Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p ₀ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p ₀ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p ₀ the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p ₀ corresponds to a limit cycle γ ₀. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ ₀. We define the notion of vanishing multiplicity of the inverse integrating factor over γ ₀. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p ₀ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.     

Graph drawings with few slopes

The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for Δ5 and all large n, there is a Δ-regular n-vertex graph with slope-number at least . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slope-number of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most . Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.     

System Dynamics Modelling for Residential Energy Efficiency Analysis and Management

A System Dynamics model to simulate the substitution of installed household appliances by more efficient ones is presented. The model allows the construction of scenarios and also the analyses of several other issues such as: alternatives for technology penetration, electricity consumption growth, gas consumption growth and effects of pricing policies on various energy demands. The proposed methodology has been applied to assist the decision process in relation to gas penetration policies. The model also supports policy making on energy efficiency and it allows the calculation of total energy savings under different scenarios. Furthermore, government underpricing policies on tariffs and appliance acquisition (longer loan terms, lower interest rates and grace periods) may also be analysed.     

Photoelastic analysis of stresses around oblique holes

A three-dimensional photoelastic analysis was conducted to determine the magnitude and distribution of stresses around oblique holes in a uniaxially loaded plate. The holes were circular and inclined at angles of 45 deg and 60 deg with the faces of the plate. The plate-thickness-to-hole-diameter ratiot/D was 2.40. One end of each hole was blended to the face of the plate through a break radius equal to the radius of the hole. The plate dimensions were sufficiently large to simulate conditions of an infinite plate. The plates were loaded perpendicular to the plane of skewness. Stress distributions were obtained on sections perpendicular to the direction of loading. Results point to two critical areas of stress concentration: one at the acute-angle intersection of the hole and the surface of the plate and the other in the break-radius area. The stress concentrations in the latter area reach values of 4.6 and 6.7 compared to 3.6 and 4.5 at the acute-angle intersection, for the inclination angles of 45 deg and 60 deg, respectively. A simplified analysis used for the break-radius area gave results in agreement with the experiment. Thus, it was shown that break radii in oblique penetrations may have deleterious rather than beneficial effects. Comparison of results for the acute-angle intersection with existing theoretical and experimental values shows a definite and pronounced dependence of the stress-concentration factor on thickness-to-diameter ratio.     

Quasi-static properties of a photoviscoelastic material

This paper describes the results of mechanical and optical measurements in plasticized polyvinyl chloride under conditions of creep and relaxation at room temperature. It covers one task of a broader investigation aimed at developing experimental methods for viscoelastic stress analysis. The moiré method of strain analysis was found well suited for continuous recording of axial and transversal deformation in creep tests. The material exhibits linear viscoelastic behavior, both mechanical and optical. Strain, stress and birefringence measured from creep and relaxation tests gave straightline plots on log-log scale and, thus simple empirical formulas were possible to derive. The theoretical prediction that birefringence in a linear viscoelastic material not exhibiting flow can be expressed as a linear relationship of stress and strain was satisfactorily substantiated.     

Synthesis and pH‐Dependent Spectroscopic Behavior of 2,4,6‐Trisubstituted Pyridine Derivatives

Seven 2,4,6‐trisubstituted pyridine derivatives with N,N‐diethylaniline substituents at the 4‐position were synthesized, and their spectroscopic properties in the absence and presence of acid were studied. The spectral effects of protonation, molar absorptivities, pK_a values, and the structural origins of the observed spectral behavior were ascertained. The pyridine nitrogen was found to be more basic than the diethylamino nitrogen atom. Protonation of the pyridine ring nitrogen is associated with the appearance of a red‐shifted intramolecular charge transfer peak in the UV‐visible spectra. Favorable color indicating properties result from electron‐donating substitution at the 2 and 6 positions of pyridine, which provide a greater absorptivity of the red‐shifted peak associated with protonation of the pyridine nitrogen. These findings will assist in the design and optimization of these compounds for ion‐indicating and pH‐sensing applications.     

Asymmetric [3+2] Annulation Approach to 3‐Pyrrolines: Concise Total Syntheses of (−)‐Supinidine, (−)‐Isoretronecanol,and (+)‐Elacomine

An asymmetric [3+2] annulation reaction to form 3‐pyrroline products is reported. Upon treatment with lithium diisopropylamide, readily available ethyl 4‐bromocrotonate is deprotonated and trapped with Ellman imines selectively at the α‐position to yield enantiopure 3‐pyrroline products. This new method is compatible with aryl, alkyl, and vinyl imines. The efficacy of the method is showcased by short asymmetric total syntheses of (−)‐supinidine, (−)‐isoretronecanol, and (+)‐elacomine. This novel annulation approach also works for an aldehyde, thus providing access to a 2,5‐dihydrofuran product in a single step from simple precursors. By modifying the structure of the carbanion nucleophile, an asymmetric vinylogous aza‐Darzens reaction can be realized.     

Role of parallel reformable bonds in the self-healing of cross-linked nanogel particles

We develop a hybrid computational approach to examine the mechanical properties and self-healing behavior of nanogel particles that are cross-linked by both stable and labile bonds. The individual nanogels are modeled via the lattice spring model (LSM), which is an effective method for probing the response of materials to mechanical deformation. The cross-links between the nanogels are simulated via the hierarchical Bell model (HBM), which allows us to capture the rupturing of multiple parallel bonds as the result of an applied force. Because the labile bonds are relatively reactive, they can reform after they have been ruptured. To incorporate the possibility of bonds reforming, we modify the HBM formalism and validate the modified HBM by considering a system of two surfaces, which are connected by multiple parallel bonds. We then use our hybrid HBM/LSM to simulate the behavior of the cross-linked nanogels under a tensile deformation. In these simulations, each labile linkage between the nanogels contains at most N parallel bonds. We vary the fraction of labile linkages and the value of N in these linkages to determine the optimal conditions for improving the robustness of the material. Although numerous parallel bonds within a linkage enhance the strength of the material, these bonds diminish the ductility and the ability of the material to undergo the structural rearrangements that are necessary for self-repair. For a relatively low fraction of labile bonds and N ≤ 4, however, we can significantly improve the strength of the material and preserve the self-healing properties. For instance, a sample with 30% labile linkages and N = 4 per linkage is roughly 200% stronger than a sample that is cross-linked solely by stable bonds and can still undergo self-repair in response to the tensile deformation. The results reveal how mechanical stress can lead not only to the appearance of cavities within the material but also to bond formation that "heals" these cavities and thus prevents the catastrophic failure of the material.