Volume 12, Issue 2

Let 𝐶 ⊂ 𝐏 𝑛 be a smooth projectively normal curve. Let 𝑍 be the scheme-theoretic base locus of 𝐻 0 (𝐏 𝑛 , 𝐼 𝐶 (2)) and 𝑍′ the connected component of 𝑍 containing 𝐶. Here we show that 𝑍′ = 𝐶 in certain cases (e.g., non-special line bundles with degree near to 2 𝑝 𝑎 (𝐶)–2 or certain special line bundles on general 𝑘-gonal curves).

We show that Cauchy's, d'Alembert's functional equations and their generalizations are the functional equations for bounded spherical functions associated to some Gel'fand pairs. Our results are very close to the ones obtained by Stetkær in [Aequationes Math. 48: 220–227, 1994].

We study some direct results for the recently introduced family of modified Baskakov type operators. In particular, we obtain local direct results on ordinary and simultaneous approximation and an estimation of error for linear combinations in terms of higher order modulus of continuity. We have applied the Steklov mean as a tool for the linear approximating method.

In this paper, theorems on the Fredholm alternative and wellposedness of the linear boundary value problem 𝑢′(𝑡) = ℓ(𝑢)(𝑡) + 𝑞(𝑡), ℎ(𝑢) = 𝑐, where ℓ : 𝐶([𝑎, 𝑏]; 𝑅 𝑛 ) → 𝐿([𝑎, 𝑏]; 𝑅 𝑛 ) and ℎ : 𝐶([𝑎, 𝑏]; 𝑅 𝑛 ) → 𝑅 𝑛 are linear bounded operators, 𝑞 ∈ 𝐿([𝑎, 𝑏]; 𝑅 𝑛 ), and 𝑐 ∈ 𝑅 𝑛 , are established even when ℓ is not a strongly bounded operator.

We prove that a function of several variables satisfying a functional differential inequality with unbounded delay can be estimated by a solution of a suitable initial problem for an ordinary functional differential equation. As a consequence of the comparison theorem we obtain a Perron-type uniqueness result and a result on continuous dependence of solutions on given functions for partial functional differential equations with unbounded delay. We consider classical solutions on the Haar pyramid.

Absolutely negligible sets in uncountable groups are considered in connection with the measure extension problem (for σ -finite invariant or quasi-invariant measures). In particular, it is proved that, for any uncountable solvable group (𝐺, ·), there exists a countable covering of 𝐺 consisting of 𝐺-absolutely negligible sets.

The present study is concerned with the plane strain problem in homogeneous orthotropic micropolar viscoelastic solid. The disturbance due to the time harmonic concentrated source is investigated by employing the eigenvalue approach. Integral transforms have been inverted by a numerical technique to obtain the components of displacement, force stress and couple stress in the physical domain. The results obtained are given and illustrated graphically.

In the present paper, we define and study the slant, bi-slant and semi-slant submanifolds of a locally product manifold. We give some characterization theorems for slant submanifolds and semi-slant submanifolds. Moreover, we obtain integrability conditions for the distributions which are involved in the definition of semi-slant submanifolds. We also get some results about mixed totally geodesic submanifolds.

For solutions of capillarity problems with the boundary contact angle being bounded away from 0 and π and the mean curvature being bounded from above and below, we show the Lipschitz continuity of a solution up to the boundary locally in any neighborhood in which the solution is bounded and ∂ Ω is 𝐶 2 ; the Lipschitz norm is determined completely by the upper bound of | cos θ |, together with the lower and upper bounds of 𝐻, the upper bound of the absolute value of the principal curvatures of ∂ Ω and the dimension 𝑛.

In this paper, we prove the boundedness for some multilinear operators generated by singular integral operators and Lipschitz functions on some Hardy and Herz type spaces.

Let 𝑋 be a non-metric continuum, and 𝐶(𝑋) be the hyperspace of subcontinua of 𝑋. It is known that there is no Whitney map on the hyperspace 2 𝑋 for non-metric Hausdorff compact spaces 𝑋. On the other hand, there exist non-metric continua which admit and ones which do not admit a Whitney map for 𝐶(𝑋). In particular, a locally connected or a rimmetrizable continuum 𝑋 admits a Whitney map for 𝐶(𝑋) if and only if it is metrizable. In this paper we investigate the properties of continua 𝑋 which admit a Whitney map for 𝐶(𝑋) or for 𝐶 2 (𝑋).

We prove that the function 𝑚( ξ ) = ψ ( ξ ) 𝑒 𝑖 ϕ ( ξ ) is not the Fourier multiplier of the space 𝐿 𝑝 , where the real phase ϕ has the property ϕ ″ ≥ 𝑐 > 0, the amplitude ψ vanishes near the origin, ψ ( ξ ) = 𝑂(| ξ | –2 ) as ξ → ∞, and ψ ′ ∈ 𝐿 1 .

Three-dimensional fluid-solid interaction problems with regard for thermal stresses are considered. An elastic structure is assumed to be a bounded homogeneous isotropic body occupying a domain , where the thermoelastic four dimensional field is defined, while in the unbounded exterior domain there is defined the scalar (acoustic pressure) field. These two fields satisfy the differential equations of steady state oscillations in the corresponding domains along with the transmission conditions of special type on the interface ∂ Ω ± . We show that uniqueness of solutions strongly depends on the geometry of the boundary ∂ Ω ± . In particular, we prove that for the corresponding homogeneous transmission problem for a ball there exist infinitely many exceptional values of the oscillation parameter (Jones eigenfrequencies). The corresponding eigenvectors (Jones modes) are written explicitly. On the other hand, we show that if the boundary surface ∂ Ω ± contains two flat, non-parallel sub-manifolds then there are no Jones eigenfrequencies for such domains.

Let 𝐸 be a set consisting of rectangular frames in . The following question connected with one problem of A. Zygmund is studied in the paper: Does there exists a function the integral of which is: 1) non-differentiable almost everywhere in a strong sense along every frame from 𝐸, and 2) strongly differentiable along every frame not belonging to 𝐸? In particular, the question is solved on the existence of a non-empty set 𝐸 different from the set of all rectangular frames for which there is a function with the properties 1) and 2).

Sufficient conditions in terms of coefficient functions have been obtained so that all nonoscillatory solutions along with their first and second derivatives of the third order delay differential equation 𝑦‴(𝑡) + 𝑎(𝑡)𝑦″(𝑡) + 𝑏(𝑡)𝑦′(𝑡) + 𝑐(𝑡)𝑦(𝑔(𝑡)) = 0 tend to zero as 𝑡 → ∞.

We study the boundedness and compactness of a special class of integral operators defined on generalized Sobolev spaces.