Soliton propagation in tapered silicon core fibers

Numerical simulations are used to investigate soliton-like propagation in tapered silicon core optical fibers. The simulations are based on a realistic tapered structure with nanoscale core dimensions and a decreasing anomalous dispersion profile to compensate for the effects of linear and nonlinear loss. An intensity misfit parameter is used to establish the optimum taper dimensions that preserve the pulse shape while reducing temporal broadening. Soliton formation from Gaussian input pulses is also observed--further evidence of the potential for tapered silicon fibers to find use in a range of signal processing applications.     

Multilayer metal/metal-oxide diffractive structure for photonic temperature sensing

We designed and fabricated multilayer metal/metal-oxide surface relief diffractive grating structures by growing alternating Pt and SnO(x) layers. Optical interrogation at 633 nm reveals the temperature dependence of their reflection and transmission diffractive effects. This function is explored here in the context of a remote, spatially localized, photonic temperature sensing operation, achieving sensitivity of 10% per °C for the zeroth-order in the transmission mode. The experimental demonstration is found to be in good agreement with the results of rigorous coupled wave analysis of the composite metal/metal-oxide element.     

On a functional equation by Baak,Boo and Rassias

In Baak et al. (J Math Anal Appl 314(1):150–161, 2006) the authors considered the functional equation

$$\begin{aligned} r f\left( \frac{1}{r}\,\sum _{j=1}^{d}x_j\right)+ & {} \sum _{i(j)\in \{0,1\} \atop \sum _{1\le j\le d} i(j)=\ell }r f\left( \frac{1}{r}\,\sum _{j=1}^d (-1)^{i(j)}x_j\right) \\= & {} \left( {d-1\atopwithdelims ()\ell }-{d-1\atopwithdelims ()\ell -1} +1\right) \sum _{j=1}^{d} f(x_j) \end{aligned}$$

where \(d,\ell \in \mathbb {N}\), \(1<\ell <d/2\) and \(r\in \mathbb {Q}{\setminus }\{0\}\). The authors determined all odd solutions \(f:X\rightarrow Y\) for vector spaces X, Y over \(\mathbb {Q}\). In Oubbi (Can Math Bull 60:173–183, 2017) the author considered the same equation but now for arbitrary real \(r\not =0\) and real vector spaces X, Y. Generalizing similar results from (J Math Anal Appl 314(1):150–161, 2006) he additionally investigates certain stability questions for the equation above, but as for that equation itself for odd approximate solutions only. The present paper deals with the general solution of the equation and the corresponding stability inequality. In particular it is shown that under certain circumstances non-odd solutions of the equation exist.

     

Trace formulas for relative Schatten class perturbations

We derive trace formulas for a pair of self-adjoint operators $H+V$ and H under the assumption that ${(H?\mathrm{i})}^{?1}V$ is in a Schatten class. This extends the trace formulas of [8], where V alone is assumed to be in a Schatten class. Our trace formulas apply, in particular, in the setting of differential operators and are based on Taylor-like approximations of operator functions. This significantly improves non-Taylor based trace formulas of [10].     

Groups in which every subgroup of infinite rank is almost permutable

In this paper, the structure of locally finite groups of infinite rank whose subgroups of infinite rank are permutable in a subgroup of finite index is investigated.     

Partition identities and quiver representations

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector.     

Measures under the flat norm as ordered normed vector space

The space of real Borel measures \(\mathcal {M}(S)\) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone \(\mathcal {M}_+(S)\) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of \(\mathcal {M}_+(S)\) are compact and semiflows on \(\mathcal {M}_+(S)\) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because \(\mathcal {M}(S)\) is rarely complete and \(\mathcal {M}_+(S)\) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on \(\mathcal {M}_+(S)\) and continuous semiflows. Both topics prepare for a dynamical systems theory on \(\mathcal {M}_+(S)\).     

Dynamics of the Douglas-Rachford Method for Ellipses and <Emphasis Type="Italic">p</Emphasis>-Spheres

We expand upon previous work that examined the behavior of the iterated Douglas-Rachford method for a line and a circle by considering two generalizations:that of a line and an ellipse and that of a line together with a p-sphere. With computer assistance we discover a beautiful geometry that illustrates phenomena which may affect the behavior of the iterates by slowing or inhibiting convergence for feasible cases. We prove local convergence near feasible points, and—seeking a better understanding of the behavior—we employ parallelization in order to study behavior graphically. Motivated by the computer-assisted discoveries, we prove a result about behavior of the method in infeasible cases.