Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford

We develop a cohomology theory for Deligne–Mumford stacks, adapted to Hirzebruch–Riemann–Roch formulas. For this, we define the cohomology with coefficients in the representations and a Chern character, and we prove a Grothendieck–Riemann–Roch formula for the associated Riemann–Roch transformation.     

A Family of Semi-Regular Divisible Difference Sets

We give a construction of semi-regular divisible difference sets with parametersm = p2a(r–1)+2b (pr – 1)/(p – 1), n = pr, k = p(2a+1)(r–1)+2b (pr – 1)/(p – 1)1 = p(2a+1)(r–1)+2b (pr–1 – 1)/(p-1), 2 = p2(a+1)(r–1)–r+2b (pr – 1)/(p – 1)where p is a prime and r a + 1.     

Effective Replaceability of the Omega—Rule for Restricted Sequents of the First–Order Linear Temporal Logic

We consider an effective replaceability of the omega–rule for the always operator in a restricted first–order linear temporal logic. This work is a continuation of two previous papers of the author where an infinitary (with the omega–rule) calculus without loop rules was constructed and founded. In the present paper, we use this calculus to construct the so–called saturated calculus consisting of four decidable deductive procedures replacing the omega–rule for the always operator.     

Some more general bounds on the residual term of the Lagrange interpolation formula

Error bounds are derived for the Lagrange interpolation formula and for the k-th derivative of the residual term of this formula in terms of the Lipschitz constant of the n-th derivative for the case with (n+1) nodes and also for the case when the functions satisfy a special condition: G x, y, z [a, b]: ¦ (x)(z–y) +(y)(x–z)+(z)(y–x)¦G¦(x–y)(y–z)(z–x)¦.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 27–32, 1992.     

The Nevanlinna—Pick Interpolation Problem in Multiply Connected Domains

We simplify and strengthen Abrahamse's result on the Nevanlinna–Pick interpolation problem in a finitely connected planar domain, according to which the problem has a solution if and only if the Pick matrices associated with character-automorphic Hardy spaces are positive semidefinite for all characters in ^n–1/^n–1, where n is the connectivity of the domain. The main aim of the paper is to reduce the indicated procedure (verification of the positive semidefiniteness) for the entire real (n–1)-torus ^n–1/^n–1 to a part of it, whose dimension is, possibly, less than n–1. Bibliography: 14 titles.     

A Characterization of Some Minihypers and its Application to Linear Codes

Any {f,r- 2+s; r,q}-minihyper includes a hyperplane in PG(r, q) if fr-1 + s 1 + q – 1 for 1 s q – 1, q 3, r 4, where i = (qi + 1 – 1)/ (q – 1 ). A lower bound on f for which an {f, r – 2 + 1; r, q}-minihyper with q 3, r 4 exists is also given. As an application to coding theory, we show the nonexistence of [ n, k, n + 1 – qk – 2 ]q codes for k 5, q 3 for qk – 1 – 2q – 1 < n qk – 1 – q – 1 when k > q – q - \sqrt q + 2$$ " align="middle" border="0"> and for when , which is a generalization of [18, Them. 2.4].     

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.     

Renormalization and Dimensional Regularization for a Scalar Field with Gauss–Bonnet-Type Coupling to Curvature

We consider a scalar field with a Gauss–Bonnet-type coupling to the curvature in a curved space–time. For such a quadratic coupling to the curvature, the metric energy–momentum tensor does not contain derivatives of the metric of orders greater than two. We obtain the metric energy–momentum tensor and find the geometric structure of the first three counterterms to the vacuum averages of the energy–momentum tensors for an arbitrary background metric of an N-dimensional space–time. In a homogeneous isotropic space, we obtain the first three counterterms of the n-wave procedure, which allow calculating the renormalized values of the vacuum averages of the energy–momentum tensors in the dimensions N=4,5. Using dimensional regularization, we establish that the geometric structures of the counterterms in the n-wave procedure coincide with those in the effective action method.     

Maximal subgroups of symmetric groups defined on projective spaces over finite fields

Let PL(n, q) be a complete projective group of semilinear transformations of the projective space P(n–1, q) of projective degree n–l over a finite field of q elements; we consider the group in its natural 2-transitive representation as a subgroup of the symmetric group S(P*(n–1, q)) on the setp*(n–1),q=p(n–1,q)/{O}. In the present note we show that for arbitrary n satisfying the inequality n>4[(qⁿ–1)/(q^n–1–1)] [in particular, for n>4(q +l)] and for an arbitrary substitutiong s (p*(n–1,q))pL(n,q) the group PL(n,q), g contains the alternating group A(P* (n–1,q)). Forq=2, 3 this result is extended to all n3.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 91–100, July, 1974.The author expresses his sincere thanks to M. M. Glukhov for his interest in his work.     

Zum Cauchy-Problem bei der verallgemeinerten W?rmeleitungsgleichung

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 recursive method of construction of resolvable BIB-designs

A theorem is proved that every resolvable BIB-design (v,k,) with =k–1 and the parameters v and k such that there exists a set of k–1 pairwise orthogonal Latin squares of order v can be embedded in a resolvable BIB-design ((k+1)v, k, k–1). An analogous theorem is established for the class of arbitrary BIB-designs. As a consequence is deduced the existence of resolvable BIB-designs (v,k,) with =k–1 and =(k–1)/2.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 707–715, May, 1977.     

