Wind identification along a flight trajectory,part 1: 3D-kinematic approach

This paper deals with the identification of the wind profile along a flight trajectory by means of a three-dimensional kinematic approach. The approach is then applied to a recent aircraft accident, that of Flight Delta 191, which took place at Dallas-Fort Worth International Airport on August 2, 1985.In the 3D-kinematic approach, the wind velocity components are computed as the difference between the inertial velocity components and the airspeed components. The airspeed profile is obtained from flight measurements. The inertial velocity profile is obtained by integration of the measured inertial acceleration. The accelerometer biases and the impact values of the inertial velocity components are determined by matching the computed flight trajectory with the measured flight trajectory, available from the digital flight data recorder (DFDR) and air traffic control radar (ATCR). This leads to a least-square problem, which is solved analytically.Key to the precision of the identified wind profile is the correct identification of the accelerometer biases and the impact velocity components. In turn, this depends on the proper selection of the integration time. Because the measured data are noise-corrupted, unstable identification occurs if the integration time is too short. On the other hand, stable identification takes place if the integration time is properly chosen.Application of the method developed to the case of Flight Delta 191 shows that the identification problem has a stable solution if the integration time is larger than 180 sec. Numerical computation shows that, for Flight Delta 191, the maximum wind velocity difference determined with the 3D-kinematic approach was W _x =124 fps in the longitudinal direction, W _y =66 fps in the lateral direction, and W _h =71 fps in the vertical direction.Notations a _b measured acceleration, ft/sec² - a _bx ,a _by ,a _bz measured acceleration components, body axes system, ft/sec² - b accelerometer bias, ft/sec² - b _x ,b _y ,b _z accelerometer bias components, body axes system, ft/sec² - g acceleration of gravity, ft/sec² - h altitude above sea level, positive upward, ft - m mass, lb sec²/ft - p air pressure, lb/ft² - R gas constant for air, ft²/sec² °R - S reference wing area, ft² - t running time, sec - T air temperature, °R - V _a airspeed, ft/sec - V _ax ,V _ay ,V _az airspeed components, Earth axes system, ft/sec - V _e inertial velocity, ft/sec - V _ex ,V _ey ,V _ez inertial velocity components, Earth axes system, ft/sec - V _ia indicated airspeed, ft/sec - W mg=weight, lb - W wind velocity, ft/sec - W _x ,W _y ,W _z wind velocity components, Earth axes system, ft/sec - x, y, z Cartesian coordinates, Earth axes system, ft - x longitudinal coordinate, positive northward, Earth axes system, ft - y lateral coordinate, positive eastward, Earth axes system, ft - z vertical coordinate, positive downward, Earth axes system, ft - angle of attack, rad - path inclination, rad - pitch angle, rad - bank angle, rad - air density, lb sec²/ft⁴ - sideslip angle, rad - integration time, sec - roll angle, rad This research was supported by Air Line Pilots Association (ALPA), United States Aviation Underwriters (USAU), and Texas Advanced Technology Program, Grant No. TATP-003604020. This paper is based on Ref. 1.     

The hyperbolic n-space as a graph in Euclidean (6n−6)-space

We construct for each n2 an isometric C-imbedding HⁿE^6n–6 of the n-dim. hyperbolic space into the Euclidean space E^6n–6 whose image is the graph of a C-map RⁿR^5n–6. This generalizes a well known result of D. Blanusa 1955 (the case n=2).     

On indefinite superlinear elliptic equations

This paper concerns semilinear elliptic equations of the form – u+m(x)u=a(x)u ^p , wherea changes sign. We discuss the question of existence of positive solutions when the linear part is not coercive.This article was processed by the author using the LATEX style file pljourlm from Springer-Verlag.     

The norm in C of orthogonal projections onto subspaces of polygonal functions

Let P be an orthogonal projection (in the sense of L₂) onto the subspace of polygonal functions over a certain partition of the segment [0, 1]. Z. Ciesielski has established the following estimate for the norm of this operators, as acting from C into C, valid for an arbitrary partition: P _CC 3. In this note it is proved that this estimate is final; more precisely, it is shown that .Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 495–502, April, 1977.     

