The following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

The theory of Yang-Mills fields is explicity formulated in terms of the theory of connections in principal and associated fiber bundles. Special attention is paid to the fiber bundles with Lorentz and Poincaré structures. Equations of the form ∇ψ = 0, where ∇ is a generalized convariant derivative, are shown to contain “mass” terms if the connection in the Poincaré fiber bundle is cononically associated to the connection in the Lorentz fiber bundle.

Ramanujan was the first mathematician to discover some of the arithmetical properties of p(n), the number of unrestricted partitions of n. His congruence,

for example, is famous [2; 3]. Some progress has been made since then; it is known that the congruence,

has an infinitude of solutions for any arbitrary value of r [4]. This is a somewhat weak relation, and one would have liked to obtain, if possible, stronger results of the type,

for ‘almost all’ values of n, which in its turn is derivable from another stronger relation, viz.,

also established by Ramanujan [2], where r(n) is Ramanujan's function defined by

We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely divisible distribution and its Lévy measure. Moreover, we study the second order asymptotic behaviour of the tail of the $t$th convolution power of an infinitely divisible distribution. The density version for a self-decomposable distribution on the real line without an exponential moment assumption is also given. Finally, the regularly varying case for a self-decomposable distribution on the half line is discussed.

In the present paper we obtain first approximation formulae for the regions of dynamical instability of linear canonical systems. These formulae are analogous to the formulae for Hamiltonian systems stated by Krein and Jakubovič [5] and proved by Pittel' and Juzefovič [8]. Special cases were considered by Malkin [7] and Jakubovič [3]. Related papers are Hale [2] and Jakubovič [4]. However, our method differs from the methods used by these authors and seems to us to be both simpler and more general.

This paper investigates complete space-like submainfold with parallel mean curvature vector in the de Sitter space. Some pinching theorems on square of the norm of the second fundamental form are given

We show that an elation generalized quadrangle that has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (that is, is a flock quadrangle).

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

If velocity and temperature profiles are known at a particular distance along a vertical heated plate, the equations of motion determine the velocity and temperature at points downstream, for a given varition of plate temperature. The problem of continuing the boundary layer solution for given initial conditions was investigated by Goldstein [2], for the isothermal case of the laminar, incompressible flow past plate, with a given streamwise varitain of pressure gradient outside the boundary layer. He showed that the solution is not always free from singularities and developed an expansion procedure to calculate the flow downstream when these occurred. Typical singularities occur, for instance, near the leading edge of the plate where the no-slip condition is imposed on the plate surface and near the trailing edge, where this condition is relaxed to one of zero stress along the axis of symmetry of the wake.

Consider a random Markovian process in which the state of the system is defined by a random variable which can take the finite set of values i=0, 1, …, N, and which is such that transition can only occur from any state i to the two nearest states i+1. This restriction brings about an essential simplification of the theory for the basic reason that in order for the system to move from i to state j (i < j say) it must first move to i+1, then i+2 and so on until it reaches j. From this it follows that the first passage distribution from i to j is the convolution of the first passage distributions from i to i+l, i+l to i+2,…, j—l to j each of which is comparatively easy to find.

There have been investigations (Janko (1966), Janko and Thompson (1966), Yamaki (1972)) of finite groups G which contain a central involution t whose centralizer (in G) has the form C(t) = 〈t〉 × F, where F is isomorphic to a non-abelian simple group. Here it is shown such a group cannot be simple when F is isomorphic to an exceptional Lie type simple group of odd characteristic. Specifically the following theorem is proved.

This paper discusses a flaw in Murasugi–Przytycki’s Memoir ‘An index of a graph with applications to knot theory’ [Mem. Amer. Math. Soc. 106 (1993)]. We point out and partly fix a gap occurring in the proof of Murasugi–Przytycki’s braid index inequalities involving the graph index. We explain why their notion of index fails to precisely reflect the reduction of Seifert circles by their diagram move, and redefine the index to account for that discrepancy.

This paper presents new proofs of some classical transcendence theorems. We use real variable methods, and hence obtain only the real variable versions of the theorems we consider: the Hermite-Lindemann theorem, the Gelfond-Schneider theorem, and the Six Exponentials theorem. We do not appeal to the Siegel lemma to build auxiliary functions. Instead, the proof employs certain natural determinants formed by evaluating n functions at n points (alternants), and two mean value theorems for alternants. The first, due to Pólya, gives sufficient conditions for an alternant to be non-vanishing. The second, due to H. A. Schwarz, provides an upper bound.

Chains of projectivities within the lattice (G) of subnormal subgroups of group G have been considered by various authors, see for example Barnes [1] and Tamaschke [2].

This paper is devoted to a study of pseudocomplements in groupoids. A characterization of an intraregular groupoid is obtained in terms of prime ideals. It is proved that the set of dense elements of an intraregular groupoid S with 0 is the intersection of all the maximal filters of S and that the set of normal elements of an intraregular groupoid closed for pseudocomplements forms a Boolean algebra under natural operations. It is shown that the pseudocomplement of an ideal of an intraregular groupoid with 0 is the intersection of all the minimal prime ideas not containing it.

We consider the case of a genetic population for which the selective advantages of the various genotypes are not constant but for each generation depend linearly on the gene frequencies in the population in the previous generation. For such populations, the effect of competition between similar genotypes may be allowed for by suitable choice of the frequency-dependent selective advantages, or, by a reversal of sign, the case where genotypes are favoured by the presence of similar genotypes may also be considered. favoured by the presence of similar genotypes may also be considered. All populations are finite and of constant size so that eventually only one type of gene will survive. The probabilities of survival for each gene are found and compared with the case where there are no frequency-dependent factors. If a small amount of mutation is allowed, gene fixation will not occur and a steady-state distribution of gene frequency will appear. The form of this distribution may be derived simply from the survival probabilities in the corresponding cases where there is no mutation. The main result is that in some cases, frequency-dependent factors have a marked effect on survival probabilities, while in other cases they can be completely ignored. The latter wifi only occur in certain cases where there exists competition between similar genotypes.

We define the notion of a best Diophantine approximation vector to a set of linear forms. This generalizes definitions of a best approximation vector to a single linear form and of a best simultaneous Diophantine approximation vector. We derive necessary and sufficient conditions for the existence of an infinite set of best Diophantine approximation vectors. Finally, we prove that such approximation vectors are spaced far apart in an appropriate sense.

We provide character-free proofs of some results on idempotents in p-adic group rings, centering around Brauer's Second Main Theorem on Blocks.

The trace T of the metrical energy-momentum tensor Tkl of fields associated with particles of zero rest mass may be zero, either identically or as a consequence of the field equations. This property of Tkl is correlated here with the behaviour of the Lagrangian of the field under arbitrary conformal transformations. Certain classes of special fields are considered explicitly. It is shown in particular that T vanishes for all non-zero spin fields which correspond respectively to the two-component neutrino field or the photon field.