Two topics are presented: synchronization games and synchronization costs. In a synchronization game on a deterministic finite automaton, there are two players, Alice and Bob, whose moves alternate. Alice wants to synchronize the given automaton, while Bob aims to make her task as hard as possible. We answer a few natural questions related to such games. Speaking about synchronization costs, we consider deterministic automata in which each transition has a certain price. The problem is whether or not a given automaton can be synchronized within a given budget. We determine the complexity of this problem.
Consider a non-negative self-adjoint operator H in L-2 (R-d). We suppose that its heat operator e(-tH) satisfies an off-diagonal algebraic decay estimate, for some exponents p(0) is an element of [0, 2). Then we prove sharp L-p -> L-p frequency truncated estimates for the Schrodinger group e(itH) for p is an element of [p(0), p'(0)]. In particular, our results apply to every operator of the form H =3D (i del + A)(2) + V, with a magnetic potential A is an element of L-loc(2)(R-d, R-d) and an electric potential V whose positive and negative parts are in the local Kato class and in the Kato class, respectively.
Wen, Fei
Liu, Peilin
Liu, Yipeng
Qiu, Robert C.
Yu, Wenxian
This paper addresses the issue of robust sparse recovery in compressive sensing (CS) in the presence of impulsive measurement noise. Recently, robust data-fitting models, such as l(1)-norm, Lorentzian-norm, and Huber penalty function, have been employed to replace the popular l(2)-norm loss model to gain more robust performance. In this paper, we propose a robust formulation for sparse recovery using the generalized l(p)-norm with 0 <=3D p < 2 as the metric for the residual error. To solve this formulation efficiently, we develop an alternating direction method (ADM) via incorporating the proximity operator of l(p)-norm functions into the framework of augmented Lagrangian methods. Furthermore, to derive a convergent method for the nonconvex case of p < 1, a smoothing strategy has been employed. The convergence conditions of the proposed algorithm have been analyzed for both the convex and nonconvex cases. The new algorithm has been compared with some state-of-the-art robust algorithms via numerical simulations to show its improved performance in highly impulsive noise.
This note is a companion to the article On the mutually non isomorphic l(p)(l(q)) spaces published in this journal, in which P. Cembranos and J. Mendoza showed that {l(p)(l(q)) : 1 <=3D p, q <=3D infinity} is a collection of mutually non isomorphic Banach spaces [5]. We now complete the picture by allowing the non-locally convex relatives to be part of their natural family and see that, in fact, no two members of the extended class {l(p)(l(q)) : 0 < p, q <=3D infinity} are isomorphic. Our approach is novel in the sense that we reach the isomorphism obstructions from the perspective of bases techniques and the different convexities of the spaces, both methods being intrinsic to quasi-Banach spaces. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
We describe a presentation for the augmented fundamental rack of a link in the lens space L(p, 1). Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in L(p, 1). In this case, the counting rack invariants also include the information about the action of pi(1) (L(p, 1)) on the augmented fundamental rack of a link.
In this letter, we formulate sparse subspace clustering as a smoothed l(p) (0 < p < 1) minimization problem (SSC-SLp) and present a unified formulation for different practical clustering problems by introducing a new pseudo norm. Generally, the use of l(p) (0 < p < 1) norm approximating the l(0) one can lead to a more effective approximation than the l(1) norm, while the l(p)-regularization also causes the objective function to be non-convex and non-smooth. Besides, better adapting to the property of data representing real problems, the objective function is usually constrained by multiple factors (such as spatial distribution of data and errors). In view of this, we propose a computationally efficient method for solving the multi-constrained non-smooth l(p) minimization problem, which smooths the l(p) norm and minimizes the objective function by alternately updating a block (or a variable) and its weight. In addition, the convergence of the proposed algorithm is theoretically proven. Extensive experimental results on real datasets demonstrate the effectiveness of the proposed method. (C) 2019 Published by Elsevier B.V.
In this paper, we show that, for each p > 1, there are continuum many Borel equivalence relations between R-omega/l(1) and R-omega/l(p) ordered by <=(B) which are pairwise Borel incomparable.
Katarzyna Bączek
Jarosław L. Przybył
Małgorzata Mirgos
Olga Kosakowska
Izabela Szymborska-Sandhu
Zenon Węglarz
Primula veris L. and Primula elatior (L.) Hill represent medicinal plants used for the production of herbal teas and preparations with antioxidant and expectorant activity. Flowers and roots of both species possess the same biological activity. In the presented study, raw materials of wild growing P. veris and P. elatior were compared in terms of the content and composition of phenolic compounds using a fast and simple HPLC-DAD method. The study showed that flowers of both species were rich in flavonoids. However, P. veris flowers were characterized with a distinctly higher content of isorhamnetin-3-O-glucoside, astragalin, and (+)-catechin, whereas P. elatior occurred to be a richer source of rutoside and isorhamnetin-3-O-rutinoside. Hyperoside was found exclusively in P. elatior flowers. Phenolic glycosides (primverin and primulaverin) were identified only in the roots. Their content was about ten times higher in P. veris in comparison with P. elatior underground organs. The obtained results clearly show that both Primula species differ distinctly in terms of the content and composition of phenolic compounds. The compounds differentiating both species to the highest degree (hyperoside, in flowers, as well as primverin and primulaverin, in the roots) may be useful chemical markers in the identification and evaluation of both species.
The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of L-P optimization. These generalized L-p-quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for 1 < p < 2. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate L-p-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme L-p-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme L-p-quantiles and elaborate the choice of p through applications to some simulated and financial real data.
Let T be a dual integrable representation of a countable discrete LCA group C, acting on a Hilbert space H. We consider the problem of characterizing l(P)(G)-linear independence (p not equal 2) of the system {T-k psi : k is an element of C} for the given psi, is an element of H, which we previously studied in the context of the integer translates of a square integrable function. The extensions of the known results for translates to this setting are obtained using a slightly different approach, through which we show that, under certain conditions, this problem is related to the 'Wiener's closure of translates' problem and the problem of the existence of p-zero divisors, arising around the zero divisor conjecture in algebra. Using this connection, we also obtain several improvements for the case of the integer translates.
Using a recent result of Batson, Spielman and Srivastava, we obtain a tight estimate on the dimension of l(p)(n), p an even integer, needed to almost isometrically contain all k-dimensional subspaces of L(p).