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In this paper, we establish the sharp L p boundedness for the bilinear integral operators with certain hypersingularities that generalize the bilinear Hilbert transform.

We propose a new interpolatory framework for model reduction of large-scale bilinear systems. The input-output representation of a bilinear system in frequency domain involves a series of multivariate transfer functions, each representing a subsystem of the bilinear system. If a weighted sum of these multivariate transfer functions associated with a reduced bilinear system interpolates a weighted sum of the original multivariate transfer functions, we say that the reduced system satisfies a Volterra series interpolation (Flagg and Gugercin, 2015). These interpolatory conditions can also ensure the necessary conditions for H 2 -optimal model reduction (Flagg and Gugercin, 2015; Benner and Breiten, 2012). We observe that, by carefully selecting the weights of the series, the Volterra series interpolatory conditions are transformed to the problem of interpolating a linear system with an affine parameter dependence. Such linear parametric systems can then be reduced by some method for parametric model order reduction.Linear systems where the affine parameter dependence is given as low-rank variation in the state matrix can be mapped into a non-parameterized multi-input multi-output linear system. This allows us to utilize the standard (non-parametric) linear IRKA (Gugercin etźal., 2008) for the problem of parameterized/bilinear interpolation. Numerical results show that the approximations are of comparable accuracy to those obtained from the bilinear iterative rational Krylov algorithm (Benner and Breiten, 2012). The proposed approach, however, has the advantage that it reduces the computational costs as it involves computations associated with solving linear systems only.

We apply the linear superposition principle to Hirota bilinear equations and generalized bilinear equations. By extending the linear superposition principle to complex field, we construct complex exponential wave function solutions first and then get complexions by taking pairs of conjugate parameters. A few examples of mixed resonant solitons and complexitons to Hirota and generalized bilinear differential equations are presented.

A Shallow Water Wave-like nonlinear differential equation is considered by using the generalized bilinear equation with the generalized bilinear derivatives D3,x and D3,t, which possesses the same bilinear form as the standard shallow water wave bilinear equation. By symbolic computation, four presented classes of rational solutions contain all rational solutions to the resulting Shallow Water Wave-like equation, which generated from a search for polynomial solutions to the corresponding generalized bilinear equation.

The fourth-order boundary value problems of one parameter gradient-elastic bar and plane strain/stress models are formulated in a variational form within an H 2 Sobolev space setting. For both problems, the existence and uniqueness of the solution is established by proving the continuity and coercivity of the associated symmetric bilinear form. For completeness, the full sets of boundary conditions of the problems are derived and, in particular, the new types of boundary conditions featured by the gradient-elastic models are given the additional attributes singly and doubly. By utilizing the continuity and coercivity of the continuous problems, corresponding error estimates are formulated for conforming Galerkin formulations. Finally, numerical results, with isogeometric C p − 1 -continuous discretizations for NURBS basis functions of order p ≥ 2 , confirm the theoretical results and illustrate the essentials of both static and vibration problems.

A submanifold M ⊂ R n lies in the sphere S n − 1 iff it carries a parallel umbilic normal vector field. We extend this theorem by replacing the sphere S n − 1 by an arbitrary extrinsic symmetric space S ⊂ R n .

With symbolic computation, Bell-polynomial scheme and bilinear method are applied to a two-dimensional Korteweg–de Vries (KdV) model, which is firstly proposed with Lax pair generating technique. Bell-polynomial expression with one auxiliary independent variable is derived and transformed into bilinear form. According to the coupled two-field conditions between the primary and replica fields, Bell-polynomial-typed Backlund transformations (BTs) are constructed and converted into the bilinear ones. Finally, soliton solutions of the two-dimensional KdV model are obtained (via solving the bilinear representation and BT, respectively) and compared. Such associated integrable properties as bilinear representation, BT (especially auxiliary-independent-variable-involved Bell-polynomial-typed ones constructed in this paper) and soliton solutions (especially the multi-soliton ones) may be useful for further study on other two-dimensional KdV and KdV-typed models.

In this article, we give some characterizations of Lipschitz space via commutators of bilinear singular integral operators and bilinear fractional integral operators, respectively.

