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The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of . As an example, a new generalization of Burde’s law is provided.
The Lang–Kobayashi system of delay differential equations describes the behavior of the complex electric field and inversion inside an external cavity semiconductor laser. This system has a family of special periodic solutions known as external cavity modes (ECMs). It is well known that these ECM solutions appear through saddle-node bifurcations, then lose stability through a Hopf bifurcation before new ECM solutions are born through a secondary saddle-node bifurcation. Employing analytical and numerical techniques, we show that for certain short external cavity lasers the loss of stability happens only after the secondary saddle-node bifurcations, which means that stable ECM solutions can coexist in these systems. We also investigate the basins of these ECM attractors.
We explore a complex extension of finite calculus on the integer lattice of the complex plane. satisfies the discretized Cauchy–Riemann equations at if and . From this principle arise notions of the discrete path integral, Cauchy’s theorem, the exponential function, discrete analyticity, and falling power series.
We present an approach to compute by changing variables in the double integral using hyperbolic trigonometric functions. We also apply this approach to present , when , as a definite improper integral of a single variable.
We present and illustrate the methodology to calculate curvature measures for continuous designs, and extend design criteria to incorporate continuous designs. These design algorithms include quadratic design procedures, a subset design criterion, a second-order mean-square error design criterion, and a marginal curvature design methodology. A discussion of confidence intervals is also provided for continuous designs.
We consider numerical semigroups , for intervals . We compute the Frobenius number and multiplicity of such semigroups, and show that we may freely restrict to be open, closed, or half-open, as we prefer.