The total mean curvature functional for submanifolds into the Riemannian product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first variational formula is presented. Later on, two second-order differential operators are defined and a nice integral inequality relating both of them is proved. Finally, we prove our main result: an integral inequality for closed stationary $\mathcal{H}$-surfaces in $\mathbb{S}^n\times\mathbb{R}$, characterizing the cases where the equality is attained.

The equation of the propagation of electric signals along cables, generally known as the equation of telegraphy, may be written

Particular solutions of this equation, adapted to various purposes have been found by Heaviside, Poincaré, A. G. Webster, T. W. Chaundy, § and others. The object of the present paper is to unify the theory of the equation by exhibiting the relations which these solutions bear to each other, and by obtaining them as particular cases of a general solution. The derivation of new particular solutions by the solution of integral equations is also discussed.

We primarily consider here the $L^2$ mapping properties of a class of strongly singular Radon transforms on the Heisenberg group $\mathbb{H}^n$; these are convolution operators on $\mathbb{H}^n$ with kernels of the form $M(z,t)=K(z)\delta_0(t)$, where $K$ is a strongly singular kernel on $\mathbb{C}^n$. Our results are obtained by using the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.

We also discuss the behaviour of related twisted strongly singular operators on $L^2(\mathbb{C}^n)$ and obtain results in this context independently of group Fourier transform methods. Key to this argument is a generalization of the results for classical strongly singular integrals on $L^2(\mathbb{R}^d)$.

When an elastic solid is subjected to a dynamical system of surface or body forces, not all of the work done by these forces is employed in deforming the material. The remainder is converted into heat energy producing a distribution of temperature throughout the body. Similarly the application of a surface temperature distribution, or the introduction of heat sources within the body, produces elastic as well as thermal effects. Thus we see that in the dynamical case there is a link between these two types of condition—thermal and elastic.

A classical result of Baer states that a finite group G which is the product of two normal supersoluble subgroups is supersoluble if and only if Gʹ is nilpotent. In this article, we show that if G = AB is the product of supersoluble (respectively, w-supersoluble) subgroups A and B, A is normal in G and B permutes with every maximal subgroup of each Sylow subgroup of A, then G is supersoluble (respectively, w-supersoluble), provided that Gʹ is nilpotent. We also investigate products of subgroups defined above when $ A\cap B=1 $ and obtain more general results.

Zubov states an elegant necessary and sufficient limit set condition for positive orbital stability of compact invariant sets in his book “Metody A. M. Lyapunova i ih Primenenie” [11]. Stated in terms of our terminology of L– for the negative limit set, Zubov's proposition is as follows: A necessary and sufficient condition for positive stability of a compact invariant set M isL–(X\M)∩M=Ø. Unfortunately, Zubov's condition L–(X\M)∩M=Ø has subsequently been shown to be necessary but not sufficient (see [9]). Bass and Ura devote considerable effort in [2] and [9[ to correcting Zubov's proposition and Desbrow obtains additional results principally concerning unstable sets in [6] and [7]. Ura gives his classical corrected prolongational version of Zubov's assertion on locally compact phase spaces in [9] and extends it to any closed invariant set with compact boundary on such spaces in [10].

A. G. D. Watson (1939-41), remarking that there are no Ricci principal directions ata world-point of space-time at which the Einstein equations are satisfied, shows how to define at any world-point a set of principal directions intrinsically related to the Riemann tensor Rijkl itself. These directions are unique except when the space-time has any kind of rotational symmetry about the world-point.

The first part of the following investigation was begun before the discovery that Mr E. Kasner had already touched upon the apolarity theory of double binary forms in an important work on the Inversion Group (Transactions of the American Mathematical Society, Vol. I (1900), pp. 471–473). The theory is carried further in what follows, with special reference to the (2, 2) form. The second part answers questions raised by Professor A. R. Forsyth in the Quarterly Journal, 1910, p. 113. It appears that the general (2, 2) form admits of three independent automorphic transformations, but the general (n, n) form admits of none, if n exceeds two.

We prove that Finsler metrics on Euclidean domains can be approximated in a certain sense by so-called Finsler-type metrics. As an application we improve upon previous estimates on the fundamental solution of higher order parabolic equations.

