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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 ...

Gravitational waves in general relativity contain two polarization degrees of freedom, commonly labeled plus and cross. Besides those two tensor modes, generic theories of gravity predict up to four additional polarization modes: two scalar and two vector. Detection of nontensorial modes in gravitational wave data would constitute a clean signature of physics beyond general relativity. Previous measurements have pointed to the unambiguous presence of tensor modes in gravitational waves, but the presence of additional generic nontensorial modes has not been directly tested. We propose a model-independent analysis capable of detecting and characterizing mixed tensor and nontensor components in transient gravitational wave signals, including those from compact binary coalescences. This infrastructure can constrain the presence of scalar or vector polarization modes on top of the tensor modes predicted by general relativity. Our analysis is morphology-independent (as it does not rely on a waveform templates), phase-coherent, and agnostic about the source sky location. We apply our analysis to data from GW190521 and simulated data and demonstrate that it is capable of placing upper limits on the strength of nontensorial modes when none are present, or characterizing their morphology in the case of a positive detection. Tests of the polarization content of a transient gravitational wave signal hinge on an extended detector network, wherein each detector observes a different linear combination of polarization modes. We therefore anticipate that our analysis will yield precise polarization constraints in the coming years, as the current ground-based detectors LIGO Hanford, LIGO Livingston, and Virgo are joined by KAGRA and LIGO India.

Generalized eigenvalue decomposition has many advantages when it is applied in modern signal processing. Compared with other methods, neural network model-based algorithms provide an efficient way to solve such problems online. Generalized feature extraction algorithms based on neural network models have been described in the literature. However, the majority of the existing algorithms can only extract the principal generalized eigenvector(s) or eigensubspace. To extract principal and minor generalized eigenvectors from two vector sequences, in this paper, two different information criteria are proposed, and a unified algorithm for the extraction of multiple components in a parallel way by simply altering the sign is derived based on these information criteria, which is feasible for generalized principal and minor component analysis. Moreover, all the corresponding principal and minor generalized eigenvalues can be extracted simultaneously because the desired equilibrium point depends on these values. Thus, the proposed algorithm can perform multiple generalized eigenpair extraction. The proposed algorithm possesses four properties: unification, self-stability, parallel extraction and generalized eigenpair extraction, that few of the existing algorithms can encompass. The global convergence and self-stability property of the proposed algorithm are proved through the Lyapunov method and ordinary differential equation method, respectively. The proposed algorithm has a fast convergence speed, high precision and strong tracking ability. Finally, numerical examples and applications are explored to further demonstrate the efficiency of the proposed algorithm.

Purpose The numerical computation of magnetization processes in moving and rotating assemblies requires the usage of vector hysteresis models. A commonly used model is the so-called Mayergoyz vector Preisach model, which applies the scalar Preisach model into multiple angles of the halfspace. The usage of several scalar models, which are optionally weighted differently, enables the description of isotropic as well as anisotropic materials. The flexibility is achieved, however, at the cost of multiple scalar model evaluations. For solely isotropic materials, two vector Preisach models, based on an extra rotational operator, might offer a lightweight alternative in terms of evaluation cost. The study aims at comparing the three mentioned models with respect to computational efficiency and practical applicability. Design/methodology/approach The three mentioned vector Preisach models are compared with respect to their computational costs and their representation of magnetic polarization curves measured by a vector vibrating sample magnetometer. Findings The results prove the applicability of all three models to practical scenarios and show the higher efficiency of the vector models based on rotational operators in terms of computational time. Originality/value Although the two vector Preisach models, based on an extra rotational operator, have been proposed in 2012 and 2015, their practical application and inversion has not been tested yet. This paper not only shows the usability of these particular vector Preisach models but also proves the efficiency of a special stageless evaluation approach that was proposed in a former contribution.

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.

We show that vector solitons can be stable in nonlinear fractional Schrodinger equations with one-dimensional parity-time-symmetric optical lattices. The families of vector solitons with two propagation constants that are present in the different gaps are investigated. It is found that the Levy index cannot change the phase transition point, but it will influence the solitons existence and stability. The effective widths of the two vector soliton components shrink as the Levy index decreases. Some unique soliton propagation properties are found, and soliton propagation simulations are performed to authenticate the results of the stability analyses.

