{"title":"Some investigations on the U-partial order induced by uninorms","authors":"Emel Aşıcı, Radko Mesiar","doi":"10.1007/s00010-024-01048-2","DOIUrl":"10.1007/s00010-024-01048-2","url":null,"abstract":"<div><p>In this paper, we study on the <span>( preceq _U )</span>-partial order induced by uninorms. We also investigate some properties direct product of two uninorms on bounded lattices. We investigate some properties of the set of incomparable elements with respect to <span>( preceq _U )</span>-partial. Then, we investigate the relation between the set of comparable elements and the distributivity property for uninorms on the unit interval [0, 1] . Finally, we define a total order induced by internal uninorms.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 4","pages":"1039 - 1052"},"PeriodicalIF":0.9,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On when the union of two algebraic sets is algebraic","authors":"Erhard Aichinger, Mike Behrisch, Bernardo Rossi","doi":"10.1007/s00010-024-01041-9","DOIUrl":"10.1007/s00010-024-01041-9","url":null,"abstract":"<div><p>In universal algebraic geometry, an algebra is called an <i>equational domain</i> if the union of two algebraic sets is algebraic. We characterize equational domains, with respect to polynomial equations, inside congruence permutable varieties, and with respect to term equations, among all algebras of size two and all algebras of size three with a cyclic automorphism. Furthermore, for each size at least three, we prove that, modulo term equivalence, there is a continuum of equational domains of that size.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"107 - 154"},"PeriodicalIF":0.9,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01041-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"General hyperstability criteria for Jensen-type equations","authors":"Dan M. Dăianu","doi":"10.1007/s00010-024-01066-0","DOIUrl":"10.1007/s00010-024-01066-0","url":null,"abstract":"<div><p>We provide two large classes of control functions that ensure the hyperstability of Jensen-type equations on restricted domains. Among other consequences, we obtain improvements on similar results known in the literature for some Aoki–Rassias-type control functions and hyperstability results for some equations whose solutions are Jensen functions compounded with radical-type functions or with trigonometric functions.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 5","pages":"1351 - 1371"},"PeriodicalIF":0.9,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01066-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On total isolation in graphs","authors":"Geoffrey Boyer, Wayne Goddard, Michael A. Henning","doi":"10.1007/s00010-024-01057-1","DOIUrl":"https://doi.org/10.1007/s00010-024-01057-1","url":null,"abstract":"<p>An isolating set in a graph is a set <i>S</i> of vertices such that removing <i>S</i> and its neighborhood leaves no edge; it is total isolating if <i>S</i> induces a subgraph with no vertex of degree 0. We show that most graphs have a partition into two disjoint total isolating sets and characterize the exceptions. Using this we show that apart from the 7-cycle, every connected graph of order <span>(nge 4)</span> has a total isolating set of size at most <i>n</i>/2, which is best possible.\u0000</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"30 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Twisted coefficients on coarse spaces and their corona","authors":"Elisa Hartmann","doi":"10.1007/s00010-024-01061-5","DOIUrl":"10.1007/s00010-024-01061-5","url":null,"abstract":"<div><p>To a metric space <i>X</i> we associate a compact topological space <span>(nu '({X}))</span> called the corona of <i>X</i>. Then a coarse map <span>(f:Xrightarrow Y)</span> between metric spaces is mapped to a continuous map <span>(nu '({f}):nu '({X})rightarrow nu '({Y}))</span> between coronas. Sheaf cohomology assigned to a coarse metric space is preserved and reflected by the corona functor. This work reveals new tools to analyze the Higson corona.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 4","pages":"1099 - 1114"},"PeriodicalIF":0.9,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01061-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Position of the centroid of a planar convex body","authors":"Marek Lassak","doi":"10.1007/s00010-024-01058-0","DOIUrl":"10.1007/s00010-024-01058-0","url":null,"abstract":"<div><p>It is well known that any planar convex body <i>A</i> permits to inscribe an affine-regular hexagon <span>(H_A)</span>. We prove that the centroid of <i>A</i> belongs to the homothetic image of <span>(H_A)</span> with ratio <span>(frac{4}{21})</span> and the center in the center of <span>(H_A)</span>. This ratio cannot be decreased.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 3","pages":"687 - 695"},"PeriodicalIF":0.9,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01058-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140669872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Manisha Devi, Jaspal Singh Aujla, Mohsen Kian, Mohammad Sal Moslehian
