Quadratic Form In Terms Research Articles

Some criteria for stably birational equivalence of quadratic forms

Let φ and ψ be quadratic forms over a field K of characteristic different from 2. In this paper, we give a criterion for isotropy of φ over the function field of ψ in terms of representations and we apply it to stably birational equivalence of φ and ψ. Then we use this criterion to investigate the case of stably birational equivalence of multiples of Pfister forms. Finally, we give a criterion of stably birational equivalence of quadratic forms in terms of isomorphisms of quotients of special Clifford groups by the kernel of the spinor norm.

Solvable lattice sums and quadratic Dirichlet $L$-values

We study how to express the Epstein zeta functions of binary quadratic forms in terms of a finite number of quadratic Dirichlet $L$-values. We prove a conjecture raised by Zucker and Robertson in 1984.

Second order directional shape derivatives of integrals on submanifolds

<p style='text-indent:20px;'>We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.</p>

Representation of non-semibounded quadratic forms and orthogonal additivity

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail.

A peridynamic approach to computation of elastic and entropic interactions of inclusions on a lipid membrane

The physics behind the interaction of inclusions on lipid membranes is relevant to the self-assembly of proteins that mediate exo- and endo-cytosis in cells. These interactions can be quantified in terms of free energy changes that can be calculated using Gaussian integrals since the potential energy of a lipid membrane with inclusions can be expressed as a quadratic form in terms of a stiffness matrix that includes the coupling of in-plane tension and out-of-plane deflections. The free energy calculation requires the evaluation of the determinant of a stiffness matrix appearing in the energy expression. However, computing the determinant of an extremely large stiffness matrix whose size is dictated by the characteristic length scale of the membrane and its molecular constituents poses computational challenges. This study presents a peridynamic (PD) approach to construct the stiffness matrix and evaluate its determinant accurately without any computational challenges. It specifically employs the PD differential operator for accurate evaluation of the determinant of the stiffness matrix which is a key step in the calculation of the partition function. The simultaneous use of fine and coarse grids along with PD interpolations eliminates the computational challenges for decreasing grid spacing. This is illustrated by reproducing the worm-like-chain force-extension relation for a fluctuating elastic rod. In the case of a membrane without any inclusions, the PD predictions converge and approach an analytical solution for the tension-area relation of a membrane as the grid spacing decreases. In the presence of inclusions, the PD predictions capture the expected deformation of the membrane as well as the behavior of relative free energy for varying spacing, shape and position of the inclusions. The magnitude of relative free energy increases with increasing number of inclusions, thus indicating higher degree of interaction in larger clusters. This approach is general and enables the exploration of the effect on membrane-inclusion interactions of different membrane geometries with curvature, boundary conditions and the contact angle between the membrane and inclusion.

Energy-stable boundary conditions based on a quadratic form: Applications to outflow/open-boundary problems in incompressible flows

We present a set of new energy-stable open boundary conditions for tackling the backflow instability in simulations of outflow/open boundary problems for incompressible flows. These boundary conditions are developed through two steps: (i) devise a general form of boundary conditions that ensure the energy stability by re-formulating the boundary contribution into a quadratic form in terms of a symmetric matrix and computing an associated eigen problem; and (ii) require that, upon imposing the boundary conditions from the previous step, the scale of boundary dissipation should match a physical scale. These open boundary conditions can be re-cast into the form of a traction-type condition, and therefore they can be implemented numerically using the splitting-type algorithm from a previous work. The current boundary conditions can effectively overcome the backflow instability typically encountered at moderate and high Reynolds numbers. In general they give rise to a non-zero traction on the entire open boundary, unlike previous related methods which only take effect in the backflow regions of the boundary. Extensive numerical experiments in two and three dimensions are presented to test the effectiveness and performance of the presented methods, and simulation results are compared with the available experimental data to demonstrate their accuracy.

Earthquake sequence simulations with measured properties for JFAST core samples.

Since the 2011 Tohoku-Oki earthquake, multi-disciplinary observational studies have promoted our understanding of both the coseismic and long-term behaviour of the Japan Trench subduction zone. We also have suggestions for mechanical properties of the fault from the experimental side. In the present study, numerical models of earthquake sequences are presented, accounting for the experimental outcomes and being consistent with observations of both long-term and coseismic fault behaviour and thermal measurements. Among the constraints, a previous study of friction experiments for samples collected in the Japan Trench Fast Drilling Project (JFAST) showed complex rate dependences: a and a-b values change with the slip rate. In order to express such complexity, we generalize a rate- and state-dependent friction law to a quadratic form in terms of the logarithmic slip rate. The constraints from experiments reduced the degrees of freedom of the model significantly, and we managed to find a plausible model by changing only a few parameters. Although potential scale effects between lab experiments and natural faults are important problems, experimental data may be useful as a guide in exploring the huge model parameter space.This article is part of the themed issue 'Faulting, friction and weakening: from slow to fast motion'.

Free energies for materials with memory in terms of state functionals

The aim of this work is to determine what free energy functionals are expressible as quadratic forms of the state functional \(I^t\) which is discussed in earlier papers. The single integral form is shown to include the functional \(\psi _F\) proposed a few years ago, and also a further category of functionals which are easily described but more complicated to construct. These latter examples exist only for certain types of materials. The double integral case is examined in detail, against the background of a new systematic approach developed recently for double integral quadratic forms in terms of strain history, which was used to uncover new free energy functionals. However, while, in principle, the same method should apply to free energies which can be given by quadratic forms in terms of \(I^t\), it emerges that this requirement is very restrictive; indeed, only the minimum free energy can be expressed in such a manner.

Nodal domains of the equilateral triangle billiard

We characterize the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems.