Vector Equilibrium Problems Under Asymptotic Analysis

Given a closed convex set K in Rⁿ; a vector function F:K×K R^m; a closed convex (not necessarily pointed) cone P(x) in ^m with non-empty interior, PP(x) Ø, various existence results to the problemfind xK such that F(x,y)- int P(x) y K under P(x)-convexity/lower semicontinuity of F(x,) and pseudomonotonicity on F, are established. Moreover, under a stronger pseudomonotonicity assumption on F (which reduces to the previous one in case m=1), some characterizations of the non-emptiness of the solution set are given. Also, several alternative necessary and/or sufficient conditions for the solution set to be non-empty and compact are presented. However, the solution set fails to be convex in general. A sufficient condition to the solution set to be a singleton is also stated. The classical case P(x)=^m ₊ is specially discussed by assuming semi-strict quasiconvexity. The results are then applied to vector variational inequalities and minimization problems. Our approach is based upon the computing of certain cones containing particular recession directions of K and F.     

Control of stochastic distributed-parameter systems

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.Notation a(t) l-vector noisy dynamic input to system - A(t, a) l-vector function - A frequency factor for first-order rate law (5.68×10⁶ sec^–1) - b distance to the centerline between two coil banks in the reactor (4.7 cm) - B k-vector function defining the control action - c(, ) concentration of styrene monomer, molel ^–1 - C _p heat capacity (0.43 cal · g^–1 · K^–1) - C _ij constants in approximate filter, Eqs. (49)–(52) - E activation energy (20330 cal · mole^–1) - expectation operator - f(t,...) n-vector function - g ₀,g ₁(t,...) n-vector functions - h(t, u) m-vector function relating observations to states - H(t) function defined in Eq. (36) - k dimensionality of control vectorv(x, t) - k _i constants in approximate filter, Eqs. (49)–(52) - K dimensionless proportional gain - l dimensionality of dynamic inputa(t) - m dimensionality of observation vectory(t) - n dimensionality of state vectoru(x, t) - P ^(vv)(x, s, t) n×n matrix governed by Eq. (9) - P ^(va)(x, t) n×l matrix governed by Eq. (10) - P ^(aa)(t) l×l matrix governed by Eq. (11) - q _i (t) diagonal elements ofm×m matrixQ(x, s, t) - Q(x, s, t) m×m weighting matrix - R universal gas constant (1.987 cal · mole^–1 · K^–1) - R(x, s, t) n×n weighting matrix - R _i (t) n×n weighting matrix - s dimensionless spatial variable - S(x, s, t) matrix defined in Eq. (11) - t dimensionless time variable - T(, ) temperature, K - u(x, t) n-dimensional state vector - u _c (t) wall temperature - u _d desired value ofu ₁(1,t) - u _c * reference control value ofu _c - u _c ^max maximum value ofu _c - u _c ^min minimum value of _c - v(x, t) k-dimensional control vector - W(t) l×l weighting matrix - x dimensionless spatial variable - y(t) m-dimensional observation vector - _i constants in approximate filter, Eqs. (49)–(52) - dimensionless parameter defined in Eq. (29) - H heat of reaction (17500 cal · mole^–1) - dimensionless activation energy, defined in Eq. (29) - (x) Dirac delta function - (x, t) m-dimensional observation noise - thermal conductivity (0.43×10^–3 cal · cm^–1 · sec^–1 · K^–1) - density (1 g · cm^–3) - time, sec - dimensionless parameter defined in Eq. (29) - spatial variable, cm - * reference value - ^ estimated value     

Asymptotic analysis of the exponential penalty trajectory in linear programming

We consider the linear program min{cx: Axb} and the associated exponential penalty functionf _r(x) = cx + rexp[(A _ix – b_i)/r]. Forr close to 0, the unconstrained minimizerx(r) off _r admits an asymptotic expansion of the formx(r) = x ^* + rd^* + (r) wherex ^* is a particular optimal solution of the linear program and the error term(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectory(r) which converges exponentially fast to a particular dual optimal solution. These results are completed by an asymptotic analysis whenr tends to : the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution.Corresponding author. Both authors partially supported by FONDECYT.     

