accepted for publication in Siberian Mathematical Journal; $\copyright$ 2019, Pleiades Publishing Ltd. http://pleiades.online/

Let E be a normalized, rank two vector bundle on P 3 . Let H be a general plane. If E is stable with c 2 ≥ 4 , we show that h 0 ( E H ( 1 ) ) ≤ 2 + c 1 . It follows that h 0 ( E ( 1 ) ) ≤ 2 + c 1 . We also show that if E is properly semi-stable and indecomposable, h 0 ( E H ( 1 ) ) = 3 .

In this paper we present a recurrent procedure to solve an inversion problem for monic bivariate Krawtchouk polynomials written in vector column form, giving its solution explicitly. As a by-product, a general connection problem between two vector column of monic bivariate Krawtchouk families is also explicitly solved. Moreover, in the non monic case and also for Krawtchouk families, several expansion formulas are given, but for polynomials written in scalar form.

The Cauchy-Kovalevskaya extension for polynomials in several variables is a crucial result in Clifford analysis which describes theoretically how to construct simplicial monogenics in m dimensions starting from certain polynomial spaces in m - 1 dimensions, hereby using the so-called branching rules. In [13] it was shown that the Cauchy-Kovalevskaya extension map is an isomorphism of the kernel of the wedge operator onto the space of spherical monogenics in two variables. The aim of this paper is to show that this extension maps simplicial polynomials of the kernel of the wedge operator just onto the simplicial monogenics.

We present a hybridization of the accelerated gradient method with two vector directions. This hybridization is based on the usage of a chosen three-term hybrid model. Derived hybrid accelerated double direction model keeps preferable properties of both included methods. Convergence analysis demonstrates at least linear convergence of the proposed iterative scheme on the set of uniformly convex and strictly convex quadratic functions. The results of numerical experiments confirm better performance profile in favor of derived hybrid accelerated double direction model when compared to its forerunners.

In a number of papers by Edge, and in a related paper by Fano, several properties are discussed of the family of scrolls of degree 8, in the complex projective space, whose plane sections are projected bicanonical models of a genus 3 curve C. This beautiful subject is implicitely related to the moduli of semistable rank two vector bundles on C with bicanonical determinant. We revisit and reconstruct this matter in modern terms with a modular point of view. This paper is the refereed version of a contribution to a volume in honor of W.L Edge, to be published by European Journal of Mathematics., Comment: 13 pages

Matrix optimization problems (MOPs) involving the Ky Fan $k$-norm arise frequently from many applications. In order to design algorithms to solve large scale MOPs involving the Ky Fan $k$-norm, we need to understand the first and second order properties of the Moreau--Yosida regularization of the Ky Fan $k$-norm function and the indicator function of the Ky Fan $k$-norm ball. According to the general theory on spectral functions, in this paper we shall conduct a thorough study on the Moreau--Yosida regularization of the vector $k$-norm function and the indicator function of the vector $k$-norm ball. In particular, we show that the proximal mappings associated with these two vector $k$-norm related functions both admit fast and analytically computable solutions. Moreover, we propose algorithms of low computational cost to compute the directional derivatives of these two proximal mappings and then completely characterize their Frechet differentiability. The work here thus builds the fundamental tools needed...

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ℙ n is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ℙ n .

We study the semistability of the tensor product of hermitian vector bundles by using the $\varepsilon$-tensor product and the geometric (semi)stability of vector subspaces in the tensor product of two vector spaces.

We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context of their previously mentioned result., accepted for publication in Forum Mathematicum

International audience; We consider the classical problem of estimating the attitude and gyro biases of a rigid body from at least two vector measurements and a triaxial rate gyro. We propose a solution based on a dynamic nonlinear estimator designed without respecting the geometry of SO(3), which achieves uniform global exponential convergence. The convergence is established thanks to a dynamically scaled Lyapunov function.

International audience; We give an explicit expression of the Hitchin Hamiltonian system for rank two vector bundles with trivial determinant bundle over a curve of genus two.

In this work, we show some global bifurcations for a class of three-dimensional discontinuous piecewise linear (DPWL) systems having a unique two-fold point of visible-invisible type. We consider the simplest case of DPWL systems, with two vector fields separated by a switching plane with a unique equilibrium point in each half-space.

