Refinement of convergence rate for the strong law of large numbers in Banach space

Let ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0001.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0001.gif"}" /> be a sequence of independent and identically distributed <i>Bi>-valued random variables, and set ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0002.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0002.gif"}" />. Let ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0003.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0003.gif"}" />, ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0004.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0004.gif"}" /> and <i>qi>>0, and putisp-formula-group id="um0001">ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_um0001.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_um0001.gif"}" />isp-formula-group>We strengthen the convergence rate for the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers in Banach space, by showing that ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0005.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0005.gif"}" />, if and only if ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0006.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_ilm0006.gif"}" /> andisp-formula-group id="um0002">ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_um0002.gif" alt=" />ipt><img src="//:0" alt=" class="no-mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2014/gssr20.v086.i06/17442508.2014.883078/20141024/images/gssr_a_883078_um0002.gif"}" />isp-formula-group>     

Limiting behaviour for arrays of rowwise widely orthant dependent random variables under conditions of R − h-integrability and its applications

ABSTRACT

In this paper, we mainly study the ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gssr20/2019/gssr20.v091.i06/17442508.2018.1563605/20190728/images/gssr_a_1563605_ilm0001.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gssr20/2019/gssr20.v091.i06/17442508.2018.1563605/20190728/images/gssr_a_1563605_ilm0001.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i>Li>i>ri> convergence, complete convergence and complete moment convergence for arrays of rowwise widely orthant dependent (WOD, for short) random variables under some conditions of <i>Ri>-<i>hi>-integrability. The results in this paper generalize and improve the corresponding ones of Sung <i>et al.i>, <i>Weak laws of large numbers for arrays under a condition of uniform integrabilityi>, Journal of the Korean Mathematical Society 45 (2008), pp. 289–300, Wang and Hu, <i>Weak laws of large numbers for arrays of dependent random variablesi>, Stochastics: An International Journal of Probability and Stochastic Processes 86 (2014), pp. 759–775, and Wu <i>et al.i>, <i>Limiting behaviour for arrays of rowwise END random variables under conditions of <i>hi>-integrabilityi>, Stochastics: An International Journal of Probability and Stochastic Processes 87 (2014), pp. 409–423. As applications of the complete convergence that we established, we present some results on complete consistency for the estimator in a nonparametric regression model based on WOD errors.     

Eight expressions for generalized inverses of a bordered matrix

A bordered matrix is a two-by-two partitioned matrix with its lower-right corner equal to a null matrix. In this article, we present eight partitioned matrices consisting of the Moore–Penrose inverses of submatrices in a bordered matrix, and give necessary and sufficient conditions for the eight partitioned matrices to be generalized inverses of the bordered matrix.     

The lower bound of the number of nonabelian subgroups of finite p-groups

In this paper, we give the lower bound of the number of nonabelian subgroups of possible index of finite <i>pi>-groups, and classify the finite <i>pi>-groups such that the number of nonabelian subgroups of possible order are exactly the lower bound.     

Spectral isoperimetric inequalities for singular interactions on open arcs

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i08/00036811.2018.1430778/20190508/images/gapa_a_1430778_ilm0001.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i08/00036811.2018.1430778/20190508/images/gapa_a_1430778_ilm0001.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i>δi>-interaction supported on an open arc with two free endpoints. Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment, the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them. As a consequence of the result for ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i08/00036811.2018.1430778/20190508/images/gapa_a_1430778_ilm0002.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i08/00036811.2018.1430778/20190508/images/gapa_a_1430778_ilm0002.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i>δi>-interaction, we obtain that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue of the Robin Laplacian on a plane with a slit along an open arc of fixed length.     

Weighted Hardy’s inequalities and Kolmogorov-type operators

We give general conditions to state the weighted Hardy inequality ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_um0001.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_um0001.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />img="eq-00001.gif">ign="right">i>ci>∫i mathvariant="double-struck">Ri>i>Ni>i mathvariant="italic">φi>2|i>xi>|2i mathvariant="normal">di>i mathvariant="italic">μi>≤∫i mathvariant="double-struck">Ri>i>Ni>|i mathvariant="normal">?i>i mathvariant="italic">φi>|2i mathvariant="normal">di>i mathvariant="italic">μi>+i>Ci>∫i mathvariant="double-struck">Ri>i>Ni>i mathvariant="italic">φi>2i mathvariant="normal">di>i mathvariant="italic">μi>,idth="1em">i mathvariant="italic">φi>∈i>Ci>i>ci>i>∞i>(i mathvariant="double-struck">Ri>i>Ni>),idth="0.166667em">i>ci>≤i>ci>0,i mathvariant="italic">μi>,

