Geometrical Interpretation of General Complex Algebra and its Application in Relativistic Mechanics

In a generalized formulation of the relativistic dynamics with internal conformation an important role is played by a quadratic polynomial, the coefficients and eigenvalues of which are generated by outer and inner momenta of the relativistic particle. This polynomial induces the general complex algebra, GC. In this paper we explore the geometrical and physical aspects of the evolution generated by the algebraic operations of the GC-algebra. It is shown that the geometrical image of the GC-number is given by a straight line passing through two given points in an euclidean plane. In this representation the straight line is characterized by a norm and an argument. The motions of the straight line are described by hyperbolic trigonometry which brings a correspondence between the Euclidean geometry and the hyperbolic one. It is proved that the evolution equation governed by the generator of the GC-algebra describes the energy conservation law of the relativistic particle. This evolution is depicted on the Euclidean plane as a rotational motion of the straight line, tangent to the circle with radius equal to the mass of the particle. In this way we come to new representation for the momenta in relativistic dynamics.     

Quadratic kernel-free non-linear support vector machine

A new quadratic kernel-free non-linear support vector machine (which is called QSVM) is introduced. The SVM optimization problem can be stated as follows: Maximize the geometrical margin subject to all the training data with a functional margin greater than a constant. The functional margin is equal to W ^T X + b which is the equation of the hyper-plane used for linear separation. The geometrical margin is equal to . And the constant in this case is equal to one. To separate the data non-linearly, a dual optimization form and the Kernel trick must be used. In this paper, a quadratic decision function that is capable of separating non-linearly the data is used. The geometrical margin is proved to be equal to the inverse of the norm of the gradient of the decision function. The functional margin is the equation of the quadratic function. QSVM is proved to be put in a quadratic optimization setting. This setting does not require the use of a dual form or the use of the Kernel trick. Comparisons between the QSVM and the SVM using the Gaussian and the polynomial kernels on databases from the UCI repository are shown.     

Canonical and Noncanonical Recursion Operators in Multidimensions

The hierarchies of evolution equations associated with the spectral operators ?_x?_y ? R?_y ? Q and ?_x?_y ? Q in the plane are considered. In both cases a recursion operator Ф₁₂, which is nonlocal and generates the hierarchy, is obtained. It is shown that only in the first case does the recursion operator satisfy the canonical geometrical scheme in 2 + 1 dimensions proposed by Fokas and Santini. The general procedure proposed allows one to derive, at the same time, the evolution equations associated with a given linear spectral problem and their Backlund transformations (if they exist), with no need to verify by long and tedious computations the algebraic properties of Ф₁₂. Two equations in the first hierarchy can be considered as two different integrable generalizations in the plane of the dispersive long wave equation. All equations in this hierarchy are shown to be both a dimensional reduction of bi-Hamiltonian n × n matrix evolution equations in multidimensions and a generalization in the plane of bi-Hamiltonian n × n matrix evolution equations on the line.     

Applications of geometric means on symmetric cones

Let V be a Euclidean Jordan algebra, and let be the corresponding symmetric cone. The geometric mean of two elements a and b in is defined as a unique solution, which belongs to of the quadratic equation where P is the quadratic representation of V. In this paper, we show that for any a in the sequence of iterate of the function defined by converges to a. As applications, we obtain that the geometric mean of can be represented as a limit of successive iteration of arithmetic means and harmonic means, and we derive the L?wner-Heinz inequality on the symmetric cone Furthermore, we obtain a formula which leads a Golden-Thompson type inequality for the spectral norm on V. Received October 5, 1999 / Revised March 6, 2000 / Published online October 30, 2000     

The quadratic plus the tensor potentials for particle with 1/2‐spin via L2 method under the spin symmetry

In the present letter, the relativistic equation for particle 1/2‐spin have been obtained for the quadratic scalar and vector potentials in the presence of the tensor interaction that depends on the radial component either linearly and inversely. Under the spin symmetry, the relativistic equation is calculated by using the idea of L² that supports a tridiagonal matrix representation of the wave operator. By this requirement, the relativistic energy spectrum and corresponding spinor wave functions are obtained. Also, the obtained analytically result is compared with other results that are in good agreement. Some of the numerical results are given, too. Copyright © 2015 John Wiley & Sons, Ltd.     

Multicomplex algebras on polynomials and generalized Hamilton dynamics

Generator of the complex algebra within the framework of general formulation obeys the quadratic equation. In this paper we explore multicomplex algebra with the generator obeying n-order polynomial equation with real coefficients. This algebra induces generalized trigonometry ((n+1)-gonometry), underlies of the nth order oscillator model and nth order Hamilton equations. The solution of an evolution equation generated by (n×n) matrix is represented via the set of (n+1)-gonometric functions. The general form of the first constant of motion of the evolution equation is established.     

A Quadratic Clipping Step with Superquadratic Convergence for Bivariate Polynomial Systems

A new numerical approach to compute all real roots of a system of two bivariate polynomial equations over a given box is described. Using the Bernstein–Bézier representation, we compute the best linear approximant and the best quadratic approximant of the two polynomials with respect to the L ² norm. Using these approximations and bounds on the approximation errors, we obtain a fat line bounding the zero set first of the first polynomial and a fat conic bounding the zero set of the second polynomial. By intersecting these fat curves, which requires solely the solution of linear and quadratic equations, we derive a reduced subbox enclosing the roots. This algorithm is combined with splitting steps, in order to isolate the roots. It is applied iteratively until a certain accuracy is obtained. Using a suitable preprocessing step, we prove that the convergence rate is 3 for single roots. In addition, experimental results indicate that the convergence rate is superlinear (1.5) for double roots.     

