Automorphic functions and the Bass-Milnor-Serre homomorphism,I

Let v=[exp2i/3], let q=(3) be an ideal in v and let SL_m(v,q) be the congruence subgroup modq in SL_m(v). For the solution of the congruence subgroup problem, Bass, Milnor, and Serre have constructed (making use of the properties of the symbol of cubic residue) a homomorphism :SL_m(v,q)^*. We consider here as a system of multipliers. The fundamental object of investigation is the Eisenstein series on SL₃()/Su(3) automorphic relative to SL₃(v,q) with the system of multipliers . For this Eisenstein series one has computed certain coefficients of the expansion with respect to the basis indicated in the author's paper: The expansion of automorphic functions on SL₃()/Su(3), Zap. Nauchn. Sem. LOMI, Vol. 125, pp. 144–153, 1983.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 85–126, 1983.     

Critical star multigraphs

A star-multigraphG is a multigraph in which there is a vertexv ⁺ which is incident with each non-simple edge. It is critical if it is connected, Class 2 and(G\e) < (G) for eache E(G). We show that, ifG is any star multigraph, then(G) (G) + 1. We investigate the edge-chromatic class of star multigraphs with at most two vertices of maximum degree. We also obtain a number of results on critical star multigraphs. We shall make use of these results in later papers.     

On the dispersion spectrum

A measure for the denseness of sequences (an) mod 1, irrational, is the dispersion constantD() introduced byH. Niederreiter. In this paper the smallest accumulation point ₁ of the set of theD() is determined and all those are explicitely given for whichD () < ₁ holds.     

Pseudo-mesures p- adiques associées aux fonctions L deQ

It is shown that some standard results concerning the p-adic L- functions, L_p(), ofQ(p-divisibilities of 1/2L_p(, s), and congruences for 1/2L_p(, t)–1/2L_p(, s), s, t_p) are direct consequences of a general structural theorem, based only on the functional properties of the p-adic pseudo-measures and distributions attached to these L_p-functions (essentially the eulerian ones). The method suggests that all such divisibilities and congruences are obtained systematically by this way, and are the best possible (in a standard point of view). In particular, these results improve significantly all the known ones.     

Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture

Letg be the Lie algebra of a connected reductive groupG over an algebraically closed field of characteristicp>0. Suppose thatG ⁽¹⁾ is simply connected andp is good for the root system ofG. Ifp=2, suppose in addition thatg admits a nondegenerateG-invariant trace form. LetV be an irreducible and faithfulg-module withp-character g ^*. It is proved in the paper that dimV is divisible byp ^1/2dim() where () stands for the orbit of under the coadjoint action ofG.Oblatum 14-III-1994 & 17-XI-1994     

Values of Dirichlet series associated with modular forms at the points S=1/2, 1

Let denote a parabolic form of even weights k which is an eigenfunction of all Hecke operators, and let be a real character. It is known that the Dirichlet series satisfies a functional equation of the Riemann type for. A number of asymptotic results on L_f(1/2,), L_f(1,) for d are proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 117–137, 1984.     

The Topology of the Coloring Complex

In a recent paper, E. Steingrímsson associated to each simple graph G a simplicial complex _G, referred to as the coloring complex of G. Certain nonfaces of _G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial _G of G, and the reduced Euler characteristic is, up to sign, equal to |_G(–1)|–1. We show that _G is constructible and hence Cohen-Macaulay. Moreover, we introduce two subcomplexes of the coloring complex, referred to as polar coloring complexes. The h-vectors of these complexes are again affine transformations of _G, and their Euler characteristics coincide with _G(0) and –_G(1), respectively. We show for a large class of graphs—including all connected graphs—that polar coloring complexes are constructible. Finally, the coloring complex and its polar subcomplexes being Cohen-Macaulay allows for topological interpretations of certain positivity results about the chromatic polynomial due to N. Linial and I. M. Gessel.Research financed by ECs IHRP Programme, within the Research Training Network Algebraic Combinatorics in Europe, grant HPRN-CT-2001-00272.     

Plane Partitions and Characters of the Symmetric Group

In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand equals the number of pairs of Littlewood-Richardson multitableaux of shape (, ), conjugate content and type . We also give lower and upper bounds for these numbers.     

Notes on Interpolation. X (Two Boundary Value Problems)

In this paper we use (0, 2) interpolational polynomials to give an approximate solution of the differential equation y(x) + A(x)y(x) = F(x), x I := [-1, 1] j in case when the boundary values are y(-1) = and y(1) = , , R.     

Series ∑F(m)qm,where F(m) is the number of odd classes of binary quadratic forms of determinant −m

Consideration of the analytic continuation of the Eisenstein series of weight 3/2 for the group ₀(4) leads to a new proof of Mordell's formula connecting the values X()= _m=1 F(m)e^im, Im > 0, and (–1/). The behavior of the function () for ₀(4) is examined by the same method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 69–79, 1976.     