Let G be a group of odd order that contains a non-central element x whose order is either a prime p ≥ 5 or 3l, with l ≥ 2. Then, in , the group of units of ℤG, we can find an alternating unit u based on x, and another unit v, which can be either a bicyclic or an alternating unit, such that for all sufficiently large integers m we have that 〈um, vm〉 = 〈um〉 ∗ 〈vm〉 ≌ ℤ ∗ ℤ

In the works known to us which contain historical or bibliographical information about circular cubics no sufficient indication of the researches of Maclaurin is found. Yet many of the classic propositions connected with these curves are due to this eminent geometer, who investigated constructions for unicursal circular cubics, for certain non-unicursal circular cubics, and for the special circular cubics now known as the trisectrix of Maclaurin, the oblique cissoid, and the strophoid. His name does not appear even in the list of writings on the strophoid published by Tortolini and Günther.

We prove that if f is an integrable function on the unit sphere S in ℝn, g is its symmetric decreasing rearrangement and u, v are the harmonic extensions of f, g in the unit ball , then v has larger convex integral means over each sphere rS, 0 < r < 1, than u has. We also prove that if u is harmonic in with |u| < 1 and u(0) = 0, then the convex integral mean of u on each sphere rS is dominated by that of U, which is the harmonic function with boundary values 1 on the right hemisphere and −1 on the left one.

For any nonzero invariant subspace M in H2(T2), set . Then Mx is also an invariant subspace of H2(T2) that contains M. If M is of finite codimension in H2(T2) then Mx = H2(T2) and if M = qH2(T2) for some inner function q then Mx = M. In this paper invariant subspaces with Mx = M are studied. If M = q1H2(T2) ∩ q2H2(T2) and q1, q2 are inner functions then Mx = M. However in general this invariant subspace may not be of the form: qH2(T2) for some inner function q. Put (M) = {ø ∈ L ∞: ø M ⊆ H2(T2)}; then (M) is described and (M) = (Mx) is shown. This is the set of all multipliers of M in the title. A necessary and sufficient condition for (M) = H∞(T2) is given. It is noted that the kernel of a Hankel operator is an invariant subspace M with Mx = M. The argument applies to the polydisc case.

We define a hyperplane group to be a finite group generated by reflections fixing a single hyperplane pointwise. Landweber and Stong proved that the invariant ring of a hyperplane group is again a polynomial ring in any characteristic. Recently, Hartmann and Shepler gave a constructive proof of this result. By their algorithm, one can always construct generators that are additive. In this paper, we study hyperplane groups of order a power of a prime p in characteristic p and give a slightly different construction of the generators than Hartmann and Shepler. We then show that such generators have a particular form. Furthermore, we show that if the group is defined by a finite additive subgroup W ⊆ , the vanishing ideal of W is generated by polynomials obtained from a set of generators of the invariant ring that are additive. Finally, we give a shorter proof of the fact that the module of the invariant differential 1-forms is free in our situation.

Bonsall and Tomiuk have shown, in (3), the connection between the local compactness of a monothetic semi-algebra and the spectral properties of a generating element. This theme was developed, in (4), to give a complete characterisation of prime, strict locally compact monothetic semi-algebras in terms of the spectrum of a generator (Theorem A). Here we extend this result to the case of a semi-simple locally compact monothetic semi-algebra (Theorem B).

In [3], [8], and [2], it was shown that if is an essentially Hermitian operator on l P, 1≦ p<∞, or on Lp[0,1], 1< p<∞, then T is a compact perturbation of a Hermitian operator. In [1], this result was established for operators on Orlicz sequence space l M, where 2∉[α M,β M] (the associated interval for M). In that same paper, it was conjectured that this result does not in general hold if 2∈[α M,β M]. In this paper, we show that this conjecture is correct by exhibiting an Orlicz sequence space l M and an essentially Hermitian operator on l M which is not a compact perturbation of a Hermitian operator.

The number of self-complementary (s.c.) graphs and digraphs with a given number of vertices was found by R. C. Read in [1]. That paper used a special case of De Bruijn's generalisation of Polya's theorem that involved the cycle-index of Gn, the group of permutations of pairs of vertices induced by permutations of the vertices. We obtain Read's formulae by using only well-known elementary facts about s.c. graphs and their complementing permutations.