Dimensional reduction for energies with linear growth involving the bending moment

A Γ-convergence analysis is used to perform a 3D–2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of functions with bounded variation.     

Counting alternating knots by genus

It is shown that the number of alternating knots of given genus g>1 grows as a polynomial of degree 6g−4 in the crossing number. The leading coefficient of the polynomial, which depends on the parity of the crossing number, is related to planar trivalent graphs with a Bieulerian path. The rate of growth of the number of such graphs is estimated.     

The non-linear Saint-Venant problem of the torsion,stretching and bending of a naturally twisted rod

Problems on large stretching, torsional and bending deformations of a naturally twisted rod, loaded with end forces and moments, are considered from the point of view of the non-linear three-dimensional theory of elasticity. Particular solutions of the equations of elastostatics are found, which are two-parameter families of finite deformations and which possess the property that, for these deformations, the initial system of three-dimensional non-linear equations reduces to a system of equations with two independent variables. The use of these equations enables one to reduce certain Saint-Venant problems for a naturally twisted rod to two-dimensional non-linear boundary-value problems for a planar domain in the form of the cross-section of a rod. Different formulations of the two-dimensional boundary-value problem for the cross-section are proposed, which differ in the choice of the unknown functions. A non-linear problem of the torsion and stretching of a circular cylinder with helical anisotropy, which is reduced to ordinary differential equations, is considered as a special case.     

A nonlocal higher-order model including thickness stretching effect for bending and buckling of curved nanobeams

An analytical approach for static bending and buckling analyses of curved nanobeams using the differential constitutive law, consequent to Eringen’s strain-driven integral model coupled with a higher-order shear deformation accounting for through thickness stretching is presented. The formulation is general in the sense that it can be deduced to examine the influence of different structural theories, for static and dynamic analyses of curved nanobeams. The governing equations derived using Hamiltons principle are solved in conjunction with Naviers solutions. The formulation is validated considering problems for which solutions are available. A comparative study is made here by various theories obtained through the formulation. The effects various structural parameters such as thickness ratio, beam length, rise of the curved beam, and nonlocal scale parameter are brought out on bending and stability characteristics of curved nanobeams.     

Obtaining graph knots by twisting unknots

Let K be a knot in the 3-sphere S³ and D a disk in S³ meeting K transversely more than once in the interior. For nontriviality we assume that |D∩K|⩾2 over all isotopies of K in S³−∂D. Let K_D,n (⊂S³) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and K_D,n is a graph knot, then |n|⩽1 or K and D form a special pair which we call an “exceptional pair”. As a corollary, if (K,D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner.     

Symmetric knots and the cabling conjecture

Dedicated to Professor G. Nöbeling on his eightyfifth birthday     

Periodic knots and the skein polynomial

This research was supported by NSERC-No. A4034 (Canada) and Polish scientific grant RP 1.10.     

Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations

In the mechanical analysis of composites containing nano-inhomogeneities, it is customary to consider only the stretching resistance of the inhomogeneity-matrix interface but neglect the bending resistance of the interface. In this paper, we consider a circular nano-inhomogeneity in an infinite elastic plane subjected to an arbitrary uniform remote in-plane loading with both stretching and bending resistance incorporated on the interface. Analytic solutions are obtained for the stress field both inside and outside the inhomogeneity by using an integral-type boundary condition representing the jump in traction across the interface. We show that the presence of interface bending resistance has no influence on the average of the mean stress in the inhomogeneity, and for certain interface stretching and bending rigidities the stress field inside the inhomogeneity can remain uniform regardless of the specific uniform remote loading. Numerical examples are presented to examine the influence of the interface bending resistance on the interfacial tractions imposed on the inhomogeneity and matrix for a uniform remote uniaxial loading. It is found that the introduction of interface bending resistance perturbs the (interfacial) tractions imposed on the inhomogeneity only slightly whether the inhomogeneity is softer or harder than the matrix, while it may influence the (interfacial) tractions imposed on the matrix significantly when the inhomogeneity is much softer than the matrix. Moreover, it is shown that the peak of the interface bending resistance-induced jump in traction across the interface initially increases and then decreases as the inhomogeneity becomes harder (from an initial state in which the inhomogeneity is softer than the matrix).     

Interpolation by periodic splines with Birkhoff knots

Summary We give a complete characterization of the Hermite interpolation problem by periodic splines with Birkhoff knots. As a dual result we derive the characterization of the Birkhoff interpolation by periodic splines with multiple knots.Sponsored by the Bulgarian Ministry of Education and Science under Contract No. MM-15     

Constrained approximation by splines with free knots

In this paper, a method that combines shape preservation and least squares approximation by splines with free knots is developed. Besides the coefficients of the spline a subset of the knot sequence, the so-calledfree knots, is included in the optimization process resulting in a nonlinear least squares problem in both the coefficients and the knots. The original problem, a special case of aconstrained semi-linear least squares problem, is reduced to a problem that has only the knots of the spline as variables. The reduced problem is solved by a generalized Gauss-Newton method. Special emphasise is given to the efficient computation of the residual function and its Jacobian. Dedicated to our colleague and teacher Prof. Dr. J. W. Schmidt on the occasion of his 65th birthday Research of the first author was supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1,2-2.     

Virtual knots and links

This paper is an introduction to the subject of virtual knot theory and presents a discussion of some new specific theorems about virtual knots. The new results are as follows: Using a 3-dimensional topology approach, we prove that if a connected sum of two virtual knots K ₁ and K ₂ is trivial, then so are both K ₁ and K ₂. We establish an algorithm for recognizing virtual links that is based on the Haken-Matveev technique. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 252, pp. 114–133.