Matrix Algebraic Infinite Product Representation for Generalized Hypergeometric Functions of Typep+1Fp

The geometric phase requires the multivaluedness of solutions to Fuchsian second-order equations. The angle, or its complement, is given by half the area of a spherical triangle in the case of three singular points, or half the area of a lune in the case of two singular points. Both are fundamental regions where the automorphic function takes a value only once, and a linear-fractional transformation tessellates the plane in replicas of the fundamental region. The condition that the homologues of the poles, representing vertices, be angles places restrictions on quantum numbers which are no longer integers, for, otherwise, the phase factors would become unity. Restriction must be made to regular singular points for only then will solutions to the differential equation be rational functions so that the covering group will be cyclic and the covering space be a ‘spiral staircase’. Many of the equations of mathematical physics, with essential singularities, become Fuchsian differential equations, with regular singularities, at zero kinetic energy. Examples of geometric phase include the phasor, the Pancharatnam phase of beams of polarized light in different states, the Aharanov-Bohm phase, and angular momentum with centripetal ‘attraction’. In the latter example, the phase is one-half the area of the lune, which disappears when the pole at infinity becomes an essential singularity thereby recovering the Schrodinger equation. The behavior of an automorphic function at a limit point on the boundary is analogous to the confluence of two regular singularities in a linear second-order differential equation to produce an essential singularity at infinity. ∗ info@bernardhlavenda.com; www.bernardhlavenda.com 1 ar X iv :1 41 1. 56 03 v1 [ ph ys ic s. ge nph ] 1 5 M ay 2 01 3

In this section, we study the behavior of solutions of a linear Pfaffian system in a neighborhood of a singular point of the coefficients, where we restrict singular points to ones corresponding to regular singular points in ordinary differential equations. Note that a singular point of analytic functions in several variables is not a point but a set of codimension one (a hypersurface).

On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.

Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence with monodromy-free ${}^LG$-opers on the projective line with regular singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues at the singular points.

Efficient accelero-summation of holonomic functions

In this chapter, we shall first explain the existence of solutions of initial value problems for differential equations and then fundamental theorems for linear differential equations in the complex domain. We explain the definition of regular and irregular singularities of linear differential equations, and the behavior of of local solutions near those singular points.

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for mean curvature flow of hypersurfaces. In particular, this implies that when a mean curvature flow has a multiplicity-one conical singularity model, the evolving surface at the singular time has an (isolated) regular conical singularity at the singular point. This should lead to a complete understanding of how to "flow through" such a singularity.

The compatible expansion in series of solutions of both the equations of P–Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P–Q pair are derived.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

This paper studies schemes to de-bias the Lasso in sparse linear regression with Gaussian design where the goal is to estimate and construct confidence intervals for a low-dimensional projection of the unknown coefficient vector in a preconceived direction a0. Our analysis reveals that previously analyzed propositions to de-bias the Lasso require a modification in order to enjoy nominal coverage and asymptotic efficiency in a full range of the level of sparsity. This modification takes the form of a degrees-of-freedom adjustment that accounts for the dimension of the model selected by the Lasso. The degrees-of-freedom adjustment (a) preserves the success of de-biasing methodologies in regimes where previous proposals were successful, and (b) repairs the nominal coverage and provides efficiency in regimes where previous proposals produce spurious inferences and provably fail to achieve the nominal coverage. Hence our theoretical and simulation results call for the implementation of this degrees-of-freedom adjustment in de-biasing methodologies. Let s0 denote the number of nonzero coefficients of the true coefficient vector and Σ the population Gram matrix. The unadjusted de-biasing scheme may fail to achieve the nominal coverage as soon as s0⋙n2/3 if Σ is known. If Σ is unknown, the degrees-of-freedom adjustment grants efficiency for the contrast in a general direction a0 when s0logp n+min{sΩlogp n,‖Σ−1 a0 ‖1 logp ‖ Σ−1/2 a0 ‖2 n}+min(sΩ,s0)logp n→0 where sΩ=‖ Σ−1a0‖0. The dependence in s0,sΩ and ‖Σ−1a0‖1 is optimal and closes a gap in previous upper and lower bounds. Our construction of the estimated score vector provides a novel methodology to handle dense directions a0. Beyond the degrees-of-freedom adjustment, our proof techniques yield a sharp ℓ∞ error bound for the Lasso which is of independent interest.

<p style='text-indent:20px;'>We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a <i>global holonomy group</i>. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.</p>

On the development of compatible elements for the solution of three-dimensional elasticity problems

Consider approximating functions based on a finite number of their samples. We show that adaptive algorithms are much more powerful than nonadaptive ones when dealing with piecewise smooth functions. More specifically, let F r 1 F_r^1 be the class of scalar functions f : [ 0 , T ] → R f:[0,T]\to \mathbb {R} whose derivatives of order up to r r are continuous at any point except for one unknown singular point. We provide an adaptive algorithm A n ad \mathcal {A}_n^\textrm {ad} that uses at most n n samples of f f and whose worst case L p L^p error ( 1 ≤ p > ∞ 1\le p>\infty ) with respect to ‘reasonable’ function classes F r 1 ⊂ F r 1 \mathcal {F}_r^1\subset F_r^1 is proportional to n − r n^{-r} . On the other hand, the worst case error of any nonadaptive algorithm that uses n n samples is at best proportional to n − 1 / p n^{-1/p} . The restriction to only one singularity is necessary for superiority of adaption in the worst case setting. Fortunately, adaption regains its power in the asymptotic setting even for a very general class F r ∞ F_r^\infty consisting of piecewise C r C^{r} -smooth functions, each having a finite number of singular points. For any f ∈ F r ∞ f\in F_r^\infty our adaptive algorithm approximates f f with error converging to zero at least as fast as n − r n^{-r} . We also prove that the rate of convergence for nonadaptive methods cannot be better than n − 1 / p n^{-1/p} , i.e., is much slower. The results mentioned above do not hold if the errors are measured in the L ∞ L^\infty norm, since no algorithm produces small L ∞ L^\infty errors for functions with unknown discontinuities. However, we strongly believe that the L ∞ L^\infty norm is inappropriate when dealing with singular functions and that the Skorohod metric should be used instead. We show that our adaptive algorithm retains its positive properties when the approximation error is measured in the Skorohod metric. That is, the worst case error with respect to F r 1 \mathcal {F}_r^1 equals Θ ( n − r ) \Theta (n^{-r}) , and the convergence in the asymptotic setting for F r ∞ F_r^\infty is n − r n^{-r} . Numerical results confirm the theoretical properties of our algorithms.

Structural reanalysis by iteration

SUMMARY Given a p × p symmetric matrix B and a p-vector c outside the range of B, a simple expression is formulated for the action of the Moore-Penrose generalized inverse B + on a vector b in the range of B, in terms of b, c and (B + cc‱)+. A special case of this formula gives a ‘downdating’ formula for the vector of least squares regression coefficients based on the Moore-Penrose inverse of the centred sum of squares and products matrix. The formula is analogous to that given by A. Albert for uncentred least squares fitting. An explicit downdating formula for the Moore-Penrose inverse B + itself is derived, and then more general downdating expressions for the inverse and the vector B + b when the p-vector c is replaced by a matrix array C of such vectors outside the range of B.