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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 ...

This article investigates the optical soliton to Lakshmanan–Porsezian–Daniel (LPD) model in birefringent fibers which incorporates two vector solitons. The optical solitons in the forms of bright, dark, singular, as well as combined form bright-dark, dark-singular, complex soliton and multiple soliton solutions are extracted successfully by the mechanism of Fan-extended sub equation approach with the effect of five free parameters. Furthermore, singular periodic wave solutions are obtained, as well as the constraint requirements for the existence of soliton solutions. The results suggest that the method is simple, direct, and effective, and that it may be applied to more complex phenomena with the help of symbolic calculations.

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.

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.

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.

International audience; The Dual Active Bridge appears as a promising DC-DC converter topology when galvanic isolation and bidirectional power flow are required. Among its advantages, Zero Voltage Switching allows the switching losses to be significantly reduced. For high power applications, the three-phase topology variant may be interesting in order to reach a higher power density, especially when a three-phase transformer is implemented instead of three single-phase transformers. Moreover, the transformer vector group offers a new degree of freedom for the designers. In this paper, the authors present the experimental validation of a 1.2 kV-100 kW-20 kHz three-phase Dual Active Bridge converter using two medium frequency transformers and different vector groups.

A large number of observables can be constructed from differential decay rate based on the polarization of final state while considering decay of a neutral meson $(P^0 \text{ or } \bar P^0)$ to two vector particles. But all of these observables are not independent to each other since there are only a few independent theoretical parameters controlling the whole dynamics and therefore various relations among observables emerge. In this paper, we have studied the behaviour of observables for neutral meson decaying to two vectors in presence of T and CPT violations in mixing accompanied by both direct and indirect CP violations. We have expressed all of the fourteen unknown theoretical parameters for this scenario in terms observables only and constructed the complete set of thirty four relations among observables whose violation would signify the existence of some new Physics involving direct violation of CPT. In addition, using this formalism we have studied three special cases too: a) SM scenario, b) SM plus direct CP violation c) SM plus T and CPT violation in mixing., 28 pages

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.

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.

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.

Using exceptional field theory we construct supersymmetric warped AdS$_7$ vacua of massive IIA and AdS$_6$ vacua of IIB, as well as their consistent truncations including vector multiplets. We show there are no consistent truncations of massive IIA supergravity around its supersymmetric AdS$_7$ vacua with vector multiplets when the Roman's mass is non-vanishing. For AdS$_6$ vacua of IIB supergravity, we find that in addition to the consistent truncation to pure $\mathrm{F}(4)$ gauged SUGRA, the only other half-maximal truncations that are consistent result in $\mathrm{F}(4)$ gauged SUGRA coupled to one or two Abelian vector multiplets, to three non-Abelian vector multiplets, leading to an $\mathrm{ISO}(3)$ gauged SUGRA, or to three non-Abelian plus one Abelian vector multiplet, leading to an $\mathrm{ISO}(3) \times \mathrm{U}(1)$ gauged SUGRA. These consistent truncations with vector multiplets exist when the two holomorphic functions that define the AdS$_6$ vacua satisfy certain differential conditions which we derive. We use these to deduce that no globally regular AdS$_6$ solutions admit a consistent truncation to $\mathrm{F}(4)$ gauged SUGRA with two vector multiplets, and show that the Abelian T-dual of the Brandhuber-Oz vacuum allows a consistent truncation to $\mathrm{F}(4)$ gauged SUGRA with a single vector multiplet., 58 pages plus appendices

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

Analytic alignment is a type of self-alignment for a Strapdown inertial navigation system (SINS) that is based solely on two non-collinear vectors, which are the gravity and rotational velocity vectors of the Earth at a stationary base on the ground. The attitude of the SINS with respect to the Earth can be obtained directly using the TRIAD algorithm given two vector measurements. For a traditional analytic coarse alignment, all six outputs from the inertial measurement unit (IMU) are used to compute the attitude. In this study, a novel analytic alignment method called selective alignment is presented. This method uses only three outputs of the IMU and a few properties from the remaining outputs such as the sign and the approximate value to calculate the attitude. Simulations and experimental results demonstrate the validity of this method, and the precision of yaw is improved using the selective alignment method compared to the traditional analytic coarse alignment method in the vehicle experiment. The selective alignment principle provides an accurate relationship between the outputs and the attitude of the SINS relative to the Earth for a stationary base, and it is an extension of the TRIAD algorithm. The selective alignment approach has potential uses in applications such as self-alignment, fault detection, and self-calibration.

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.

Nowadays many cities and countries are creating their 3D building models for a better daily management and smarter decision making. The newly created 3D models are required to be consistent with existing 2D footprint maps. Thereby the 2D maps are usually combined with height data for the task of 3D reconstruction. Many buildings are often composed by parts that are discontinuous over height. Building parts can be reconstructed independently and combined into a complete building. Therefore, most of the state-of-the-art work on 3D building reconstruction first decomposes a footprint map into parts. However, those works usually change the footprint maps for easier partitioning and cannot detect building parts that are fully inside the footprint polygon. In order to solve those problems, we introduce two methodologies, one more dependent on height data, and the other one more dependent on footprints. We also experimentally evaluate the two methodologies and compare their advantages and disadvantages. The experiments use Airborne Laser Scanning (ALS) data and two vector maps, one with 1:10,000 scale and another one with 1:500 scale.

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.

In this work we present an extension of the L\"uscher formalism to include the interaction of particles with spin, focusing on the scattering of two vector particles. The derived formalism will be applied to Scalar QED in the Higgs Phase, where the U(1) gauge boson acquires mass., Comment: Lattice 2017 proceeding contribution

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

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

We study maximally supersymmetric Anti-de Sitter backgrounds in consistent N=2 truncations of type IIB supergravity compactified on the Sasaki-Einstein manifold $T^{1,1}$. In particular, we focus on truncations that contain fields coming from the nontrivial second and third cohomology forms on $T^{1,1}$. These give rise to N=2 supergravity coupled to two vector- and two hypermultiplets (Betti-vector truncation) or one vector- and three hypermultiplets (Betti-hyper truncation), respectively. We find that both truncations admit AdS$_5$ backgrounds with the gauge group always being broken but containing at least an $U(1)_R$ factor. Moreover, in both cases we show that the moduli space of AdS vacua is nontrivial and of maximal dimension. Finally, we explicitly compute the metrics on these moduli spaces., 20 pages; v2: updated references; v3: 21 pages, as published in JHEP

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.