This paper addresses the numerical solution of the linear-quadratic-Gaussian (LQG) for a new class of singularly perturbed (SP) system using fixed-point recursive algorithms. Traditionally, SP systems have been studied for one vector input controlling both slow and fast subsystems. We show that the LQG problem can be solved efficiently for systems containing two vector inputs governing the slow and fast subsystems independently. The recursive algorithm solution helps avoid numerical stiffness problems that can arise by solving the regulator and/or filter continuous algebraic Riccati equations (CAREs) using established methods. In addition, the solutions given in terms of reduced-order Lyapunov equations and CAREs simplify controller and Kalman filter designs.

The moduli space $M(c_2)$, of stable rank two vector bundles of degree one on a very general quintic surface $X\subset {\mathbb P}^3$, is irreducible for all $c_2\geq 4$ and empty otherwise., Comment: Adds a review of initial cases

A novel method, called Opposite Maps, is introduced with the aim of generating reduced sets for efficiently training of support vector machines (SVM) and least squares support vector machines (LSSVM) classifiers. The main idea behind the proposed approach is to train two vector quantization (VQ) algorithms (one for each class, \({\fancyscript{C}}_{-1}\) and \({\fancyscript{C}}_{+1}\), in a binary classification context) and then, for the patterns of one class, say \({\fancyscript{C}}_{-1}\), to find the closest prototypes among those belonging to the VQ algorithm trained with patterns of the other class, say \({\fancyscript{C}}_{+1}\). The subset of patterns mapped to the selected prototypes in both VQ algorithms form the reduced set to be used for training SVM and LSSVM classifiers. Comprehensive computer simulations using synthetic and real-world datasets reveal that the proposed approach is very efficient and independent of the type of VQ algorithm used.

The six-phase permanent magnet synchronous motor (PMSM) is designed with two sets of Y-connected windings phase shifted by 30 electrical degrees. Two mathematic models of six-phase PMSM are established by different methods, which are based on dual d-q transformation and decoupling vector space respectively. By the dual d-q transformation, the electromagnetic torque is calculated based on the two sets of Y-connected windings separately. By the decoupling vector transformation, extra vector subspace is established, in which the current components have no effect on the electromagnetic torque. According to the different models, two vector control systems for six-phase PMSM are presented. The relationship and equivalence of the two control methods are indicated through comparison analysis and open-loop simulation. And the closed-loop simulation and experiment results show that the two methods have consistent control performance

We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.

A rotational fluid model which can be used to describe broad vortical flows ranging from large scale to the atmospheric mesoscale and the oceanic submesoscale is studied by the symmetry group theory. After introducing one scalar-, two vector-, and two tensor potentials, we find that the Lie symmetries of the extended system include many arbitrary functions ofzand{z,t}. The obtained Lie symmetries are used to find some types of exact solutions. One of exact solutions can be used to qualitatively describe the three-dimensional structure of hurricanes.

In this paper, a generalized $\epsilon-$subdifferential, which was defined by a norm, is first introduced for a vector valued mapping. Some existence theorems and the properties of the generalized $\epsilon-$subdifferential are discussed. A relationship between the generalized $\epsilon-$subdifferential and a directional derivative is investigated for a vector valued mapping. Then, the calculus rules of the generalized $\epsilon-$subdifferential for the sum and the difference of two vector valued mappings were given. The positive homogeneity of the generalized $\epsilon-$subdifferential is also provided. Finally, as applications, necessary and sufficient optimality conditions are established for vector optimization problems.

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO ( n ) in SU ( n ) , the asymptotic of the restrictions of representations of SL ( n ) to SO ( n ) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.

Here we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover we show that every rank $r>2$ weakly uniform vector bundle with splitting type $a_{1,1}=...=a_{r,s}=0$ is trivial and every rank $r>2$ uniform vector bundle with splitting type $a_1>...>a_r$, splits., 6 pages no figures

Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

This paper describes a relationship between essentially finite groupoids and two-vector spaces. In particular, we show to construct two-vector spaces of Vect-valued presheaves on such groupoids. We define two-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation—a weak functor—from Span(FinGpd) (the bicategory of essentially finite groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.