Unbounded quantum graphs with unbounded boundary conditions

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self‐adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self‐adjoint operators. We then characterize the lower bounded self‐adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions.

Thrust and Efficiency Optimization of a Harmonically Deforming Thin Airfoil for MAV Design

This article presents analytical and numerical approaches to optimizing thrust and thrust efficiency of a harmonically deforming thin airfoil. An unsteady aerodynamic model for a deforming thin airfoil undergoing harmonic oscillations has been developed previously. Here the thrust and efficiency functions are presented in a quadratic form in terms of the deformation design variables. A zero leading-edge suction constraint is developed which can be used to avoid leading edge separation in thin airfoils. Pareto fronts in terms of thrust magnitude and efficiency are generated showing optimum deformation conditions (magnitude and phase) at various reduced frequencies for various constraints.

Stability analysis for axially-symmetric magnetic field levitation of a superconducting sphere

For a magnetically levitated superconducting sphere, the stability analysis in both axial and radial directions is analyzed theoretically using a direct boundary-value problem approach. The external magnetic field is produced by a system of current-carrying coils with axial symmetry. We also assume complete exclusion of magnetic fields from the interior of the superconductor. Use of the P l 1 associated Legendre functions as a basis set allows for a simple series solution for the magnetic stiffness coefficients as tridiagonal quadratic forms in terms of the expansion coefficients. Analysis shows that stability for this problem is not guaranteed in general. Stability is guaranteed though, if coil windings are designed in such away that all but one of the expansion coefficients are zero. Further cases with two and three non-zero coefficients are also solved. In particular the important 1, 3 and 2, 4 cases are treated in detail, and stability maps have been produced. Finally, the case where levitation is attempted by one pair of discrete current coils has been solved and stability mapping has been thoroughly explored.

Weight of quadratic forms and graph states

We prove a connection between Schmidt rank and weight of quadratic forms. This provides a new tool for the classification of graph states based on entanglement. Our main tool arises from a reformulation of previously known results concerning the weight of quadratic forms in terms of graph states properties. As a byproduct, we obtain a straightforward characterization of the weight of functions associated with pivot-minor of bipartite graphs.

The onset of mixed-mode intralaminar cracking in a cross-ply composite laminate

The intralaminar fracture toughness of a unidirectionally reinforced glass/epoxy composite is determined experimentally at several mode I and mode II loading ratios. The crack propagation criterion, expressed as a quadratic form in terms of single-mode stress intensity factors (alternatively, linear in terms of energy release rates), approximates the test results reasonably well. The mixed-mode cracking criterion obtained is used to predict the intralaminar crack on set in a cross-ply glass/epoxy composite under off-axis tensile loading.

Quantum states transfer between coupled fields

We study the exchange of states in coupled fields along their time evolution. The coupling is described by a quadratic form in terms of annihilation and creation operator in the field Hamiltonian. An analytical approach is employed to describe the time evolution of the field state in Fock's space and the conditions for an arbitrary initial states to be transferred with 100% fidelity is determined. We show that only for initial states C0|0>+CN|N>, this situation can occurs. The important |1〉↔|0〉 qubits transfer is a particular case of this transference of number state. The relation between the coupling constant and characteristic field frequencies for complete state transference is also determined.

Energy transfer in turbulent polymer solutions

The paper addresses a set of new equations concerning the scale-by-scale balance of turbulent fluctuations in dilute polymer solutions. The main difficulty is the energy associated with the polymers, which is not of a quadratic form in terms of the traditional descriptor of the micro-structure. A different choice is however possible, which, at least for mild stretching of the polymeric chains, directly leads to an L2 structure for the total free-energy density of the system thus allowing the extension of the classical method to polymeric fluids. On this basis, the energy budget in spectral space is discussed, providing the spectral decomposition of the energy of the system. New equations are also derived in physical space, to provide balance equations for the fluctuations in both the kinetic field and the micro-structure, thus extending, in a sense, the celebrated Kármán–Howarth and Kolmogorov equations of classical turbulence theory. The paper is limited to the context of homogeneous turbulence. However the necessary steps required to expand the treatment to wall-bounded flows of polymeric liquids are indicated in detail.

The number of representations of a number by various forms

We start with classical results due to Jacobi, Dirichlet and Lorenz which give the number of representations of a number by various quadratic forms in terms of divisor functions. We express the results in terms of products and series, then systematically dissect the products and series to obtain further results of similar type. In particular, we obtain results of Legendre and Ramanujan, as well as many others.

A note on the theta characteristics of a compact Riemann surface

AbstractLet X be a compact connected Riemann surface and ξ a square root of the holomorphic contangent bundle of X. Sending any line bundle L over X of order two to the image of dim H0(X, ξ ⊗ L) − dim H0(X, ξ) in Z/2Z defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X.

A criterion for equivalence of quadratic forms on a special class of riemannian manifolds

We consider two quadratic forms on a class of manifolds conformally equivalent to the euclidean space. We give necessary and sufficient conditions for the equivalence of these quadratic forms in terms of a weighted Hardy inequality.

Transformation of quadratic forms to perfect squares for broken extremals

In this paper we study a quadratic form which corresponds to an extremal with piecewise continuous control in variational problems. This form, compared with the classical one, has some new terms connected with the set Θ of all points of discontinuity of the control. Its positive definiteness is a sufficient optimality condition for broken extremals. We show that if there exists a solution to corresponding Riccati equation satisfying some jump condition at each point of the set Θ, then the quadratic form can be transformed to a perfect square, just as in the classical case. As a result we obtain sufficient conditions for positive definiteness of the quadratic form in terms of the Riccati equation and hence, sufficient optimality conditions for broken extremals.