Integral Invariants Research Articles

Integral invariants and decay of temporally developing grid turbulence

We present a study of a large-scale energy spectrum and integral invariants in temporally developing grid turbulence at mesh Reynolds numbers of ReM = 10 000 and 20 000 by employing direct numerical simulations (DNSs) in a periodic box. The simulations are initialized with a velocity field that approximates the wakes induced by the bars of conventional square grids. The turbulence statistics obtained in the temporal DNS agree well with those of the previous experiments in both the production and decay regions. The temporally developing grid turbulence also has a so-called non-equilibrium region, which is consistent with its spatially developing counterpart, where the normalized dissipation rate of turbulence kinetic energy (TKE), Cε, increases as the turbulence decays. The decay exponent n of TKE is n = 1.22 at ReM = 20 000 and n = 1.35 at ReM = 10 000, which are close to the values for the Saffman turbulence [i.e., 6/5 for ReM = 20 000 and 6(1 + p)/5 ≈ 1.36 for ReM = 10 000 with p ≈ 0.13 obtained by Cε ∼ tp at large t]. The longitudinal integral length scale and the TKE dissipation rate also exhibit temporal evolutions consistent with the Saffman turbulence for both ReM. The Saffman integral directly evaluated in the grid turbulence tends to be time-independent after the turbulence evolves for about 200 times of characteristic time scale defined by mesh size divided by the mean velocity of a fluid passing the grid. A direct evaluation of the TKE spectrum E(k) shows that E(k) ≈ Lk2/4π2 is valid for a finite range of low wavenumbers.

On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

Ruled Surfaces According to Bishop Frame in the Euclidean 3-Space

Attempts have been made to introduce ruled surfaces generated from any vector X, Bishop Darboux vector and Bishop vectors. Some properties with respect to developable and minimal and the relations between the integral invariants of these surfaces are discussed. Also the fundamental forms, geodesic curvatures, normal curvatures and geodesic torsions are calculated, and some results are obtained.

Invariants of the Axisymmetric Flows of an Inviscid Gas and Fluid with Variable Density

Abstract Material conservation laws and integral invariants are constructed for the axisymmetric flows of an inviscid compressible gas and an ideal incompressible fluid with variable density ρ ( x , t ) $\rho(\mathbf{x},t)$ . The functional independence of the new invariants from helicity is proven.

Existence and nonexistence of solutions for the heat equation with a superlinear source term

We develop a classification theory for the existence and non-existence of local in time solutions for initial value problems for nonlinear heat equations. By focusing on some quasi-scaling property and its invariant integral, we reveal the explicit threshold integrability of initial data that classifies the existence and nonexistence of solutions. Typical nonlinear terms, for instance polynomial type, exponential type and their sums, products and compositions can be treated as applications.

Some p-Adic Integrals on ℤp Associated with Trigonometric Functions

In this paper, we study some p-adic invariant and fermionic p-adic integrals on ℤp associated with trigonometric functions. By using these p-adic integrals we represent several trigonometric functions as a formal power series involving either Bernoulli or Euler numbers. In addition, we obtain some identities relating various special numbers like zigzag, some ‘trigonometric’, Bernoulli, Euler numbers, and Euler numbers of the second kind.

The double pentaladder integral to all orders

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar mathcal{N} = 4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.

On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nosé–Hoover oscillator

We consider the well-known Sprott A system, which is a special case of the widely studied Nosé–Hoover oscillator. The system depends on a single real parameter a, and for suitable choices of the parameter value, it is shown to present chaotic behavior, even in the absence of an equilibrium point. In this paper, we prove that, for $$a\ne 0,$$ the Sprott A system has neither invariant algebraic surfaces nor polynomial first integrals. For $$a>0$$ small, by using the averaging method we prove the existence of a linearly stable periodic orbit, which bifurcates from a non-isolated zero-Hopf equilibrium point located at the origin. Moreover, we show numerically the existence of nested invariant tori surrounding this periodic orbit. Thus, we observe that these dynamical elements and their perturbation play an important role in the occurrence of chaotic behavior in the Sprott A system.

Vector Hamiltonians in Nambu Mechanics

We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in ℝ n can be represented as a generalized Nambu mechanics with n − 1 integral invariants. For the case when the phase flow in ℝ n has n − 3 or less first integrals, we introduce the Cartan concept of mechanics. As an example we give the fifth integral invariant of Euler top.

Differential and Integral Projective Invariants for the Groups of Diffeomorphisms

In this paper we study the differential and integral invariants for the action of the projective group PGL(n + 1) on the group of diffeomorphisms Diff(ℝP n ) by conjugations. Cases n = 1 and n = 2 are considered. For n = 1 the algebra of differential invariants is found and the criterion of the local equivalence of two diffeomorphisms is obtained. Also several integral invariants for n = 1 and n = 2 are calculated, the analogy with Calaby integral invariant for the symplectic groups is established.

The quaternionic expression of ruled surfaces

In this paper, firstly, the ruled surface is expressed as a spatial quaternionic. Also, the spatial quaternionic definitions of the Striction curve, the distribution parameter, angle of pitch and the pitch are given. Finally, integral invariants of the closed spatial quaternionic ruled surfaces drawn by the motion of the Frenet vectors {t,n1,n2} belonging to the spatial quaternionic curve ? are calculated.

