کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4599643 | 1631149 | 2014 | 46 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
Extended-valued topical and anti-topical functions on semimodules
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
![عکس صفحه اول مقاله: Extended-valued topical and anti-topical functions on semimodules Extended-valued topical and anti-topical functions on semimodules](/preview/png/4599643.png)
چکیده انگلیسی
In the papers [16] and [17] we have studied functions defined on a b-complete idempotent semimodule X over a b-complete idempotent semifield K=(K,â,â), with values in K, where K may (or may not) contain a greatest element supK, and the residuation x/y is not defined for xâX and y=infX. In the present paper we assume that K has no greatest element, then adjoin to K an outside “greatest element” â¤=supK and extend the operations â and â from K to K¯:=Kâª{â¤}, so as to obtain a meaning also for x/infX, for any xâX, and study functions with values in K¯. In fact we consider two different extensions of the product â from K to K¯, denoted by â and âË respectively, and use them to give characterizations of topical (i.e. increasing homogeneous, defined with the aid of â) and anti-topical (i.e. decreasing anti-homogeneous, defined with the aid of âË) functions in terms of some inequalities. Next we introduce and study for functions f:XâK¯ their conjugates and biconjugates of Fenchel-Moreau type with respect to the coupling functions Ï(x,y)=x/y, âx,yâX, and Ï(x,(y,d)):=inf{x/y,d}, âx,yâX, âdâK¯, and use them to obtain characterizations of topical and anti-topical functions. In the subsequent sections we consider for the coupling functions Ï and Ï some concepts that have been studied in Rubinov and Singer (2001) [11] and Singer (2004) [15] for the so-called “additive min-type coupling functions” Ïμ:RmaxnÃRmaxnâRmax and Ïμ:AnÃAnâA respectively, where A is a conditionally complete lattice ordered group and Ïμ(x,y):=inf1⩽i⩽n(xi+yi), âx,yâRmaxn (or An). Thus, we study the polars of a set GâX for the coupling functions Ï and Ï, and we consider for a function f:XâK¯ the notion of support set of f with respect to the set TË of all “elementary topical functions” tËy(x):=x/y, âxâX, âyâX\{infX} and two concepts of support set of f at a point x0âX. The main differences between the properties of the conjugations with respect to the coupling functions Ï,Ï and Ïμ and between the properties of the polars of a set G with respect to the coupling functions Ï,Ï and Ïμ are caused by the fact that while Ïμ is symmetric, with values only in Rmax (resp. A), Ï and Ï are not symmetric and take values also outside Rmax (resp. A).
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 446, 1 April 2014, Pages 25-70
Journal: Linear Algebra and its Applications - Volume 446, 1 April 2014, Pages 25-70
نویسندگان
Ivan Singer, Viorel Nitica,