An Analytic Condition for the Finite Degree-of-Freedom of Linkages and Its Computational Evaluation

The finite degree of freedom (DOF) of a mechanism is determined by the number of independent loop constraints. In this paper a method is introduced to determine the maximal number of loop closure constraints (which is independent of a specific configuration) of multi-loop linkages and is applied to calculate the finite DOF. It rests on an algebraic condition on the joint screws and the corresponding computational algorithm to determine the maximal rank of the constraint Jacobian in an arbitrary (possibly singular) reference configuration, making use of the analytic condition that minors of certain rank and their higher derivatives vanish. Unlike the Lie group methods for estimation the DOF of so-called exceptional linkages, this method does not rely on partitioning kinematic loops into partial kinematic chains, and it is applicable to multi-loop linkages. The DOF computed with this method is at least as accurate as the DOF computed with the Lie group methods. It gives the correct DOF for any (possibly overconstrained) linkage where the constraint Jacobian has maximal rank in regular configurations. The so determined maximal rank has further significance for classifying linkages as being exceptional or paradoxical, but also for detecting singularities and shaky linkages.

Rotating vector solving method applied for nonlinear oscillator

Significant number of procedures for solving of the finite degree-of-freedom forced nonlinear oscillator are developed. For all of them it is common that they are based on the exact solution of the corresponding linear oscillator. For technical reasons, the aim of this paper is to develop a simpler solving procedure. The rotating vector method, developed for the linear oscillator, is adopted for solving of the nonlinear finite degree-of-freedom oscillator. The solution is assumed in the form of trigonometric functions. Assuming that the nonlinearity is small all terms of the series expansion of the function higher than the first are omitted. The rotating vectors for each mass are presented in the complex plane. In the paper, the suggested rotating vector procedure is applied for solving of a three-degree-of-freedom periodically excited oscillator. The influence of the nonlinear stiffness of the flexible elastic beam, excited with a periodical force, on the resonant properties of the system in whole is investigated. It is obtained that the influence of nonlinearity on the amplitude and phase of vibration is more significant for smaller values of the excitation frequency than for higher ones.

On the degeneracy of spin ice graphs, and its estimate via the Bethe permanent

The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. We map spin ice graphs to the Ising model on a graph and clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the one-dimensional Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. We show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such a technique can also be used to obtain an upper bound, we find that in all finite degree examples we studied, another upper bound based on Schrijver inequality is tighter.

Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds

We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$, which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of $Q$ satisfies the Jordan property.

Convergence theorems on multi-dimensional homogeneous quantum walks

We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved.

Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

Maxwell tipping points: the hidden mechanics of an axially compressed cylindrical shell

Numerical results for the axially compressed cylindrical shell demonstrate the post-buckling response snaking in both the applied load and corresponding end-shortening. Fluctuations in load, associated with progressive axial formation of circumferential rings of dimples, are well known. Snaking in end-shortening, describing the evolution from a single dimple into the first complete ring of dimples, is a recent discovery. To uncover the mechanics behind these different phenomena, simple finite degree-of-freedom cellular models are introduced, based on hierarchical arrangements of simple unit cells with snapback characteristics. The analyses indicate two fundamentally different variants to this new form of snaking. Each cell has its own Maxwell displacement, which are either separated or overlap. In the presence of energetic background disturbance, the differences between these two situations can be crucial. If the Maxwell displacements of individual cells are separated, then buckling is likely to occur sequentially, with the system able to settle into different localized states in turn. Yet if Maxwell displacements overlap, then a global buckling pattern triggers immediately as a dynamic domino effect. We use the term Maxwell tipping point to identify the point of switching between these two behaviours.

Multipolar Intuitionistic Fuzzy Hyper BCK-Ideals in Hyper BCK-Algebras

In 2020, Kang et al. introduced the concept of a multipolar intuitionistic fuzzy set of finite degree, which is a generalization of a k-polar fuzzy set, and applied it to a BCK/BCI-algebra. The specific purpose of this study was to apply the concept of a multipolar intuitionistic fuzzy set of finite degree to a hyper BCK-algebra. The notions of the k-polar intuitionistic fuzzy hyper BCK-ideal, the k-polar intuitionistic fuzzy weak hyper BCK-ideal, the k-polar intuitionistic fuzzy s-weak hyper BCK-ideal, the k-polar intuitionistic fuzzy strong hyper BCK-ideal and the k-polar intuitionistic fuzzy reflexive hyper BCK-ideal are introduced herein, and their relations and properties are investigated. These concepts are discussed in connection with the k-polar lower level set and the k-polar upper level set.

On the depth overhead incurred when running quantum algorithms on near-term quantum computers with limited qubit connectivity

This paper addresses the problem of finding the depth overhead that will be incurred when running quantum circuits on near-term quantum computers. Specifically, it is envisaged that near-term quantum computers will have low qubit connectivity: each qubit will only be able to interact with a subset of the other qubits, a reality typically represented by a qubit interaction graph in which a vertex represents a qubit and an edge represents a possible direct 2-qubit interaction (gate). Thus the depth overhead is unavoidably incurred by introducing swap gates into the quantum circuit to enable general qubit interactions. This paper proves that there exist quantum circuits where a depth overhead in Omega(\log n) must necessarily be incurred when running quantum circuits with n qubits on quantum computers whose qubit interaction graph has finite degree, but that such a logarithmic depth overhead is achievable. The latter is shown by the construction of a 4-regular qubit interaction graph and associated compilation algorithm that can execute any quantum circuit with only a logarithmic depth overhead.