By introducing the inner-product matrix of two vector functions and using conversion matrix, explicit formulas for the dual basis functions of Wang-Bezier type generalized Ball bases (WBGB) with respect to the Jacobi weight function are given. The dual basis functions with and without boundary constraints are also considered. As a result, the paper includes the weighted dual basis functions of Bernstein basis, Wang-Ball basis and some intermediate bases. Dual functionals for WBGB and the least square approximation polynomials are also obtained.

A linear map between two vector spaces has a very important characteristic: a determinant. In modern theory two generalizations of linear maps are intensively used: to linear complexes (the nilpotent chains of linear maps) and to non-linear mappings. Accordingly, determinant of a linear map has two generalizations: to determinants of complexes and to resultants. These quantities are in fact related: resultant of a non-linear map is determinant of the corresponding Koszul complex. We give an elementary introduction into these notions and interrelations, which will definitely play a role in the future development of theoretical physics.

It is proved that for any free A-modules F and ∊ of finite rank on some C-algebraized space (X,A) a degenerate A-bilinear morphism Φ : F × ∊ → A induces a non-degenerate A-bilinear morphism Φ : F/∊ ⊥ × ∊/F ⊥ → A, where ∊ ⊥ and F ⊥ are the orthogonal sub-A-modules associated with ∊ and F, respectively. This result generalizes the finite case of the classical result, which states that given two vector spaces W and V, paired into a field k, the induced vector spaces W/V ⊥ and V/W ⊥ have the same dimension. Some related results are discussed as well.

Given an m ∈ N and two vector spaces V and W , a function f : V m → W is called multi-Jensen if it satisfies Jensen’s equation in each variable separately. In this paper we unify these m Jensen equations to obtain a single functional equation for f and prove its stability in the sense of Hyers-Ulam, using the so-called direct method.

In this article we introduce a new notion of differential forms to describe the cohomology associated to the sheaf of regular functions in several quaternionic variables. We then use these differential forms to introduce and describe concretely a sheaf of quaternionic hyperfunctions as boundary values of regular functions in two quaternionic variables. We show how these ideas can be generalized to the case of monogenic functions in two vector variables with values in a Clifford algebra.

In this paper, we consider two vector versions of Minty's Lemma and obtain existence theorems for three kinds of vector variational-like inequalities. The results presented in this paper are extension and improvement of the corresponding results of other authors.

A function being the sum of two bilinear functions with one and the same first vector argument belonging to a polyhedron and the other two vector arguments belonging to another polyhedron is considered. It is shown that a certain minimum function of this sum and the maximin function of the sum (on the second polyhedron of connected variables) are continuous on corresponding polyhedra, which can be used in solving a maximin problem that is considered in the article.

SUMMARY The notion that the self-gravitation problem of a perfectly elastic solid body involves small strains from a reference state with arbitrary large initial stresses is reconfirmed and extended. It is shown that the Lagrangian, from which the equations of motion, the boundary conditions and Poisson's equation can be derived in Eulerian or Lagrangian coordinates by the Variational Principle, can be written in a general form which incorporates the various stress measures (Piola-Kirchhoff, Cauchy). Especially for an Eulerian description it is shown that the derived equations of motion and boundary conditions lead to a complete set of mutually orthogonal seismic normal modes. The Lagrangian which Geller (1988) gives is re-evaluated. It appears that in his considerations Geller neglected finite pre-stresses and only accounted for gravitational contributions. This, in general, is not correct. Though the static equilibrium equation has no unique solution for the initial stress components, this does not imply that in physical situations there would not be a specified initial pre-stress. Geller's statement that the asymmetric stress tensor Woodhouse & Dahlen (1978) employ in their Lagrangian necessarily leads to non-conservation of angular momentum is not valid. This stress tensor is a first Piola-Kirchhoff stress tensor, which is a so-called two-point tensor, associating two vector fields defined in different coordinate systems.

In the paper a mixed Nash equilibrium (NE) in bimatrix games is considered. Developing an algebraic approach, a necessary and sufficient condition for existence of the NE, the equilibrium profile, the value of its payoff are expressed with saddle point matrices of a special form, i.e. the block matrices with two vector blocks of ones. Mathematics Subject Classification: 91A05, 91A10, 15A24