Adaptive control of linear stochastic evolution systems

An adaptive control problem for some linear stochastic evolution systems in Hilbert spaces is formulated and solved in this paper. The solution includes showing the strong consistency of a family of least squares estimates of the unknown parameters and the convergence of the average quadratic costs with a control based on these estimates to the optimal average cost. The unknown parameters in the model appear affinely in the infinitesimal generator of the C ₀ semigroup that defines the evolution system. A recursive equation is given for a family of least squares estimates and the bounded linear operator solution of the stationary Riccati equation is shown to be a continuous function of the unknown parameters in the uniform operator topology     

Riesz Potentials and Liouville Operators on Fractals

An analogue to the theory of Riesz potentials and Liouville operators in R ⁿ for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of Euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodifferential equations the fractional heat-type equation is solved.     

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

Asymptotics and analytic modes for the wave equation in similarity coordinates

We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues with analytic eigenfunctions. Our results show that, for sufficiently regular data, the long-time behaviour of the solution is governed by the analytic eigenfunctions. The same techniques are applied to the linear stability problem for the fundamental self-similar solution χ _T of the wave equation with a focusing power nonlinearity. Analogous to the free wave equation, we show that the long-time behaviour (in similarity coordinates) of linear perturbations around χ _T is governed by analytic mode solutions. In particular, this yields a rigorous proof for the linear stability of χ _T with the sharp decay rate for the perturbations.      

Fast projection onto the ordered weighted ℓ1 norm ball

In this paper,we provide a finitely terminated yet efficient approach to compute the Euclidean projection onto the ordered weighted?₁(OWL1)norm ball.In particular,an efficient semismooth Newton method is proposed for solving the dual of a reformulation of the original projection problem.Global and local quadratic convergence results,as well as the finite termination property,of the algorithm are proved.Numerical comparisons with the two best-known methods demonstrate the efficiency of our method.In addition,we derive the generalized Jacobian of the studied projector which,we believe,is crucial for the future designing of fast second-order nonsmooth methods for solving general OWL1 norm constrained problems.     

Properties of Three-Dimensional Median Line Location Models

We consider the problem of locating a line with respect to some existing facilities in 3-dimensional space, such that the sum of weighted distances between the line and the facilities is minimized. Measuring distance using the l _p norm is discussed, along with the special cases of Euclidean and rectangular norms. Heuristic solution procedures for finding a local minimum are outlined.     

Representation Theorem for Generators of Quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in z. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic g-expectation.     

Residue minimization technique to analyze the efficiency of convective straight fins having temperature-dependent thermal conductivity

In this article, we propose and implement a numerical technique based on residue minimization to solve the nonlinear differential equation, which governs the temperature distribution in straight convective fins having temperature-dependent thermal conductivity. The form of temperature distribution is approximated by a polynomial series, which exactly satisfies the boundary conditions of the problem. The unknown coefficients of the assumed series are optimized using the Nelder–Mead simplex algorithm such that the squared L₂ norm of the residue attains its minimum value within a specified tolerance limit. The near-exact solution thus obtained is further used to calculate the fin efficiency. For the case of constant thermal conductivity, the obtained results are validated with the analytical solutions, while for the case of variable thermal conductivity, the obtained results are corroborated with those previously published in the literature. An excellent agreement in each case consolidates the effectiveness of the proposed numerical technique.     

Note on the computational complexity ofj-radii of polytopes in ℝ n

We show that, for fixed dimensionn, the approximation of inner and outerj-radii of polytopes in ℝ ⁿ , endowed with the Euclidean norm, is in ℙ. Our method is based on the standard polynomial time algorithms for solving a system of polynomial inequalities over the reals in fixed dimension.     

Projecting <Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> onto Classical Spaces

We describe an explicit construction of a linear projection of a symmetric conical section of the n-dimensional cube onto a (1+ε)-isomorphic version of the Euclidean ball of proportional dimension, or more generally onto a (1+ε)-isomorphic image of an l _p ^m -ball. Allowing non-linear projections (of logarithmic polynomial nonlinearity) we may even project the full n-dimensional cube onto the same images. This is done by gluing together explicit projections onto two-dimensional spaces, interpreting and modifying a paper of Ben-Tal and Nemirowski on polynomial reductions of conic quadratic programming problems to linear programming problems in terms of Banach spaces.      

A posteriori error estimates for optimal distributed control governed by the evolution equations

We describe a technique for a posteriori error estimates suitable to the optimal control problem governed by the evolution equations solved by the method of lines. It is applied to the control problem governed by the parabolic equation, convection-diffusion equation and hyperbolic equation. The error is measured with the aid of the L²-norm in the space-time cylinder combined with a special time weighted energy norm.     

Inversion of degree <Emphasis Type="Italic">n</Emphasis> + 2

By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where the corresponding points lie on the rays of the first order, nth class congruence C _n ¹ and are conjugate with respect to a proper quadric Ψ. We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in the Euclidean space the pedal surfaces of the congruences C _n ¹ can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces with the program Mathematica. They are shown in four examples.