Fourth order exceptional points with spinning resonators

Investigation of exceptional points mostly focuses on the second order case and employs the gain-involved parity-time (PT) symmetric systems. Here, we propose an approach to implementing fourth order exceptional points (FOEPs) using directly coupled optical resonators with rotation. On resonance, the system manifests FOEP through tuning the spinning velocity to targeted values. Eigenfrequency bifurcation and enhanced sensitivity for nanoparticle have been presented. Also, near FOEP, nonreciprocal light propagation exhibits great boost and dramatic change, which may be applied to high-efficiency isolators and circulators.

From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By specialisations of his formula, we deduce kernel identities for deformed Macdonald–Ruijsenaars (MR) and Noumi–Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.

Consensus Indices of Two-Layered Multi-Star Networks: An Application of Laplacian Spectrum

In this article, the convergence speed and robustness of the consensus for several dual-layered star-composed multi-agent networks are studied through the method of graph spectra. The consensus-related indices, which can measure the performance of the coordination systems, refer to the algebraic connectivity of the graph and the network coherence. In particular, graph operations are introduced to construct several novel two-layered networks, the methods of graph spectra are applied to derive the network coherence for the multi-agent networks, and we find that the adherence of star topologies will make the first-order coherence of the dual-layered systems increase some constants in the sense of limit computations. In the second-order case, asymptotic properties also exist when the index is divided by the number of leaf nodes. Finally, the consensus-related indices of the duplex networks with the same number of nodes but non-isomorphic structures have been compared and simulated, and it is found that both the first-order coherence and second-order coherence of the network D are between A and B, and C has the best first-order robustness, but it has the worst robustness in the second-order case.

Corrigendum to “Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode”

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.57 for the incommensurate order case. Also, the period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

The curl–curl conforming virtual element method for the quad-curl problem

In this paper, we present the [Formula: see text]-conforming virtual element (VE) method for the quad-curl problem in two dimensions. Based on the idea of de Rham complex, we first construct three families of [Formula: see text]-conforming VEs, of which the simplest one has only one degree of freedom associated to each vertex and each edge in the lowest-order case, respectively. An exact discrete complex is established between the [Formula: see text]-conforming and [Formula: see text]-conforming VEs. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the coercivity and inf–sup condition of the corresponding discrete formulation. We show that the conforming VEs have the optimal convergence. Some numerical examples are given to confirm the theoretical results.

Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

Reversible Regular Languages: Logical and Algebraic Characterisations

We present first-order (FO) and monadic second-order (MSO) logics with predicates ‘between’ and ‘neighbour’ that characterise the class of regular languages that are closed under the reverse operation and its subclasses. The ternary between predicate bet(x, y, z) is true if the position y is strictly between the positions x and z. The binary neighbour predicate N(x, y) is true when the the positions x and y are adjacent. It is shown that the class of reversible regular languages is precisely the class definable in the logics MSO(bet) and MSO(N). Moreover the class is definable by their existential fragments EMSO(bet) and EMSO(N), yielding a normal form for MSO formulas. In the first-order case, the logic FO(bet) corresponds precisely to the class of reversible languages definable in FO(<). Every formula in FO(bet) is equivalent to one that uses at most 3 variables. However the logic FO(N) defines only a strict subset of reversible languages definable in FO(+1). A language-theoretic characterisation of the class of languages definable in FO(N), called locally-reversible threshold-testable (LRTT), is given. In the second part of the paper we show that the standard connections that exist between MSO and FO logics with order and successor predicates and varieties of finite semigroups extend to the new setting with the semigroups extended with an involution operation on its elements. The case is different for FO(N) where we show that one needs an additional equation that uses the involution operator to characterise the class. While the general problem of characterising FO(N) is open, an equational characterisation is shown for the case of neutral letter languages.

Categoricity by convention

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem for second-order —these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic.