Heavy handed quest for fixed points in multiple coupling scalar theories in the ε expansion

The tensorial equations for non trivial fully interacting fixed points at lowest order in the ε expansion in 4 − ε and 3 − ε dimensions are analysed for N-component fields and corresponding multi-index couplings λ which are symmetric tensors with four or six indices. Both analytic and numerical methods are used. For N = 5, 6, 7 in the four-index case large numbers of irrational fixed points are found numerically where ‖λ‖2 is close to the bound found by Rychkov and Stergiou [1]. No solutions, other than those already known, are found which saturate the bound. These examples in general do not have unique quadratic invariants in the fields. For N ⩾ 6 the stability matrix in the full space of couplings always has negative eigenvalues. In the six index case the numerical search generates a very large number of solutions for N = 5.

What Makes a Partner Ideal, and for Whom? Compatibility Tests, Filter Tests, and the Mating Stability Matrix

We introduce a typological characterization of possible human heterosexual couples in terms of the concordance-opposition of the orientations of their active and receptive areas as defined by the tie-up theory. We show that human mating incentives, as characterized by widely adopted approaches, such as Becker’s marriage market approach, only capture very specific instances of actual couples thus characterized. Our approach allows us to instead explore how super-cooperation among partners vs. convenience vs. constriction may be regarded as alternatives modes of couple formation and cohesion, leading to very different types of couples with different implications in terms of stability and resilience. Our results may have interesting implications for future experimental research and for individual and family counseling.

New perspectives on polygonal and polyhedral finite element methods

Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the virtual element method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understanding of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and visualization in the VEM, and on the other hand, provides a means to exactly satisfy the patch test with efficient numerical integration in polygonal and polyhedral finite elements. We present numerical examples that demonstrate the sound accuracy and performance of the proposed method. For Poisson problems in ℝ2and ℝ3, we establish that linearly complete generalized barycentric interpolants deliver optimal rates of convergence in the L2-norm and the H1-seminorm.