This paper presents a construction of a digital convex 2k–gon (for a given natural number k), which can be inscribed into a square grid of the minimal possible size F(2k). An explicit expression for the function F(2k) is also given.

Let $\ell >0$ be arbitrary. We introduce the extremal quantities

$$\begin{eqnarray}G(\ell ):=\sup _{f}\int _{-\ell }^{\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\quad C(\ell ):=\sup _{f}\sup _{a\in \mathbb{R}}\int _{a-\ell }^{a+\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\end{eqnarray}$$

$\ell =2$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}G(k+\unicode[STIX]{x1D700})$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}C(k+\unicode[STIX]{x1D700})$

$(k\in \mathbb{N})$

$\ell$

In this paper the diffraction of a planetary wave by a semi-infinite plate of arbitrary inclination is investigated using a β-plane approximation. The Wiener-Hopf technique is used to obtain an integral representation of the solution and an asymptotic description of the diffracted wave is obtained by the method of steepest descent.

Let B be the class of functions ω(z) regular in |z| < 1 and satisfying ω(0) = 0, |ω(z)|<1 in |z|<1. We denote by P(A, B), −1 ≤ B < A ≤1, the class of functions p(z) = l+p1z+… regular in |z| < 1 and such that p(z) = [1+Aω(z)]/[1+Bω(z)] for some ω(z) ∈ Β. This paper establishes sharp lower and upper bounds on |z| = r<1 for the functional

where p(z) varies in P(A, B). The results are then used to study certain geometric properties of the corresponding class of meromorphic starlike univalent functions

A natural number $n$ is called multiperfect or $k$-perfect for integer $k\ge 2$ if $\sigma (n)=kn$, where $\sigma (n)$ is the sum of the positive divisors of $n$. In this paper, we establish a theorem on odd multiperfect numbers analogous to Euler’s theorem on odd perfect numbers. We describe the divisibility of the Euler part of odd multiperfect numbers and characterise the forms of odd perfect numbers $n=\pi ^\alpha M^2$ such that $\pi \equiv \alpha ~({\rm mod}~8)$, where $\pi ^\alpha $ is the Euler factor of $n$. We also present some examples to show the nonexistence of odd perfect numbers of certain forms.

The propogation of disturbance when a shear flow with a free surface, in a channel of infinite horizontal extent and finite depth, is disturbed by the application of time-oscillatory pressure, is studied. The initial value problem is solved by using transform techniques and the steady state solution is obtained therefrom in the limit t → ∞. The effect of the initial shear on the development of the wave system is investigated.

Let A = {a1, a2,…}(a1 < a2 < …) be an infinite sequence of positive integers. Let k ≥ 2 be a fixed integer and denote by Rk(n) the number of solutions of . Erdős, Sárközy and Sós studied the boundness of |R2(n + 1) − R2(n)| and the monotonicity property of R2(n). In this paper, we extend some results to k > 2.

The main object of this paper is to extend the theorem of W.H. Young on convergence of the conjugate Fourier series to Wiener's class. We prove that the conjugate Fourier series of a function of Wiener's class converges to for every x for which exists.

Let $p(z)= z{f}^{\prime } (z)/ f(z)$ for a function $f(z)$ analytic on the unit disc $\mid z\mid \lt 1$ in the complex plane and normalised by $f(0)= 0, {f}^{\prime } (0)= 1$. We provide lower and upper bounds for the best constants ${\delta }_{0} $ and ${\delta }_{1} $ such that the conditions ${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\mid \lt {e}^{{\delta }_{0} / 2} $ and $\mid p(w)/ p(z)\mid \lt {e}^{{\delta }_{1} } $ for $\mid z\mid , \mid w\mid \lt 1$ respectively imply univalence of $f$ on the unit disc.

Let G be a graph, and k a positive integer. Let h:E(G)→[0,1] be a function. If ∑ e∋xh(e)=k holds for each x∈V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh={e∈E(G)∣h(e)>0}. In this paper we use neighbourhoods to obtain a new sufficient condition for a graph to have a fractional k-factor. Furthermore, this result is shown to be best possible in some sense.

We detail a construction of totally symmetric surfaces of constant mean curvature 0≤H<1 in hyperbolic 3-space of sectional curvature −1 via the generalized Weierstrass type representation.

Motivated by constructions of Witt rings in the algebraic theory of quadratic forms, the authors construct new classes of finite commutative rings and explore some of their properties. These rings are constructed as quotient rings of a special class of integral group rings for which the group is an elementary 2-group. The new constructions are compared to other rings in the literature.

Let a group G be covered by finitely many disjoint cosets of subgroups Gi. We study conditions which imply that the subgroups Gi are conjugate in G.

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

In this paper, we establish a convergence theorem for fixed points of generalised weak contractions in complete metric spaces under some new control conditions on the functions. An illustrative example of a generalised weak contraction is discussed to show how the new conditions extend known results.

An example is found of a nonreflexive Banach space X such that the union of {0} and the set of non-norm-attaining functionals on X contains no two-dimensional subspace.

In this work we are concerned with the problem of the existence of an exponential dichotomy for the linear singularly perturbed system εx′ = A(t)x, where the matrix A(t) is piecewise uniformly continuous, that is, A(t) admits points of discontinuity but is uniformly continuous in any interval where it is continuous. We shall prove that the classical result regarding the existence of an exponential dichotomy extends to this case, when there is a constant γ > 0 such that |Reλ(t)| ≥ γ > 0 for any eigenvalue λ(t) of A(t). The proofs are obtained by means of the quasidiagonalisation of a non-constant matrix: For A(t), a piecewise uniformly continuous matrix and σ > 0 there exists a bounded, piecewise constant function L(t): J → ℂn×n, and a bounded matrix Δ(t, σ) such that L-1(t)A(t)L(t) = Λ(t) + Δ(t, σ), |Δ(t, σ)| ≤ σ, where Λ(t) is the diagonal matrix consisting of eigenvalues of A(t).

We show that the ring of locally bounded Nash meromorphic functions on a connected d-dimensional Nash submanifold of ℝn is a Prüfer domain and every finitely generated ideal in this ring can be generated by d + 1 elements.

Moreover, every finitely generated ideal can be generated by d elements if and only if the Nash manifold is noncompact.

Let G be a graph of order n, and let k≥1 be an integer. Let h:E(G)→[0,1] be a function. If ∑ e∋xh(e)=k holds for any x∈V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh ={e∈E(G):h(e)>0}. A graph G is called a fractional (k,m) -deleted graph if for every e∈E(H) , there exists a fractional k-factor G[Fh ] of G with indicator function h such that h(e)=0 , where H is any subgraph of G with m edges. The minimum degree of a vertex in G is denoted by δ(G) . For X⊆V (G), NG(X)=⋃ x∈XNG(x) . The binding number of G is defined by In this paper, it is proved that if then G is a fractional (k,m) -deleted graph. Furthermore, it is shown that this result is best possible in some sense.