with respect to a probability measure ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0001.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0001.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i mathvariant="normal">di>i mathvariant="italic">μi>. Moreover, the optimality of the constant ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0002.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0002.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i>ci>0,i mathvariant="italic">μi> is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_um0002.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_um0002.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />img="eq-00004.gif">ign="right">i>Li>i>ui>+i>Vi>i>ui>=i mathvariant="normal">Δi>i>ui>+i mathvariant="normal">?i>i mathvariant="italic">μi>i mathvariant="italic">μi>·i mathvariant="normal">?i>i>ui>+i>ci>|i>xi>|2i>ui>

for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0003.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0003.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i mathvariant="normal">di>i mathvariant="italic">μi> allow the drift term to be of type ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0004.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0004.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i mathvariant="normal">?i>i mathvariant="italic">μi>i mathvariant="italic">μi>=-|i>xi>|i>mi>-2i>xi> with ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0005.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i07/00036811.2017.1419200/20190410/images/gapa_a_1419200_ilm0005.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i>mi>>0.     

Asymptotic analysis of a size-structured population model with infinite states-at-birth

In this paper, we study a size-structured population model with infinite states-at-birth and distributed delay in birth process. We are interested in the asymptotic behavior of the considered system and, particularly, in the effect of time lag on the long-term dynamics. To this end, we formally linearize the system around a steady state and study the linearized system using ipt><img src="/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i05/00036811.2017.1408075/20190226/images/gapa_a_1408075_ilm0001.gif" alt=" />ipt><img src="//:0" alt=" class="mml-formula" data-formula-source="{"type" : "image", "src" : "/na101/home/literatum/publisher/tandf/journals/content/gapa20/2019/gapa20.v098.i05/00036811.2017.1408075/20190226/images/gapa_a_1408075_ilm0001.gif"}" /><img src="//:0" alt=" data-formula-source="{"type" : "mathjax"}" />if">i>Ci>0-semigroup framework and spectral analysis methods. In this manner, we achieve some results on linearized stability, instability, and asynchronous exponential growth for the system under some conditions. The obtained results are also illustrated by some examples and simulations.     

A few level approach for the electronic partition function of atomic systems

A simplified model to calculate partition functions and thermodynamic properties of atomic species is presented. This model consists in grouping the atomic states in few virtual levels. Their statistical weights and energies are calculated summing or mediating over the states belonging to each group. The partition function is calculated considering the virtual levels to follow the Boltzmann distribution. A theoretical foundation of the model has been described for a general case and verified for hydrogen, oxygen and nitrogen atoms. Two- and three-level models are adequate to keep the error within a few percent. A second order correction term can also be added to further reduce the error without changing the advantages of the model.     

离子液体中一步合成14-芳基-1,6,7,14-四氢二苯并[a, i]吖啶-1,6-二酮衍生物

室温条件下在离子液体[bmim]BF4中合成了14-芳基-1,6,7,14-四氢二苯并[a, i]吖啶-1,6-二酮类衍生物,该方法具有产率高,反应介质可回收使用,反应底物范围广,反应时间短,操作简单等优点。     

On perfect Γ‐decompositions of the complete graph

Generalizing the well‐known concept of an <i>ii>‐perfect cycle system, Pasotti [Pasotti, in press, Australas J Combin] defined a Γ‐decomposition (Γ‐factorization) of a complete graph <i>Ki>_<i>vi> to be <i>i‐perfecti> if for every edge [<i>xi>, <i>yi>] of <i>Ki>_<i>vi> there is exactly one block of the decomposition (factor of the factorization) in which <i>xi> and <i>yi> have distance <i>ii>. In particular, a Γ‐decomposition (Γ‐factorization) of <i>Ki>_<i>vi> that is <i>ii>‐perfect for any <i>ii> not exceeding the diameter of a connected graph Γ will be said a Steiner (Kirkman) Γ‐system of order <i>vi>. In this article we first observe that as a consequence of the deep theory on <i>decompositions of edge‐colored graphsi> developed by Lamken and Wilson [Lamken and Wilson, 2000, J Combin Theory Ser A 89, 149–200], there are infinitely many values of <i>vi> for which there exists an <i>ii>‐perfect Γ‐decomposition of <i>Ki>_<i>vi> provided that Γ is an <i>i‐equidistance graphi>, namely a graph such that the number of pairs of vertices at distance <i>ii> is equal to the number of its edges. Then we give some concrete direct constructions for elementary abelian Steiner Γ‐systems with Γ the wheel on 8 vertices or a circulant graph, and for elementary abelian 2‐perfect cube‐factorizations. We also present some recursive constructions and some results on 2‐transitive Kirkman Γ‐systems. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 197–209, 2009