Noetherian Equations for Matrices

Let L|K be a finite Galois extension. Using central simple algebras we deal with the crossed representations of G = Gal(L|K) over L which are defined as mappings X of G into GL_n(L) satisfying X = X X. The last equation is the Noetherian equation in case n=1. Furtheron, more general crossed projective representations are considered which obey an equation X X = Xf_, where f_, L.     

On Generalized (σ,τ)-Derivations

We generalize the notion of (,)-derivation of Nakajima and Bresar. We define the generalized (,)-derivations, generalized Jordan (,)-derivations, and generalized Lie (,)-derivations, We study interrelations between these classes of derivations as well as their homological properties.     

Self-intersections of 1-dimensional random walks

Summary Consider a random walk S _n on the integers, where the steps _i have mean 0 and variance ². Let T be the time of first self-intersection of the random walk. It is shown that, as , T grows at rate ^2/3. More precisely, T^2/3 has a non-degenerate limit distribution which can be described in terms of Brownian motion local time.Research supported by National Science Foundation Grant MCS80-02698.     

A cuspidal edge of first kind with a point of asymptotic type

We shall say that an analytic surface supports a line with a singularity of order 2 if is a singular line of the surface (the first fundamental form of anyC -parametrization of is degenerate on ) and there exists aC -parametrization(s, v) of the surface such that the second derivative of in the direction orthogonal to is nonzero. We shall assume that has no singular points or points of straightening or flattening, that the Gaussian curvature of is nonpositive, and that the Gaussian curvature of has a finite limita(s) on (s is arc length on ), and the osculating plane of is tangent to in the sense that this plane contains as a half-plane the contingency of at the corresponding point of . Finally supposeb(0)=0,b(s)0 fors0, andb _ss (0)0, whereb(s)=a(s)+ ²(s) and is the torsion of . It is shown that under these hypotheses envelopes the asymptotes of one of the families of. All the asymptotes of the other family are tangent to except the asymptote passing through the points=0 of the singular line.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 66–76.     

Messbare Mengen von Massen und Inhalten

Let X be a completely regular space. The customary -field is the coarsest -field on the space of Bairemeasures on X which makes (A) measurable for any Baire set A. We compare the customary -field with the Baire and Borel -field induced by the weak* topology which lies on the dual space C(X). In (2.3) it is shown that the customary -field is just the Baire -field. In part 3 necessary and sufficient conditions are given under which the set of -smooth measures is measurable with respect to the Borel -field which lies on the positive cone of the space of finitely additive, regular measures C(X). Finally, a decomposition theorem for generalized kernels is proved. The -smooth part of a generalized kernel is a kernel again if certain conditions are fulfilled.     

Weak convergence of probability measures relative to incompatible topology and σ-field with applications to renewal theory

Summary Necessary and sufficient conditions are given in order that a sequence of probability measures, weakly convergent relative to a given topology ₀ and associated -field ( ₀), are weakly convergent (and satisfy a continuity theorem) relative to the ( ₀)-measurable functions which are continuous in some finer topology ₁, even if does not extend to ( ₀). These conditions are shown to be applicable to a sequence of translated renewal measures. Alternate conditions (tightness, uniformity of weak convergence) are investigated and shown to be inappropriate.This research was partially supported by UMC Summer Faculty Research Fellowships     

Diffusions réfléchies réversibles dégénérées

On a domain D in ^d, for a smooth enough probability density and a diffusion matrix which can degenerate, we construct the law Q ^s of a (x)d -symmetric reflecting process in D with matrix . Therefore, we use the associated Dirichlet form and a sequence of approximating processes already used by Pardoux and R. Williams in [23]. Under mild conditions on the boundary ofD (finite Minkowski content), we prove that Q ^s is the law of a semi-martingale and provide its decomposition. Comparing with the decomposition in additive functionals, we conclude that the process is reflected in the conormal direction ^* n where n denotes Chen's normal (cf [10]), that is, the reflection direction of the Brownian motion in Kuramochi compactification.     

A note on the position of the centre of flexure of a beam of isotropic material having a section bounded by a parabola and a straight line

Zusammenfassung Die vorliegende Notiz behandelt das Problem der Auffindung des Biegungszentrums eines aus isotropem Material bestehenden Balkens, der einen durch eine parabolische und eine geradlinige Kante begrenzten Querschnitt besitzt. Die Last ist parallel zur geradlinigen Kante angenommen. Der Verfasser benützt zur Lösung der Aufgabe dasselbe Verfahren, dasSokolnikoff [1] für die Auffindung des Biegungszentrums eines halbkreisförmigen Balkens verwendete.