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.

We present analytical results for massless correlators of two vector, tensor, and scalar composite vertices with the Bjorken fractions x and y of order $$ {\alpha}_s^2 $$ α s 2 β0 of QCD. The structure of these correlators ΠV,T,S(x, y; p2) and properties of its main elements are discussed in detail. Special attention is paid to verifying the results and comparing them with known particular cases. We apply the correlators to evaluate radiative corrections to the distribution amplitudes of light mesons within the QCD sum rules.

The inner product argument is an effective tool to reduce communication complexity in many cryptographic protocols. Bootle et al. (EUROCRYPT’16) presented an inner product argument with a statement including two vector commitments to two vectors and the inner product of the two vectors equals to a public scalar. Bunz et al. (S&P’18) then presented an inner product argument with a statement including only one vector commitment to two vectors. In this paper, we first summarize the scenarios to use inner product arguments based on Bootle et al. and Bunz et al. Then we propose and implement an improved inner product argument for the same statement as Bootle et al. Our argument has a lower communication complexity of \(4\log _2n\) which improves by about 30% when \(n=8192\). Moreover, as most existing inner product argument protocols have a recursive structure, we find the most appropriate recursive round that decides a better communication complexity.

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, the coupled Sasa-Satsuma equations are investigated for the propagation of the ultrashort pulse in such optical media as multi-mode and birefringent fibers. With the Hirota method and introduction of three auxiliary functions, a different bilinear form is obtained. Vector one- and two-soliton solutions in the bright-bright form are constructed. Based on the vector one-soliton solutions, parametric conditions for the existence of the vector single- and double-hump solitons are presented. From the vector two-soliton solutions, elastic and inelastic interactions can both be observed between the (i) two vector single-hump solitons, (ii) two vector double-hump solitons, and (iii) vector single- and double-hump solitons.

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

The most general form of linear boundary conditions in electromagnetics can be expressed in terms of two scalar-valued equations of the form ${\mathrm{f}_{1}(\mathbf{E},\mathbf{H})=0\$}$ and $\mathrm{f}_{2}(\mathbf{E},\mathbf{H})=0$ , where f 1 and f 2 are two bilinear functions, each defined by two vector quantities.

The conventional model predictive current control (MPCC) suffers from the errors caused by parameter uncertainties, which will deteriorate the motor performance. This paper proposes a two-vector-based predictive current control with the improved parameter robustness. Two arbitrary voltage vectors are applied in each sampling period to enhance steady-state performance. By adding the prediction error to the prediction stage of the proposed two-vector-based MPCC, the errors related to the prediction stage and vector duration calculation can be effectively eliminated. The simulated and experimental results are given to verify the proposed method.

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

Aiming at the problems of the traditional model predictive control (MPTC) of permanent magnet synchronous motor (PMSM) that the weight coefficient setting is cumbersome and the torque and flux pulsation are large, an improved PMSM two-vector model is proposed to predict the torque. This method expands the virtual vector to increase the voltage candidate vector, combines the voltage vector to act on the torque and flux linkage principle, selects the voltage vector, and reduces the calculation amount of the candidate vector. The evaluation mechanism of the selected vector is set by combined with the duty cycle method, so the weight factor of cost function in the traditional method is eliminated. The simulation results show that the improved method proposed can not only maintain high dynamic performance, but also improve the control effect of torque and flux linkage in the steady state of the system.

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.

In this paper the decays of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] have been investigated. The available experimental results for these decays are (in units of [Formula: see text]): [Formula: see text] By applying the theoretical value of the [Formula: see text] that span the range of [Formula: see text], the results for QCDF approach are about [Formula: see text] times smaller than experimental one. Therefore, it is decided to calculate the theoretical branching ratio by applying the final state interaction (FSI) through the [Formula: see text] (crossed and uncrossed) channels. The FSI effects are very sensitive to the changes in the phenomenological parameter [Formula: see text]. This parameter appears in the FSI form factors that increase strong interaction share. In most calculation changing two units in this parameter, makes the final result multiply in the branching ratio, therefore the decision to use FSI is not unexpected. In this study there are thirteen intermediate states for [Formula: see text] decay, fifteen intermediate states for [Formula: see text] decay and four intermediate states for [Formula: see text] and [Formula: see text] decays, in which the contribution of each one is calculated and summed in the final amplitude. Considering [Formula: see text] and fixing [Formula: see text] between four and five acceptable results have been obtained.

