For p ∈ [2,∞) the metric Xp inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this result include the following sharp statements about embeddings of Lq into Lp when 2 < q < p <∞: the maximal θ ∈ (0, 1] for which Lq admits a bi-θ-Hölder embedding into Lp equals q/p, and for m,n ∈ N the smallest possible bi-Lipschitz distortion of any embeddi...

Sumatra Island is the third largest island in Indonesia which has extraordinary natural resource potential. 10 provinces are islands with most Indonesia. Each province different resources and geographical conditions so that inequality occurs an uneven economy. This study aims to identify leading sectors prospective each Sumatra. The analysis was carried out on economic between uses Location Quo...

A stochastic linear quadratic (LQ) control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Indefinite stochastic LQ theory has been extensively developed and has found interesting applications in finance. However, there remains an outstanding open problem, which is to identify an appropriate Riccati-type equation whose solvabili...

The classic Poincaré inequality bounds the Lq-norm of a function f in a bounded domain Ω ⊂ Rn in terms of some Lp-norm of its gradient in Ω. We generalize this in two ways: In the first generalization we remove a set Γ from Ω and concentrate our attention on Λ = Ω \ Γ. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it m...

We give a simple direct proof of the interpolation inequality ‖∇f‖2 L2p C‖f‖BMO‖f‖W 2, p , where 1 < p < ∞. For p = 2 this inequality was obtained by Meyer and Rivière via a different method, and it was applied to prove a regularity theorem for a class of Yang–Mills fields. We also extend the result to higher derivatives, sharpening all those cases of classical Gagliardo– Nirenberg inequalities...

In this paper we deal with a definition of Lp-Fourier transform on locally compact groups. Recall that, for locally compact abelian groups, the Hausdorff-Young inequality reads: Let 1 < p < 2 and q = p/(p − 1). If g ∈ L1(G) ∩ Lp(G), then ĝ ∈ Lq(Ĝ), with ‖ĝ‖q ≤ ‖g‖p. The inequality allows to extend the Fourier transform to a continuous operator Fp : Lp → Lq by continuity. It was generalized to t...

Given a dictionary of Mn initial estimates of the unknown true regression function, we aim to construct linearly aggregated estimators that target the best performance among all the linear combinations under a sparse q-norm (0 ≤ q ≤ 1) constraint on the linear coefficients. Besides identifying the optimal rates of aggregation for these lq-aggregation problems, our multi-directional (or adaptive...

The Standard Model is extended by a SU(2) L singlet of vector leptoquarks. An additional SU(4) gauge symmetry between right–handed up quarks and right– handed leptons is introduced to render the model renormalizable. The arrangement is made in such a way that no conflict with low energy restrictions is encountered. The SU(2) L singlet mediates interactions between the right–handed leptons and u...

In this paper, the extreme problem for Orlicz and Lq torsional rigidity is discussed. Moreover, we introduce geominimal rigidity, which defined as being motivated through L' mixed rigidity. Also, invariance of under orthogonal matrices proved, isoperimetric type inequality circular are established well.

Let Sβ := {z ∈ C : |Imz| < β} be a strip in complex plane. H̃r ∞,β denotes those 2π-periodic, real-valued functions on R which are analytic in the strip Sβ and satisfy the condition |f (r)(z)| ≤ 1, z ∈ Sβ . Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel′fand, and information n-widths of H̃r ∞,β in L∞[0, 2π], r = 0, 1, 2, . . ., and 2n-widths of H̃r ∞,β in Lq [0, ...