Properties of continuous mappings of unbounded metric spaces

Suppose a closed unbounded set F R_n is a union of a finite number p of closed unbounded sets F_i that are pairwise disjoint, and suppose f is a continuous mapping of F into the metric space R⁽²⁾. With each set F_i there is associated a point at infinity _i, at which it is assumed that f has a finite limit A_i R⁽²⁾, i=1, 2, ..., p. It is proved that: 1) f is bounded on F; 2) if f is a real functional, then the set contains a smallest and a largest value; 3) if the distance between F_i and F_j is greater than zero whenever i j, then f is uniformly continuous on F.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 422–427, March, 1991.     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Number of no nisomorphic subgraphs in an n-point graph

LetG(n) be the set of all nonoriented graphs with n enumerated points without loops or multiple lines, and let v_k(G) be the number of mutually nonisomorphic k-point subgraphs of G G(n). It is proved that at least |G(n)| (1–1/n) graphs G G(n) possess the following properties: a) for any k [6log₂n], where c=–c log₂c–(1–c)×log₂(1–c) and c>1/2, we havev _k(G) > C _n ^k (1–1/n²); b) for any k [cn + 5 log₂n] we havev _k(G) = C _n ^k . Hence almost all graphs G G(n) containv(G) 2ⁿ pairwise nonisomorphic subgraphs.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 263–273, March, 1971.     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.     

Integrals involving Gegenbauer and Hermite polynomials on the imaginary axis

Summary The integral _- [C _2n (it)]^–2(1+t ²)^–-1/2 dt is evaluated for > –1/2 whereC _2n is the Gegenbauer polynomial of degree 2n. Letting gives the value _- [H _2n (it)]^–2 e ^{1-1/2t 2} dt involving the Hermite polynomialH _2n of degree 2n. The result is obtained using Gegenbauer functions of the second kind.     

Signature Operators and Surgery Groups over C*-algebras

Let A be a unital complex C^* algebra, L^*(A) the projective symmetric surgery groups, and K_*(A) topological K theory. We define groups B^*(A) of bordism classes of Fredholm complexes over A with Poincaré duality. These generalize the de Rham complex. It is shown that there are isomorphisms B^*(A)K_* (A) and B^*(A) L_*(A) given by abstract versions of the signature operator and symmetric signature. The remaining side of a triangle is formed by an isomorphism due to Mienko.     

Zum Cauchy-Problem bei der verallgemeinerten Wärmeleitungsgleichung

Cauchy's problem for the equationu _xx +x ^–1 u _x =u _t ( real) was discussed byD. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods ofD. Colton and of this paper are different but the results are similar.     

A characterization of brownian motion in a lipschitz domain by its killing distributions

Let (Y _t, Q^x) be a strong Markov process in a bounded Lipschitz domainD with continuous paths up to its lifetime , and let (X _t, P^x) be a Brownian motion inD. IfY _– exists in D andQ ^x(Y_– C)=P^x(X_– C) for all Borel subsetsC of D and allx, thenY is a time change ofX.     

Boundedness in the mean of orthonormalized polynomials

For the polynomials {p_n(t)} ₀ , orthonormalized on [–1, 1] with weightp(t) = (1–t) (1+t) _v=1 ^m , we obtain necessary and sufficient conditions for boundedness of the sequences of norms: 1) 2) and 3) with the conditions that on [–1, 1] and (H,)^–1 L²(0, 2), where(H,) is the modulus of continuity in C(–1, 1) of function H.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 759–770, May, 1973.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

Factorization and reflexivity on Fock spaces

The framework of the paper is that of the full Fock space and the Banach algebraF which can be viewed as non-commutative analogues of the Hardy spacesH ² andH respectively.An inner-outer factorization for any element in as well as characterization of invertible elements inF are obtained. We also give a complete characterization of invariant subspaces for the left creation operatorsS ₁ ,..., S _n of . This enables us to show that every weakly (strongly) closed unital subalgebra of {(S ₁ ,..., S _n ) F } is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.The first author was supported in part by NSF DMS 93-21369 1991Mathematics Subject Classification. Primary 47D25, Secondary 32A35, 47A67.     

Self-Converse Directed BIBDs with Block Size Four

A directed balanced incomplete block design (or D B(k,;v)) (X,) is called self-converse if there is an isomorphic mapping f from (X,) to (X,^–1), where ^–1={B ^–1:B} and B ^–1=(x _k ,x _{k –1},,x ₂,x ₁) for B=(x ₁,x ₂,,x _{k –1},x _k ). In this paper, we give the existence spectrum for self-converse D B(4,1;v). AMS Classification:05BResearch supported in part by NSFC Grant 10071002 and SRFDP under No. 20010004001