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/

In this work, we present an explicit form of the Luescher equation and consider the construction of the operators in different irreducible representations for the case of scattering of two vector particles. The formalism is applied to scalar QED in the Higgs Phase, where the $U(1)$ gauge boson acquires mass., Update to match published version

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.

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

Conventional model predictive control (MPC) presents high steady state ripples due to the application of only one voltage vector during one control period. Furthermore, the enumeration-based evaluation of the cost function poses high computational burden. Recently two-vector-based MPC has been proposed to improve the steady-state performance, but it still requires many calculations to obtain the two optimal vectors and their durations. This paper proposes an efficient two-vector-based MPC for induction motor (IM) drives, which achieves the same performance as prior method but greatly reduces the control complexity and computational burden. The inherent relationship between the proposed method and deadbeat control based on space vector modulation (SVM) is also revealed in this paper, which has not been reported before. The effectiveness of the proposed method is confirmed by the presented simulation and experimental results from a 2.2 kW IM drive platform.

In this paper, I construct a class of everywhere regular geometric sigma models that possess bouncing solutions. Precisely, I show that every bouncing metric can be made a solution of such a model. My previous attempt to do so by employing one scalar field has failed due to the appearance of harmful singularities near the bounce. In this work, I use four scalar fields to construct a class of geometric sigma models which are free of singularities. The models within the class are parametrized by their background geometries. I prove that, whatever background is chosen, the dynamics of its small perturbations is classically stable on the whole time axes. Contrary to what one expects from the structure of the initial Lagrangian, the physics of background fluctuations is found to carry 2 tensor, 2 vector and 2 scalar degrees of freedom. The graviton mass, that naturally appears in these models, is shown to be several orders of magnitude smaller than its experimental bound. I provide three simple examples to demonstrate how this is done in practice. In particular, I show that graviton mass can be made arbitrarily small., 34 pages, 7 figures

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.

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.

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.

It is known that the correlator of one axial and two vector currents, that receives leading contributions through one-loop fermion triangle diagrams, is not modified by QCD radiative corrections at two loops. It was suggested that this non-renormalization of the 〈 VVA 〉 correlator persists in higher orders in perturbative QCD as well. To check this assertion, we compute the three-loop QCD corrections to the 〈 VVA 〉 correlator using the technique of asymptotic expansions. We find that these corrections do not vanish and that they are proportional to the QCD β-function.

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.

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

The composite Stokes singularities formed in the coherent and incoherent superposition of two vector monochromatic coherent optical fields are studied and compared analytically and numerically. It is shown that the dynamic behavior of composite Stokes singularities depends on the superposition scheme (coherent or incoherent superposition), initial spatial distributions of the two superimposed fields, and propagation distance, and for the coherent superposition additionally depends on the relative phase of the two fields. There exist S 12 (C-points), S 31 , S 23 composite singularities, and L-lines, in particular, V-points and C-lines at the transverse plane. Generally, both V-points and C-lines are unstable. For the coherent superposition the degree of polarization P =1, whereas for the incoherent superposition P may be less than 1 and is variable upon propagation. The results are illustrated analytically and numerically.

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.

For two real Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, let $K_p$ be the projective cone in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$. Using this we define a cone norm on the algebraic tensor product of two vector spaces over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ and discuss some properties. We derive some fixed point theorems in this projective cone normed tensor product space over Banach algebra with a suitable example. For two self mappings $S$ and $T$ on a cone Banach space over Banach algebra, the stability of the iteration scheme $x_{2n+1}=Sx_{2n}$, $x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$ converging to the common fixed point of $S$ and $T$ is also discussed here.