In this paper, we extend Su-Zhang’s Cheeger-Muller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary. Our result also extends Bruning-Ma’s Cheeger-Muller type theorem for Ray-Singer metric on manifolds with boundary to symmetric bilinear torsions in product case. We also compare it with the Ray-Singer analytic torsion on manifolds with boundary.

We investigate the bilinear Hilbert transform with oscillatory factors and the truncated bilinear Hilbert transform. The main result is that theLp1×Lp2→Lq-boundedness of the two operators is equivalent with1≤p1,p2<∞, and1/q=1/p1+1/p2. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase.

Binary Bell Polynomials play an important role in the characterization of bilinear equation. The bilinear form, bilinear B?cklund transformation and Lax pairs for the modified Kadomtsev-Petviashvili equation are derived from the Binary Bell Polynomials.

First we present the explicit formula for the norm of a bilinear form on the 2-dimensional real predual of the Lorentz sequence space . Using this formula, we classify the extreme points of the unit ball of .

A class of bilinear differential operators is introduced by assigning specific signs to derivatives. The corresponding bilinear differential equations, containing Hirota bilinear equations as particular examples, are characterized by Bell polynomials. Resonant solutions of exponential waves are analyzed via recognizing the superposition principle using Bell polynomials. Two illustrative examples are computed by an algorithm adopting weights of dependent variables.

In this paper, we study the root system of rank 2 hyperbolic Kac-Moody algebras. We give some sufficient conditions for the existence of imaginary roots of square length 2k(k 2 Z>0). We also give several relations between the integral points on the hyperbola h to show that the value of the symmetric bilinear form of any two integral points depends only on the number of integral points between them. We also give some generalizations of Binet formula and Catalan's identity.

First we present the explicit formula for the norm of a symmetric bilinear form on the 2-dimensional real predual of the Lorentz sequence space . Using this formula, we classify the extreme points of the unit ball of .

The aim of this paper is to give a complete answer to the isotropy of bilinear forms of dimension less than or equal to 5 over the function field of a quadric in characteristic 2.

A three-degree-of-freedom vibro-impact system with symmetric two-sided rigid constraints is considered. Since the symmetric period n−2 motion of the vibro-impact system corresponds to the symmetric fixed point of the Poincare map of the vibro-impact system, we investigate bifurcations of the symmetric period n−2 motion by researching into bifurcations of the associated symmetric fixed point. The Poincare map of the system has symmetry property, and can be expressed as the second iteration of another unsymmetric implicit map. Based on both the Poincare map and the unsymmetric implicit map, the center manifold technique and the theory of normal forms are applied to deduce the normal form of the Neimark–Sacker-pitchfork bifurcation of the symmetric fixed point. By numerical analysis, we obtain the Neimark–Sacker-pitchfork bifurcation of the symmetric fixed point of the Poincare map in the vibro-impact system.

Binary Bell polynomials are applied to construct two kinds of bilinear derivative equations of a $(3+1)$ -dimensional nonlinear equation. Based on one of the bilinear forms, we derive a Backlund transformation, the corresponding Lax pair, infinite conservation laws, and explicit solutions with an arbitrary function in y. In the meantime, from the other bilinear form, we get another bilinear Backlund transformation and exact solutions by utilizing the exchange formulas for Hirota’s bilinear operators.

We consider the stabilization problem for the zero equilibrium of bilinear and affine systems in canonical form. We obtain necessary and sufficient conditions for the stabilizability of second-order bilinear and affine systems in canonical form and generalize these conditions to the case of simultaneous stabilization of a family of bilinear systems and to nth-order bilinear systems.

We explore when Hirota bilinear equations possess linear subspaces of solutions. First, we establish a sufficient and necessary criterion for the existence of linear subspaces of exponential traveling wave solutions to Hirota bilinear equations. Second, we show that multivariate polynomials whose zeros form a vector space can generate the desired Hirota bilinear equations with given linear subspaces of solutions, and formulate such multivariate polynomials by using multivariate polynomials which have one and only one zero. Third, applying an algorithm using weights, we present parameterizations of wave numbers and frequencies achieved by using one parameter to compute the desired Hirota bilinear equations.

