Some Aspects of Positive Kernel Method of Quantization

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups $$\tau _t \in \text{ Aut }(P,\vartheta )$$ τ t ∈ Aut ( P , ϑ ) of automorphisms of a G-principal bundle $$P(G,\pi ,M)$$ P ( G , π , M ) with a fixed connection form $$\vartheta $$ ϑ on its total space P. We show that the generator $${\hat{F}}$$ F ^ of the unitary flow $$U_t = e^{it {\hat{F}}}$$ U t = e i t F ^ being the quantization of $$\tau _t $$ τ t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when $$G=\hbox {GL}(N,{\mathbb {C}})$$ G = GL ( N , C ) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow $$\tau _t^{\mathrm{hol}} \in \text{ Aut }(P,\vartheta )$$ τ t hol ∈ Aut ( P , ϑ ) . For the above case, we present the integral decompositions of the positive kernels on $$P\times P$$ P × P invariant with respect to the flows $$\tau _t^{\mathrm{hol}}$$ τ t hol in terms of the spectral measure of $${\hat{F}}$$ F ^ . These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on $${\mathbb {C}} \times {\mathbb {C}}$$ C × C invariant with respect to the one-parameter groups of translations of complex plane.

HYERS-ULAM STABILITY FOR NONLOCAL DIFFERENTIAL EQUATIONS

In this paper, we present a result on Hyers-Ulam stability for a class of nonlocal differential equations in Hilbert spaces by using the theory of integral equations with completely positive kernels together with a new Gronwall inequality type. The paper develops some recent results on fractional differential equations to the ones involving general nonlocal derivatives. Instead of Mittag-Leffler functions, we must utilize the characterization of relaxation function.

Some new multidimensional hardy-type inequalities with kernels via convexity

We prove some new multidimensional Hardy-type inequalities involving general Hardy type operators with positive kernels for functions ? which may not necessarily be convex but satisfy the condition A?(x) ? ?(x) ? B? (x), where ? is convex. Our approach is mainly the use of convexity argument and the results obtained are new even for the one-dimensional case and also unify and extend several inequalities of Hardy type known in the literature.

Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models

Suppose that {Xt} is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {Xt} typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models

Suppose that {X t } is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {X t } typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.