With fourth-order derivative theories leading to propagators of the generic ghostlike 1/(k2−M12)−1/(k2−M22) form, it would appear that such theories have negative norm ghost states and are not ...unitary. However on constructing the associated quantum Hilbert space for the free theory that would produce such a propagator, Bender and Mannheim found that the Hamiltonian of the free theory is not Hermitian but is instead PT symmetric, and that there are in fact no negative norm ghost states, with all Hilbert space norms being both positive and preserved in time. Even though perturbative radiative corrections cannot change the signature of a Hilbert space inner product, nonetheless it is not immediately apparent how such a ghostlike propagator would not then lead to negative probability contributions in loop diagrams. Here we obtain the relevant Feynman rules and show that all states obtained in cutting intermediate lines in loop diagrams have positive norm. Also we show that due to the specific way that unitarity (conservation of probability) is implemented in the theory, negative signatured discontinuities across cuts in loop diagrams are canceled by a novel and unanticipated contribution of the states in which tree approximation (no loop) graphs are calculated, an effect that is foreign to standard Hermitian theories. Perturbatively then, the fourth-order derivative theory with propagator 1/(k2−M12)−1/(k2−M22) is viable. Implications of our results for the pure massless 1/k4 propagator are also discussed.
We consider the random-design least-squares regression problem within the reproducing kernel Hubert space (RKHS) framework. Given a stream of independent and identically distributed input/output ...data, we aim to learn a regression function within an RKHS ℋ, even if the optimal predictor (i.e., the conditional expectation) is not in ℋ. In a stochastic approximation framework where the estimator is updated after each observation, we show that the averaged unregularized least-mean-square algorithm (a form of stochastic gradient descent), given a sufficient large step-size, attains optimal rates of convergence for a variety of regimes for the smoothnesses of the optimal prediction function and the functions in ℋ. Our results apply as well in the usual finite-dimensional setting of parametric least-squares regression, showing adaptivity of our estimator to the spectral decay of the covariance matrix of the covariates.
In this paper, we give a complete form of bijective (not necessarily linear) maps Formula omitted. where H,K are Hilbert spaces with Formula omitted. that satisfy Formula omitted. where Formula ...omitted. is the λ-Aluthge transform of T and the operation Formula omitted. is the Formula omitted. -Jordan-triple product. We show that there exists a unitary or anti-unitary operator Formula omitted. and a constant Formula omitted. , with Formula omitted. such that Formula omitted.
in this paper, we give a concept of szl -widering mapping which is independent of nonspreading mappings, k - strictly pseudo nonspring mappings and nonexpansive mappings. Also we introduce a new ...proximal point schemes of resolvent and szl -widering mappings. On the other hand, we discuss that the weak and strong convergence for these proximal point schemes in real Hilbert space under different conditions to asymptotic common fixed point of szl -widering mappings.
The aim of this article is to propose a new definition of fuzzy fractional derivative, so-called fuzzy conformable. To this end, we discussed fuzzy conformable fractional integral softly. Meanwhile, ...uniqueness, existence, and other properties of solutions of certain fuzzy conformable fractional differential equations under strongly generalized differentiability are also utilized. Furthermore, all needed requirements for characterizing solutions by equivalent systems of crisp conformable fractional differential equations are debated. In this orientation, modern trend and new computational algorithm in terms of analytic and approximate conformable solutions are proposed. Finally, the reproducing kernel Hilbert space method in the conformable emotion is constructed side by side with numerical results, tabulated data, and graphical representations.
We use self-consistent Hartree-Fock calculations performed in the full π-band Hilbert space to assess the nature of the recently discovered correlated insulator states in magic-angle twisted bilayer ...graphene (TBG). We find that gaps between the flat conduction and valence bands open at neutrality over a wide range of twist angles, sometimes without breaking the system's valley projected C_{2}T symmetry. Broken spin-valley flavor symmetries then enable gapped states to form not only at neutrality, but also at total moiré band filling n=±p/4 with integer p=1, 2, 3, when the twist angle is close to the magic value at which the flat bands are most narrow. Because the magic-angle flat band quasiparticles are isolated from remote band quasiparticles only for effective dielectric constants larger than ∼20, the gapped states do not necessarily break C_{2}T symmetry and as a consequence the insulating states at n=±1/4 and n=±3/4 need not exhibit a quantized anomalous Hall effect.
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