By using the generalized Hirrota bilinear operators a kind of bilinear differential equations is established and examined when the linear super position principle can apply to the resulting generalized bilinear differential equations. Examples of generalized bilinear differential equations together with an algorithm using weights are computed using a 1+1 and 2+1 dimensional equations in order to shed more lights on the presented general scheme for the construction of the bilinear differential equations which posses linear subspaces of solutions.

In this note, we establish an inequality between the integral norm and Euclidean norm of a symmetric bilinear form f on a Euclidean space E n , i.e., 1 volSn−1 � Sn−1 f 2 (θ,θ)dθ 1 n |f | 2 Euc

This article studies the realisability property of discrete-time bilinear and quadratic input–output (i/o) equations in the classical state-space form. Constraints on the parameters of the i/o model are suggested that lead to realisable models. Using new formulae for computing basis vectors of certain vector spaces of differential one-forms, we present in this article the complete list of the third- and fourth-order realisable i/o bilinear models, and a new realisable subclass of an arbitrary order is suggested. Moreover, we provide the sufficient conditions of the second- and third-order realisable i/o quadratic models, respectively. All the developed theory and algorithms are illustrated by means of several examples.

A linear superposition principle of exponential traveling waves is analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves. Applications are made for the 3+1 dimensional KP, Jimbo–Miwa and BKP equations, thereby presenting their particular N-wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated N-wave solutions, and a few illustrative examples are presented, together with an algorithm using weights.

In this paper, we extend the complex-valued Ray-Singer torsion, introduced by Burghelea-Haller, to compact connected Riemannian manifolds with boundary. We also compare it with the refined analytic torsion.

In this paper, we characterize symmetric Sobolev bilinear forms defined on P × P , where P is the space of polynomials. More specifically we show that symmetric Sobolev bilinear forms, like symmetric matrices, can be re-written with a diagonal representation. As an application, we introduce the notion of a ghost matrix, extending some classic work of Stieltjes.

We establish some estimates on the hyper bilinear Hilbert transform on both Euclidean space and torus. We also use a transference method to obtain a Kenig-Stein's estimate on bilinear fractional integrals on the n-torus.

A class of bilinear systems of integral Volterra equations of the first kind related to the problem of automatic control of a nonlinear dynamic system (object) with unknown structure and vector input and output is studied. Algorithms for an analytic solution to corresponding bilinear systems and its numerical approximation are developed. A special character of the algorithms is illustrated by model examples.

Several results about convolution and about Fourier coefficients for X-valued functions defined on the torus satisfying the condition sup‖y‖=1 ∫−ππ ‖ B(f(eiϑ), y)‖dϑ/2π < ∞ for a bounded bilinear map B: X × Y → Z are presented and some applications are given.

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform.

Let Gr L ⊂ Gr ( n , V ) be the space of all Lagrangian subspaces of C 2 n equipped with the standard symplectic form. Let X ˜ be a universal cover of a compact connected Riemann surface X . We consider all immersions f : X ˜ ⟶ Gr L satisfying the following two conditions: (1) the map f is equivariant with respect to some homomorphism into Sp ( 2 n , C ) of the Galois group of the covering X ˜ ⟶ X , and (2) the symmetric bilinear form on the pullback, to X ˜ , of the tautological vector bundle over Gr L is fiberwise nondegenerate. Two such maps are called equivalent if they differ by the action of some fixed element of Sp ( 2 n , C ) . We prove that the equivalence classes of all such maps are bijectively parametrized by pairs of the form ( P , ( F , ∇ ) ) , where P is a projective structure on X and ( F , ∇ ) is an equivalence class of flat O ( n , C ) -connection on X . Two flat O ( n , C ) -bundles are equivalent if the corresponding flat PO ( n , C ) -bundles are isomorphic.

The authors establish a Cheeger-Muller type theorem for the complex valued analytic torsion introduced by Burghelea and Haller for flat vector bundles carrying nondegenerate symmetric bilinear forms. As a consequence, they prove the Burghelea-Haller conjecture in full generality, which gives an analytic interpretation of (the square of) the Turaev torsion.