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Nomenclature x, y, z rectangular co-ordinates - l, m, n direction cosines of outward drawn normal - _x , _y , _z normal components of stress parallel tox, y, z axes - _xy , _yz , _zx shearing stress components in rectangular co-ordinates - u, v, w components of displacement - T (x, y) stress function in rectangular coordinates - Poisson's ratio - shear modulus - _yz , _zx , _xy shearing strain components in rectangular coordinates     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

A Note on Base Change,Identities Involving τ(n), and a Congruence of Ramanujan

We use the theory of quadratic base change to derive some new identities involving the Ramanujan -function, and show how the Ramanujan congruence (n) ₁₁(n) (mod 691) follows.     

Transforming binary sequences using priority queues

A priority queue transforms an input sequence into an output sequence which is a re-ordering of the sequence . The setR of all such related pairs is studied in the case that is a binary sequence. It is proved thatR is a partial order and that ¦R¦=c _n+1, the (n+1)th Catalan number. An efficient (O(n ²)) algorithm is given for computing the number of outputs achievable from a given input.     

Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net of ideals of R such that, where E() is the subgroup generated by all transvections of the net subgroup G(). and is the normalizer of G(). The subgroup E() is normal in. To study the factor group we introduce an intermediate subgroup F(), E() F() G(). The group is finite and is connected with permutations in the symmetric group. The factor group G()/F() is Abelian — these are the values of a certain determinant. In the calculation of F()/E() appears the SK₁-functor. Results are stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 13–20, 1979.     

A skew-product which is Bernoulli

We show that if is the shift on sequences of {0,1} and is the entropy zero transformation used by Ornstein in constructing a counter-example toPinsker's conjecture, then the skew-product transformationT defined byT(x,y)=(x, ^x0 y) is Bernoulli. ThisT is conditionally mixing with respect to the independent generator for , a partition with full entropy.This research was done while the first author was a visitor at Stanford, supported in part by NSF Grant MP-575-08324.     

Flag-Transitive 2-Designs Arising from Line-Spreads in PG(2n-1,2)

A Singer cycle in GL(n,q) is an element of order q permuting cyclically all the nonzero vectors. Let be a Singer cycle in GL(2n,2). In this note we shall count the number of lines in PG (2n-1,2) whose orbit under the subgroup of index 3 in the Singer group is a spread. The lines constituting such a spread are permuted cyclically by the group ³, hence gives rise to a flag-transitive 2-(2²ⁿ ,4,1) design.     

Harmonic functions on nilpotent groups

For a probability measure on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL (G) is called -harmonic if F(st)d(t)=F(s), for almost alls inG. Constant functions are -harmonic and it is known that for abelianG all -harmonic functions are constant. For other groups it is known that non constant -harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant -harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.     

Convergence of automorphisms of compact projective planes

We show that a pointwise convergent sequence ( _n) _nN of continuous collineations of a compact projective plane converges uniformly if and only if the pointwise limit of ( _n) _nN has a quadrangle in its image. Moreover is then a continuous collineation. Furthermore, we derive that a homomorphism between topological projective planes is continuous if and only if it is continuous at some point.Supported by DFG/Graduiertenkolleg.     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

PAT – a Reliable Path-Following Algorithm

This paper presents a new technique for the reliable computation of the -pseudospectrum defined by (A)={zC : _min(A–zI)} where _min is the smallest singular value. The proposed algorithm builds an orbit of adjacent equilateral triangles to capture the level curve (A)={zC : _min(A–zI)=} and uses a bisection procedure on specific triangle vertices to compute a numerical approximation to . The method is guaranteed to terminate, even in the presence of round-off errors.     

Comparison of the activation energies of the processes of thermal degradation and mechanical fracture of polymethyl methacrylate

On the basis of a comparison of the activation energies of the processes of thermal degradation and mechanical fracture of PMM with various additives it is concluded that in the presence of a "pole shift effect" it is more correct to calculate the activation energy U₂ of the mechanical fracture process on the assumption that ₀ = const and =f(, T). The energy U₀₂ is then correlated with the weak-bond rupture energy. In the general case, the activation energy U₀₁, calculated on the assumption that ₀=f(, T) const, i. e., from the slopes of the lg -1/T graphs, cannot be identified with the interatomic-bond rupture energy.Mekhanika Polimerov, Vol. 4, No. 3, pp. 467–473, 1968