Three formulas for eigenfunctions of integrable Schrödinger operators

We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherlandsintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the Etingof–Kirillov tracesof intertwining operators, the second an integral representationof hypergeometric type, and the third is a formula of Bethe ansatz type.The last two formulas are degenerations of elliptic formulasobtained previously in connection with theKnizhnik–Zamolodchikov–Bernardequation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctionsare parametrized by a Hermite–Bethe variety, a generalizationof the spectral variety of the Lamé operator.We also give the q-deformed version of ourfirst formula. In the scalar slN case, this gives common eigenfunctionsof the commuting Macdonald–Rujsenaars difference operators.     

Locally nilpotent groups with the min−∞−n condition

The study of locally nilpotent groups with the weak minimality condition for normal subgroups, the min^––n condition, is continued. The following results are obtained.THEOREM 1. A locally nilpotent group with the min^––n condition is countable.THEOREM 2. If G is a locally nilpotent group with the min^––n condition whose periodic part is nilpotent and the orders of the elements of the periodic part are bounded in the aggregate, then G=t(G)A, where the subgroup A is minimax.THEOREM 3. Suppose G is a locally nilpotent group with the min^––n condition and T is its periodic part. If T is nilpotent and G/T is Abelian, then G=TA, where the subgroup A is minimax.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 340–346, March, 1990.     

Über meromorphe Modifikationen

Ohne ZusammenfassungUnter dem gemeinsamen Titel Über meromorphe Modifikationen sind eine Reihe von Arbeiten erschienen, und zwar in der Mathematischen Zeitschrift: Teil I (§ 1–§ 3) Bd. 61, S. 206–234, Teil II (§ 4–§ 6) Bd. 61, S. 467–488, Teil III (§ 7–§ 9) Bd. 62, S. 189–210; in den Mathematischen Annalen: Teil IV (§ 10–§ 12) Bd. 130, S. 147–182. Die Literatur und eine ausführliche Einleitung findet sich in Teil I.     

A splitting theorem for certain cohomology theories associated to BP* (−)

Let P(n)^*(–) be Brown-Peterson cohomology modulo I_n and put B(n)^*(–)=P(n)^*(–)[1/v_n]. In this note we construct a canonical multiplicative and idempotent operation _n in a suitable completion (n)^*(–) of B(n)^*(–) which has the property that its image is canonically isomorphic to the n-th Morava K-theory K(n)^*(–). In particular, the ring theory K(n)^*(–) is contained as a direct summand in the theory (n)^*(–). A similar result is not true before completing. pleting. Because the completion map B (n)^*(–) (n)^*(–) is injective, the above splitting theorem contains also information about B(n)^*(–). The proof of the theorem depends on a result about the behaviour of formal groups of finite height over complete graded F_p.     

Zerfallende Quadrikenschnitte und stereographische Projektion

If the intersection of two quadrics Q ₁ ^n–1 , Q ₂ ^n–1 in projective space Pⁿ contains a quadric Q ₃ ^n–2 which is different from its singularity space, then the intersection Q ₁ ^n–1 Q ₂ ^n–1 is the union of two quadrics Q ₃ ^n–2 , Q ₄ ^n–2 . This theorem leads to a geometric characterization of those hyperquadrics of a hyperplane of Pⁿ that are images of quadrics on a given quadric Q^n–1 by a given stereographic projection of Q^n–1 onto .

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Herrn Prof. Dr.Oswald Giering zum 50. Geburtstag gewidmet     

Measurable solutions of a (2, 2)-type sum form functional equation

Summary In this paper we find the general measurable solutions of the functional equationF(xy) + F(x(1 – y)) – F((1 – x)y) – F((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[) whereF, G, H:]0, 1[ C are unknown functions. The solution of this equation is part of our program to determine the measurable solutions of the functional equationF ₁₁ (xy) + F ₁₂ (x(1 – y)) + F ₂₁ ((1 – x)y) + F ₂₂ ((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[). Our method of solution is based on the structure theorem of sum form equations of (2, 2)-type and on a result of B. Ebanks and the author concerning the linear independence of certain functions.     

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

Edges of Canonical Decompositions for 2-Bridge Knots and Links

Let L=C(a1,..., ak) be a hyperbolic 2-bridge knot or link. We give a family of arcs in S3–L which are ambient isotopic to edges of the canonical decomposition of S3–L if each aj is sufficiently large. This result supports the conjecture that the decomposition of S3–L given by Sakuma–Weeks in [12] is the canonical one.     

Hereditary properties of partitions, ordered graphs and ordered hypergraphs

In this paper we use the Klazar–Marcus–Tardos method (see [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory Ser. A 107 (2004) 153–160]) to prove that, if a hereditary property of partitions has super-exponential speed, then, for every k-permutation π, contains the partition of [2k] with parts {{i,π(i)+k}:i[k]}. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs.Our results generalize the Stanley–Wilf conjecture on the number of n-permutations avoiding a fixed permutation, which was recently proved by the combined results of Klazar [M. Klazar, The Füredi–Hajnal conjecture implies the Stanley–Wilf conjecture, in: D. Krob, A.A. Mikhalev, A.V. Mikhalev (Eds.), Formal Power Series and Algebraic Combinatorics, Springer, Berlin, 2000, pp. 250–255] and Marcus and Tardos [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley–Wilf conjecture, J. Combin. Theory Ser. A 107 (2004) 153–160]. Our main results follow from a generalization to ordered hypergraphs of the theorem of Marcus and Tardos.