Let V be a vector space of finite dimension over the field of real or complex numbers, q a nondegenerate quadratic form on V, φ the symmetric bilinear form such that q(x) = φ(x, x) for all x ∈ V, and Cl(V, q) the Clifford algebra provided with a Clifford multiplication and an exterior multiplication such that xy = x ∧ y + φ(x, y) for all x, y ∈V. First I shall explain that the theorem of Lipschitz about exterior exponentials of bivectors leads to an easy calculation of the Clifford exponential of a bivector u by means of the exterior exponential of another bivector v.

Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, by virtue of symbolic computation, the investigation is made on a generalized variable-coefficient higher-order nonlinear Schrodinger equation with varying higher-order effects and gain or loss, which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide. A modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable-coefficient bilinear form. Based on the formal parameter expansion technique, the multi-soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints. A Backlund transformation in bilinear form is also obtained for the first time in this paper. Finally, discussions on the analytic soliton solutions are given and various propagation situations are illustrated.

We study the structure of the minimum set of the normal curvature for a symmetric bilinear map on Euclidean or Hilbert space, the conditions when this set contains strongly umbilical, conformal nullity, etc. linear subspaces. The main goals are estimates from above of the codimension of these subspaces for a symmetric bilinear map with positive normal curvature and the inequality type restriction on the extrinsic qth scalar curvature. We estimate from above the codimension of asymptotic and relative nullity subspaces for a symmetric bilinear map with nonpositive extrinsic qth scalar curvature.

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known Lp bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on Lp1 × Lp2 when either 1/p1 + 1/p2 = 3/2 or one of p1, p2 is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.

We introduce an embedding of real or complex n-dimensional space K n as an algebraic variety V which is determined by the action of a linear one-parameter group. Every analytic vector field on K n corresponds to some embedded vector field on V. For a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non-autonomous linear systems. Examples and applications are mostly concerned with Poincare–Dulac normal forms. Embeddings provide a natural setting for perturbations of symmetric systems, in particular of systems in normal form up to some degree.

We prove that uniform global asymptotic stability of bilinear infinite-dimensional control systems is equivalent to their integral input-to-state stability. Next we present a method for construction of iISS Lyapunov functions for such systems if the state space is a Hilbert space. Unique issues arising due to infinite-dimensionality are highlighted.

To settle the L2 bounded curvature conjecture for the Einstein-vaccum equations one needs to prove bilinear type estimates for solutions of the homogeneous wave equation on a fixed background with H2 local regularity. In this paper we introduce a notion of primitive parametrix for the homogeneous wave equation for which we can prove, under very broad assumptions, the desired bilinear estimates.

We study finite-dimensional bilinear spaces and their isometry groups. To each bilinear space V V we associate two canonical filtrations, which yield structural results on V V . Prominent among these is an explicit formula for the number of indecomposable and degenerate blocks of V V of a given dimension. Equipped with this material, we proceed to characterize those bilinear spaces whose isometry group is contained in the special linear group. This characterization can easily be implemented in practice by means of an algorithm. As an application, we determine the real n n -by- n n matrices whose congruence class is disconnected.

We study symmetric powers of classes of symmetric bilinear forms in the Witt–Grothendieck ring of a field of characteristic not equal to 2, and derive their basic properties and compute their classical invariants. We relate these to earlier results on exterior powers of such forms.

A symmetric (2+1)-dimensional Lotka–Volterra equation is proposed. By means of a dependent variable transformation, the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables. A bilinear Backlund transformation is found and then the corresponding Lax pair is derived. Explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric (2+1)-dimensional Lotka–Volterra equation are given. As a special case of the pfaffian solutions, we obtain soliton solutions and dromions.

In this paper we analyze a family of discontinuous Galerkin methods, parameterized by two real parameters, for elliptic problems in one dimension. Our main results are: (1) a complete inf-sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof, (2) an analysis of the error in L2 where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case. The analysis is consistent with numerical results and similar behavior is observed in two dimensions.

An essentially analytical approach for the solution of state equations of QLS systems is presented, QLS being an extension of the bilinear models which keeps the same approximation properties while yielding a lower model dimension.