Reguläre Graphen beliebigen Grades mit vorgegebenen Eigenschaften

Zusammenfassung der Sätze 2, 3, 11 und 16 das Hauptresultat dieser Arbeit formulieren:Satz 17. Wenn eine endliche Gruppe und zwei natürliche Zahlen n und so vorgegeben sind, daß 3n< und 2n ist, dann gibt es stets eine unendliche Menge endlicher und eine nicht abzählbare Menge unendlicher regulärer Graphen X vom Grade n(X)=n mit der Farbenzahl (X)= und einer AutomorphismengruppeG(X). Da die Graphen, von denen wir bei der Konstruktion ausgegangen sind, für frei von Fixpunkten und Fixkanten waren, ist im Lichte der verwendeten Substitutionstechnik leicht einzusehen, daß auch die neu erhaltenen Graphen diese Eigenschaften haben müssen. Denn zu jeder Kante eines neuen Graphen, die aus einer Kante des alten Graphen hervorgegangen ist, gibt es wegen der Fixkantenfreiheit des alten ein Element der Gruppe, das diese Kante verändert. Jene Knotenpunkte und Kanten aber, die aus den Knotenpunkten des alten Graphen hervorgehen, können nicht invariant sein, weil der alte Graph keine Fixpunkte hatte. Zum Schluß sei noch vermerkt, daß die konstruierten Graphen zusammenhängend sind und weder Einecke noch Zweiecke enthalten.     

Reconstruction of a wiener field from its realizations on a curve

Let w(x, y), x 0 and y 0 be a Wiener field on the plane; be a curve given parametrically x=x() and y=y(), [0, 1], where x() is a positive, continuous, nondecreasing function; y() is a positive, continuous, nonincreasing function. A best estimate in the mean-square sense is constructed for w(u, v)(u, v) , based on the values w(x, y), (x, y) and its error is found.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 87–93, 1988.     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

The addition formula for theta functions

Summary In this paper we solve the functional equationx(u + v)(u – v) = f ₁(u)g₁(v) + f₂(u)g₂(v) under the assumption thatx, , f ₁, f₂, g₁, g₂ are complex-valued functions onR ⁿ ,n N arbitrary, and 0 and 0 are continuous. Our main result shows that, apart from degeneracy and some obvious modifications, theta functions of one complex variable are the only continuous solutions of this functional equation.     

The crossing number of posets

The purpose of this paper is to investigate some properties of the crossing number (P) of a posetP. We first study the crossing numbers of the product and the lexicographical sum of posets. The results are similar to the dimensions of these posets. Then we consider the problem of what happens to the crossing number when a point is taken away from a poset. We show that ifP is a poset such that P and (P–)1, then 1/2 (P)(P–)(P). We don't know yet how to improve the lower bound. We also determine the crossing numbers of some subposets of the Boolean latticeB _n which consist of some specified ranks. Finally we show that _n is crossing critical where _n is the subposet ofB _n which is restricted to rank 1, rankn–1 and middle rank(s). Some open problems are raised at the end of this paper.     

On the discrete mean square of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions at 1

There have been many results obtained so far for the mean square of the (absolute) value of the Dirichlet L-function L(s,) in the critical strip 0<<1, especially on the critical line , but relatively few results were known for discrete mean value of |L(1,)|² till W. Zhang had published papers improving the error term step by step, which have recently been superseded by M. Katsurada and K.Matsumoto in which they succeeded in deriving an asymptotic formula for ₀|L(1,)|². The object of our paper is to point out a structural property contained in the formation of the mean square, to find out the niryana–the true body of the above sum.Dedicated to Professor Jean Louis Nicolás on his sixtieth birthdayin final form: 7 October 2003     

Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by ^*(G). Later, Zhu called it circular chromatic number and denoted it by _c(G). Let (G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then _c(G)=(G). Denote by M(G) the Mycielski graph of G. Recursively define M^m(G)=M(M^m–1(G)). It was conjectured that if mn–2, then _c(M^m(K_n))=(M^m(K_n)). Suppose that G is a graph on n vertices. We prove that if , then _c(M(G))=(M(G)). Let S be the set of vertices of degree n–1 in G. It is proved that if |S| 3, then _c(M(G))=(M(G)), and if |S| 5, then _c(M²(G))=(M²(G)), which implies the known results of Chang, Huang, and Zhu that if n3, _c(M(K_n))=(M(K_n)), and if n5, then _c(M²(K_n))=(M²(K_n)).* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.     

Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

---

Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.

     

On two mean value properties and functional equations associated with them

Summary Motivated by different mean value properties, the functional equationsf(x) – f(y)/x–y=[(x, y)], (i)xf(y) – yf(x)/x–y=[(x, y)] (ii) (x y) are completely solved when, are arithmetic, geometric or harmonic means andx, y elements of proper real intervals. In view of a duality between (i) and (ii), three of the results are consequences of other three.The equation (ii) is also solved when is a (strictly monotonic) quasiarithmetic mean while the real interval contains 0 and when is the arithmetic mean while the domain is a field of characteristic different from 2 and 3. (A result similar to the latter has been proved previously for (i).)     

Sûto's method applied to a problem of Färe and Mitchell

Summary A real solution of the functional equation(x + (y – x)) = f(x) + g(y) + h(x)k(y) on a set ² is a 6-tuple (f, g, h, k, , ) of real valued functions such that the equation is identically fulfilled on. Except for cases known before—e.g. when is linear—we present all real solutions in an arbitrary region where the functions have derivatives of second order.