Using Generalized Modus Ponens reasoning, we examine the values of the inferred conclusion by using Fodor's implication in order to interpret a fuzzy if-then rule with a single input single output and the tnorm ( ) ( )( ) ( xy y x max y , x t ) λ λ − − + + = 1 1 , 1 − ≥ λ , for composition operation. This t-norm is important to use because for 1 − = λ and 0 = λ it gives the commonly used t-norm...

No problem is ever permanently closed. The editor is always pleased to consider for publication new solutions or new insights o n p a s t p r oblems. (1) Determine the set of all c > 0 such that whenever (1) holds, then we have x + y + z xy + yz+ zx. The positive r eals c asked for are those satisfying c 4. From (1), we have xy < c and z = c ; xy x + y + xy. Substituting this value of z into th...

We find that the classical one-dimensional XY model, with angular-momentum-conserving Langevin dynamics, mimics the non-Newtonian flow regimes characteristic of soft matter when subjected to counterrotating boundaries. An elaborate steady-state phase diagram has continuous and first-order transitions between states of uniform flow, shear-banding, solid-fluid coexistence and slip planes. Results...

We consider the family of relative Thue equations x − (t− 1)xy − (t+ 2)xy − y = μ, where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. We prove that there are only trivial solutions (with |x|, |y| ≤ 1), if |t| is large enough or if the discriminant of the quadratic number field is large enough or if Re t = −1/2 (there a...

Abstract We investigate an XY spin-glass model in which both spins and interactions (or couplings) evolve in time, but with widely separated time-scales. For large times this model can be solved using replica theory, requiring two levels of replicas, one level for the spins and one for the couplings. We define the relevant order parameters, and derive a phase diagram in the replica-symmetric ap...

Large N geometric transitions and the Dijkgraaf-Vafa conjecture suggest a deep relationship between the sum over planar diagrams and Calabi-Yau threefolds. We explore this correspondence in details, explaining how to construct the Calabi-Yau for a large class of M matrix models, and how the geometry encodes the correlators. We engineer in particular two-matrix theories with potentials W (X,Y ) ...

Thermal transport in the mixed state of a d-wave superconductor is considered within the weak-field regime. We express the thermal conductivity, kappa(xx), and the thermal Hall conductivity, kappa(xy), in terms of the cross section for quasiparticle scattering from a single vortex. Solving for the cross section (neglecting the Berry phase contribution and the anisotropy of the gap nodes), we ob...

In this paper we prove that if a weighted symmetric digraph (~ G, c) has a mapping T : E( ~ G) → {0, 1} with T (xy) + T (yx) = 1 for all arcs xy in ~ G such that for each dicycle C satisfying 0 < |C|c(mod r) < maxxy∈E(~ G) c(xy) + c(yx) we have |C|c/|C|T ≤ r, then (~ G, c) has a circular r-coloring. Our result generalizes the work of Zhu (J. Comb. Theory, Ser. B, 86 (2002) 109-113) concerning t...