{"title":"Matrix inequalities between $$f(A)sigma f(B)$$ and $$Asigma B$$","authors":"Manisha Devi, Jaspal Singh Aujla, Mohsen Kian, Mohammad Sal Moslehian","doi":"10.1007/s00010-024-01059-z","DOIUrl":"https://doi.org/10.1007/s00010-024-01059-z","url":null,"abstract":"<p>Let <i>A</i> and <i>B</i> be <span>(ntimes n)</span> positive definite complex matrices, let <span>(sigma )</span> be a matrix mean, and let <span>(f: [0,infty )rightarrow [0,infty ))</span> be a differentiable convex function with <span>(f(0)=0)</span>. We prove that </p><span>$$begin{aligned} f^{prime }(0)(A sigma B)le frac{f(m)}{m}(Asigma B)le f(A)sigma f(B)le frac{f(M)}{M}(Asigma B)le f^{prime }(M)(Asigma B), end{aligned}$$</span><p>where <i>m</i> represents the smallest eigenvalues of <i>A</i> and <i>B</i> and <i>M</i> represents the largest eigenvalues of <i>A</i> and <i>B</i>. If <i>f</i> is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if <i>f</i>(<i>x</i>)/<i>x</i> is increasing, then </p><span>$$begin{aligned} |||f(A)+f(B)|||le frac{f(M)}{M} |||A+B|||le |||f(A+B)||| end{aligned}$$</span><p>holds for all <i>A</i> and <i>B</i> with <span>(Mle A+B)</span>. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski’s determinant inequality.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"67 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
José Pedro Gaivão, Michel Laurent, Arnaldo Nogueira
{"title":"Rotation number of 2-interval piecewise affine maps","authors":"José Pedro Gaivão, Michel Laurent, Arnaldo Nogueira","doi":"10.1007/s00010-024-01064-2","DOIUrl":"https://doi.org/10.1007/s00010-024-01064-2","url":null,"abstract":"<p>We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps <span>(f_{varvec{p}})</span> are parametrized by a quintuple <span>(varvec{p})</span> of real numbers satisfying inequations. Viewing <span>(f_{varvec{p}})</span> as a circle map, we show that it has a rotation number <span>(rho (f_{varvec{p}}))</span> and we compute <span>(rho (f_{varvec{p}}))</span> as a function of <span>(varvec{p})</span> in terms of Hecke–Mahler series. As a corollary, we prove that <span>(rho (f_{varvec{p}}))</span> is a rational number when the components of <span>(varvec{p})</span> are algebraic numbers.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Set valued pexiderized quadratic functional equation","authors":"Elham Mohammadi, Abbas Najati, Kazimierz Nikodem","doi":"10.1007/s00010-024-01067-z","DOIUrl":"10.1007/s00010-024-01067-z","url":null,"abstract":"<div><p>Consider a real vector space denoted as <i>X</i>, and let <i>cc</i>(<i>Y</i>) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space <i>Y</i>. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation </p><div><div><span>$$begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), end{aligned}$$</span></div></div><p>for unknown functions <span>(f_1,f_2,f_3,f_4:Xrightarrow cc(Y))</span>. This functional equation incorporates many functional equations including the quadratic, Cauchy’s and Drygas’ equations. A characterization for set-valued solutions of this functional equation is presented in this paper.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"243 - 255"},"PeriodicalIF":0.9,"publicationDate":"2024-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalizing the concept of bounded variation","authors":"Angshuman R. Goswami","doi":"10.1007/s00010-024-01050-8","DOIUrl":"https://doi.org/10.1007/s00010-024-01050-8","url":null,"abstract":"<p>Let <span>([a,b]subseteq mathbb {R})</span> be a non-empty and non singleton closed interval and <span>(P={a=x_0<cdots <x_n=b})</span> is a partition of it. Then <span>(f:Irightarrow mathbb {R})</span> is said to be a function of <i>r</i>-bounded variation, if the expression <span>(sum nolimits ^{n}_{i=1}|f(x_i)-f(x_{i-1})|^{r})</span> is bounded for all possible partitions like <i>P</i>. One of the main results of the paper deals with the generalization of the classical Jordan decomposition theorem. We establish that for <span>(rin ]0,1])</span>, a function of <i>r</i>-bounded variation can be written as the difference of two monotone functions. While for <span>(r>1)</span>, under minimal assumptions such a function can be treated as an approximately monotone function which can be closely approximated by a nondecreasing majorant. We also prove that for <span>(0<r_1<r_2)</span>, the function class of <span>(r_1)</span>-bounded variation is contained in the class of functions satisfying <span>(r_2)</span>-bounded variations. We go through approximately monotone functions and present a possible decomposition for <span>(f:I(subseteq mathbb {R}_+)rightarrow mathbb {R})</span> satisfying the functional inequality </p><span>$$f(x)le f(x)+(y-x)^{p}quad (x,yin I text{ with } x<y text{ and } pin ]0,1[ ).$$</span><p>A generalized structural study has also been done in that specific section. On the other hand, for <span>(ell [a,b]ge d)</span>, a function satisfying the following monotonic condition under the given assumption will be termed as <i>d</i>-periodically increasing </p><span>$$f(x)le f(y)quad text{ for } text{ all }quad x,yin Iquad text{ with }quad y-xge d.$$</span><p>We establish that in a compact interval any function satisfying <i>d</i>-bounded variation can be decomposed as the difference of a monotone and a <i>d</i>-periodically increasing function. The core details related to past results, motivation, structure of each and every section are thoroughly discussed below.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"116 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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