Global Conformal Invariants of Submanifolds

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis (more precisely we assume the integrand depends separately on the intrinsic and extrinsic curvatures, and not on their derivatives) the integrand can only consist of an intrinsic scalar conformal invariant, an extrinsic scalar conformal invariant and the Chern-Gauss-Bonnet integrand. In particular, for codimension one surfaces, we show that the Willmore energy is the unique global conformal invariant, up to the addition of a topological term (the Gauss curvature, giving the Euler Characteristic by the Gauss Bonnet Theorem). A similar statement holds also for codimension two surfaces, once taking into account an additional topological term given by the Chern-Gauss-Bonnet integrand of the normal bundle. We also discuss existence and properties of natural higher dimensional (and codimensional) generalizations of the Willmore energy.

On localizing Futaki–Morita integrals at isolated degenerate zeros

We study the localization of Futaki–Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implementing the localization procedure for a holomorphic vector field on CPn with a maximally degenerate zero. This yields an essentially unique formula for these Futaki–Morita integral invariants without using a summation over multiple points.

Low momentum expansion of one loop amplitudes in heterotic string theory

We consider the low momentum expansion of the four graviton and the two graviton-two gluon amplitudes in heterotic string theory at one loop in ten dimensions, and analyze contributions upto the {D}^2{mathrm{mathcal{R}}}^4 interaction from the four graviton amplitude, and the {D}^4{mathrm{mathcal{R}}}^2{mathrm{mathcal{F}}}^2 interaction from the two graviton-two gluon amplitude. The calculations are performed by obtaining equations for the relevant modular graph functions that arise in the modular invariant integrals, and involve amalgamating techniques used in the type II theory and the calculation of the elliptic genus in the heterotic theory.

A generalization of vortex lines

Abstract Helmholtz theorem states that, in ideal fluid, vortex lines move with the fluid. Another Helmholtz theorem adds that strength of a vortex tube is constant along the tube. The lines may be regarded as integral surfaces of a 1-dimensional integrable distribution (given by the vorticity 2-form). In general setting of theory of integral invariants, due to Poincare and Cartan, one can find d -dimensional integrable distribution (given by a possibly higher-rank form) whose integral surfaces show both properties of vortex lines: they move with (abstract) fluid and, for appropriate generalization of vortex tube, strength of the latter is constant along the tube.

On the Properties of q-Bernstein-type Polynomials

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified $q$-Bernoulli numbers and $q$-Euler numbers applying $p$-adic $q$-integral representation on $\mathbb {Z}_p$ and $p$-adic fermionic $q$-invariant integral on $\mathbb {Z}_p$, respectively, to the inverse of $q$-Bernstein polynomials.

Orbital angular momentum eigenfunctions for fast and numerically stable evaluations of closed-form pseudopotential matrix elements.

The computation of s-type Gaussian pseudopotential matrix elements involving low powers of the distance from the pseudopotential center using Gaussian orbitals can be reduced to familiar integrals. They may be directly expressed as either simple three-center overlap integrals for even powers of the radial distance from the pseudopotential center or related to the three-center nuclear integrals of a Gaussian charge distribution for odd powers. Orbital angular momentum about each atom is added to these integrals by solid-harmonic differentiation with respect to its center. The solid-harmonic addition theorem allows all the integrals to be factored into products of invariant one-dimensional integrals involving the Gaussian exponents and angular factors that contain the azimuthal quantum numbers but are independent of all Gaussian exponents. Precomputing the angular factors allow looping over all Gaussian exponents about the three centers. The fact that solid harmonics are eigenstates of angular momentum removes the singularities seen in previous treatments of pseudopotential matrix elements.

Renormalized Volume

We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.

Invariant integral: The earliest works and most recent application

The present paper embraces mainly the three-year period of 1966 to 1968 when the invariant integral of fracture mechanics appeared and became popular, and the last two years of 2015 to 2016 when the neoclassic cosmology based on the invariant integral came up. A mention is given to the previous works of Euler, Cauchy, Maxwell, Nother, Gunther and Eshelby who dealt with invariant integrals in mathematics, hydrodynamics, electrodynamics, and the theory of dislocations. A brief review is given of the creation of the invariant integral of fracture mechanics under static and dynamic conditions for a solid continuum including elastic, plastic and viscoelastic materials, as well as of some of its most important applications, ramifications and generalizations for other physical fields. The initial phase of the expansion and revolution of the large-scale universe is studied in the framework of the neoclassic approach, including the Big Bang and the Dark Energy; it is shown that the spheroidal shape of the universe assumed at the Big Bang retains its eccentricity constant in the initial phase. The assumption of a superphoton as a primordial universe was analyzed.

A Tensor-Train accelerated solver for integral equations in complex geometries

We present a framework using the Quantized Tensor Train (qtt) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the qtt decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O(log⁡N) and once the inverse is computed, it can be applied in O(Nlog⁡N).We analyze the qtt ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as fmm and hss. We prove that the qtt ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals.For volume integrals, the qtt decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the qtt-based solver when applied to both translation and non-translation invariant volume integrals in 3d.For boundary integral equations, we demonstrate that using a qtt decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the qtt preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.