Maximal Partial Spreads and Flocks

Denote by a flock of a quadratic cone of PG(3,q) by S() the spread of PG(3,q) associated with and by l the common line of the base reguli. Suppose that there are two lines not transversal to a base regulus which share the same lines of S() Then we prove that is either linear or a Kantor-Knuth semifield flock. Using this property we can extend the result of J3 on derivable flocks proving that, if a set of q + 1 lines of S() defines a derivable net different from a base regulus-net, then is either linear or a Kantor-Knuth semifield flock. Moreover if l is not a component of the derivable net, then is linear.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

Galois theory for the family of partial differential equationsΔ α ψ=0

Summary The partial differential fields most suited for the purpose of construction of Galois theory for the family (1) are endowed with the symmetric bilinear form (2iv) and are called -differential fields. In Section 1 are defined certain algebraic notions related to the symmetric bilinear form (2iv) and which are necessary for the construction of any Galois theory. Necessary and sufficient condition for the extension of the domain of the operator (this operator is not a derivation although it commutes with the partial derivations of the -differential field) from an -differential fieldK to a finitely generated -differential extension field is given in Theorem 1.Section 2 defines the notion of -differential mapping as linear mappings which preserve the symmetric bilinear form and commute with the partial derivations. The group properties of the set of -differential mappings are discussed and the Galois correspondence theorems set up for -differential fields.Section 3 sets up the notion of -Liouvillian extensions of -differential fields and briefly discusses the Galois groups associated with these -Liouvillian extension fields.Section 4 points to the procedure for the algebraic characterization of -simple--differential field extensions by elementary solutions of the partial differential equation _m =0.     

Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential

The properties of stationary solutions of the one-dimensional fractional Einstein--Smoluchowski equation with a potential of the form x ^2m+2, m=1,2,..., and of the Riesz spatial fractional derivative of order , 12, are studied analytically and numerically. We show that for 1<2, the stationary distribution functions have power-law asymptotic approximations decreasing as x ^–(+2m+1) for large values of the argument. We also show that these distributions are bimodal.     

Existenzaussagen für gewisse Zweipunkt-Randwertaufgaben

Zusammenfassung Für Randwertaufgaben der Form–u–l ₀ ...u–l ₀ ...u=f(x, u) mitl ₀R,lR,f definiert und stetig auf {a<-x<-b, |u|<} wird eine Existenzaussage gewonnen, fallsf inu linear durch die aufeinanderfolgenden Eigenwerte der zugehörigen linearen Aufgabe beschränkt ist. Zum Beweis betrachtet man die äquivalente Hammersteinsche Integralgleichung mit nichtsymmetrischem Kern. Mit Hilfe des Schauderschen Fixpunktsatzes erhält man für diese Integralgleichung Existenzaussagen, welche Ergebnisse von Dolph verallgemeinern.

---

Summary This note contains an existence theorem for a two-point boundary value problem of the form–u–l ₀ u–l ₀ u=f(x, u) wherel ₀R,l ₀R,f defined and continuous on {a<-xb, |u|<} iff is linear bounded inu by the successive eigenvalues of the corresponding linear problem. To prove this result we consider the equivalent Hammerstein integral equation with non-symmetric kernel. Schauders fixpoint principle supplies existence theorems for integral equations of this type which generalize results of Dolph in some sense.

     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

Asymptotic stability at infinity of planar vector fields

Let >0 andX be aC ¹ vector field on the plane such that: (i) for allq², Det(DX(q))>0; and (ii) for allp², with p, Trace(D(X(p))<0. IfX has a singularity and ² Trace(DX)dxdy is less than 0 (resp. greater or equal than 0), then the point at infinity of the Riemann sphere ²{} is a repellor (resp. an attractor) ofX.     

Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

Subdivision algorithms with nonnegative masks generally converge

In this paper, a subdivision scheme consists of an operator froml () tol () determined by a doubly infinite sequence, called the mask. This operator convolutes, in a certain sense, sequences l () with the mask, thus producing a new sequence inl (). Moreover, this new sequence is placed on a finer grid. If we iterate this process with a positive mask infinitely many times, it is known that this process will produce a continuous function, which we callf . In this paper, we consider the extent to which non-negative masks yield similar results. An important application of subdivision schemes in computer graphics is the generation of curves and surfaces from an initial sequence.     

Convergence proofs and error estimates for an iterative method for conformal mapping

Summary The iterative method as introduced in [8] and [9] for the determination of the conformal mapping of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve ofG is parametrized by a function with Lipschitz continuous derivative then the method converges locally in the Sobolev spaceW of 2-periodic absolutely continuous functions with square integrable derivative. If is in a Hölder classC ²⁺, the order of convergence is at least 1+. If is inC ^l+1+ withl1, 0<<1, then the iteration converges inC ^l+. For analytic boundary curves the convergence takes place in a space of analytic functions.For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N ^1–l–) if is inC ^l+1+, andO (exp (–N/2)) if is an analytic curve. The number in the latter formula is bounded by logR, whereR is the radius of the largest circle into which can be extended analytically such that'(z)0 for |z|<R. The results of some test calculations are reported.     

On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G (, R) of twisted type =A_l,l odd, D_l, E₆ over a commutative semilocal ring R with involution are in one-to-one correspondence with the -invariant parabolic nets of ideals of R of type , i.e., with the sets, of ideals of R such that: (l) whenever; (2) = for all ; (3) =R for > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 21–36, 1979.     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

A note on the illumination of convex bodies

LetKE ^d be a convex body and letl _r(K) denote the minimum number ofr-dimensional affine subspaces ofE ^d lying outsideK with which it is possible to illuminateK, where 0rd–1. We give a new proof of the theorem thatl _r(K)(d+1)/(r+1) with equality for smoothK.The work was supported by Hung. Nat. Found. for Sci . Research No. 326-0213 and 326-0113.     

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.