15 January 2019, Volume 35 Issue 1

    

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  A Partial First-Order Affine-Scaling Method

  Ran GU, Ya Xiang YUAN

  Acta Mathematica Sinica. 2019, 35(1): 1-16. https://doi.org/10.1007/s10114-017-7097-z7

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  We present a partial first-order affine-scaling method for solving smooth optimization with linear inequality constraints. At each iteration, the algorithm considers a subset of the constraints to reduce the complexity. We prove the global convergence of the algorithm for general smooth objective functions, and show it converges at sublinear rate when the objective function is quadratic. Numerical experiments indicate that our algorithm is efficient.
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  Statistical Estimation of the Shannon Entropy

  Alexander BULINSKI, Denis DIMITROV

  Acta Mathematica Sinica. 2019, 35(1): 17-46. https://doi.org/10.1007/s10114-018-7440-z

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  The behavior of the Kozachenko-Leonenko estimates for the (differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L²-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy-Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.
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  Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

  Peter BEELEN, Mrinmoy DATTA, Sudhir R. GHORPADE

  Acta Mathematica Sinica. 2019, 35(1): 47-63. https://doi.org/10.1007/s10114-018-8024-7

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  We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gröbner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre's inequality for the number of points of projective hypersurfaces over finite fields is included.
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  Modelling, Analysis and Computation in Plasticity

  B. Daya REDDY

  Acta Mathematica Sinica. 2019, 35(1): 64-82. https://doi.org/10.1007/s10114-018-7477-z

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  The typical problem in the mechanics of deformable solids comprises a mathematical model in the form of systems of partial differential equations or inequalities. Subsequent investigations are then concerned with analysis of the model to determine its well-posedness, followed by the development and implementation of algorithms to obtain approximate solutions to problems that are generally intractable in closed form. These processes of modelling, analysis, and computation are discussed with a focus on the behaviour of elastic-plastic bodies; these are materials which exhibit path-dependence and irreversibility in their behaviour. The resulting variational problem is an inequality that is not of standard elliptic or parabolic type. Properties of this formulation are reviewed, as are the conditions under which fully discrete approximations converge. A solution algorithm, motivated by the predictor-corrector algorithms that are common in elastoplastic problems, is presented and its convergence properties summarized. An important extension of the conventional theory is that of straingradient plasticity, in which gradients of the plastic strain appear in the formulation, and which includes a length scale not present in the conventional theory. Some recent results for strain-gradient plasticity are presented, and the work concludes with a brief description of current investigations.
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  Analytic Fragmentation Semigroups and Classical Solutions to Coagulation-fragmentation Equations-a Survey

  Jacek BANASIAK

  Acta Mathematica Sinica. 2019, 35(1): 83-104. https://doi.org/10.1007/s10114-018-7435-9

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  In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation-coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentation-coagulation equations.
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  Periodic Solutions of Third-order Differential Equations with Finite Delay in Vector-valued Functional Spaces

  Shang Quan BU, Gang CAI

  Acta Mathematica Sinica. 2019, 35(1): 105-122. https://doi.org/10.1007/s10114-018-8001-1

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  In this paper, we study the well-posedness of the third-order differential equation with finite delay (P₃):αu"'(t) + u"(t)=Au(t) + Bu' (t) + Fu_t + f(t)(t ∈ T:=[0, 2π]) with periodic boundary conditions u(0)=u(2π), u' (0)=u' (2π), u"(0)=u" (2π), in periodic Lebesgue-Bochner spaces L^p(T; X) and periodic Besov spaces B_p,q^s (T; X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B)≠ {0}, α≠ 0 is a fixed constant and F is a bounded linear operator from L^p([-2π, 0]; X) (resp. B_p,q^s ([-2π, 0]; X)) into X, u_t is given by u_t(s)=u(t + s) when s ∈[-2π, 0]. Necessary and sufficient conditions for the L^p-well-posedness (resp. B_p,q^s -well-posedness) of (P₃) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.
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  Stability Characterizations of ∈-isometries on Certain Banach Spaces

  Li Xin CHENG, Long Fa SUN

  Acta Mathematica Sinica. 2019, 35(1): 123-134. https://doi.org/10.1007/s10114-018-8038-1

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  Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-isometry f:X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y^*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ∈-isometry to be stable in assuming that N is ω^*-closed in Y^*. Making use of this result, we improve several known results including Figiel's theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f) ≡ span[f(X)] contains a complemented linear isometric copy of X; Moreover, if X=Y, then for every ∈-isometry f:X → X, there exists a surjective linear isometry S:X → X such that f -S is uniformly bounded by 2∈ on X.
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  A Tree-valued Markov Process Associated with an Admissible Family of Branching Mechanisms

  Hong Wei BI, Hui HE

  Acta Mathematica Sinica. 2019, 35(1): 135-160. https://doi.org/10.1007/s10114-018-8049-y

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  Let T ⊂ R be an interval. By studying an admissible family of branching mechanisms {ψ_t, t ∈ T} introduced in Li[Ann. Probab., 42, 41-79 (2014)], we construct a decreasing Lévy-CRTvalued process {T_t, t ∈ T} by pruning Lévy trees accordingly such that for each t ∈ T, T_t is a ψ_t-Lévy tree. We also obtain an analogous process {T_t^*, t ∈ T} by pruning a critical Lévy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {T_t, t ∈ T} at the ascension time A:=inf{t ∈ T; T_t is finite} can be represented by {T_t^*, t ∈ T}. The results generalize those studied in Abraham and Delmas[Ann. Probab., 40, 1167-1211 (2012)].