Aequationes Mathematicae最新文献

{"title":"A class of series representations for Catalan’s constant","authors":"HORST ALZER,&nbsp;MAN KAM KWONG","doi":"10.1007/s00010-026-01278-6","DOIUrl":"10.1007/s00010-026-01278-6","url":null,"abstract":"<div><p>Let </p><div><div><span>$$ S(j)= sum _{nu =1}^infty frac{nu }{16^nu (2nu -1)^2 (2nu +1)(2nu +j)}{2nu atopwithdelims ()nu }^2, quad jin {2,3,4,... }. $$</span></div></div><p>In 2022, N. Bhandari showed that for <span>(jin {3,4,5})</span> there are rational numbers <span>(a_j)</span> and <span>(b_j)</span> such that </p><div><div><span>$$ 4 pi S(j)=a_j-b_ jG, $$</span></div></div><p>where <i>G</i> denotes the Catalan constant. He conjectured that this representation holds for all <span>(jge 3)</span>. Here, we prove this conjecture. More precisely, we offer recursion formulas to determine the numbers <span>(a_j)</span> and <i>bj</i> <span>((jge 3))</span> explicitly.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147621130","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":"A generalization of the P-core–EP inverse","authors":"Dijana Mosić","doi":"10.1007/s00010-026-01282-w","DOIUrl":"10.1007/s00010-026-01282-w","url":null,"abstract":"<div><p>The aim of this paper is to generalize the system of matrix equations for defining the P-core–EP inverse based on a minimal rank weak Drazin inverse. Solving new system, we present the concept of the weak P-core–EP inverse as a new wider class of generalized inverses involving Drazin inverse, P-core–EP inverse and CPO-inverse. We investigate many characterizations and expressions of the weak P-core–EP inverse. Applying the weak P-core–EP inverse, we solve certain linear equations. As consequences of our results, we get new properties of some special types of weak P-core–EP inverse.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147621136","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":"Solutions to the operator equation (f(XX^*)=g(X^*X))","authors":"Rasoul Eskandari,&nbsp;Mohammad Sal Moslehian,&nbsp;Ali Talebi","doi":"10.1007/s00010-026-01286-6","DOIUrl":"10.1007/s00010-026-01286-6","url":null,"abstract":"<div><p>The characterization of special operators (or matrices), such as normal and self-adjoint (Hermitian) ones, through operator (or matrix) equations has been a topic of significant interest for a long time. In this paper, we explore equations involving products of an operator <i>X</i> and its adjoint <span>(X^*)</span>. We provide new characterizations of normal and Hermitian matrices using general matrix equations. We then extend and improve these results to operators on Hilbert spaces of any dimension. Moreover, we establish further characterizations of normal and self-adjoint operators (and matrices) based on our findings.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147606690","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":"A characterization of the set of all integers by a composite functional equation","authors":"Soichi Ikeda","doi":"10.1007/s00010-026-01284-8","DOIUrl":"10.1007/s00010-026-01284-8","url":null,"abstract":"<div><p>Let <span>(f: mathbb {R} rightarrow mathbb {R})</span>. The Heaviside function <i>H</i> is defined by </p><div><div><span>$$ H(x) = {left{ begin{array}{ll} 1 quad (x &gt; 0) 0 quad (x le 0). end{array}right. } $$</span></div></div><p>In this paper, we study the composite functional equation </p><div><div><span>$$begin{aligned} f(x + f(y)) = (x + f(y)) G_f (x), end{aligned}$$</span></div></div><p>where </p><div><div><span>$$begin{aligned} G_f (x) = f(H(x) + f(x) - x) + f(H(1 - x) + f(1 - x) - (1 - x)). end{aligned}$$</span></div></div><p>Define the function <span>(I: mathbb {R} rightarrow mathbb {R})</span> by setting </p><div><div><span>$$ I(x) = {left{ begin{array}{ll} x quad (x in mathbb {Z}) 0 quad (x in mathbb {R} setminus mathbb {Z}). end{array}right. } $$</span></div></div><p>We prove that <span>(f = I)</span> holds in the case <span>(f(1) ne 0)</span>. We can regard this fact as a characterization of <span>(mathbb {Z}.)</span></p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01284-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147607228","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":"Global asymptotic stability of one time-continuous model and its time-discrete variant for basic virus dynamics","authors":"Benjamin Wacker","doi":"10.1007/s00010-026-01285-7","DOIUrl":"10.1007/s00010-026-01285-7","url":null,"abstract":"<div><p>In this article, we investigate global asymptotic stability of both equilibria, namely the virus-free and the virus-endemic equilibria, of a classical within-host HIV dynamical model. At first, we show global asymptotic stability of these equilibria for the time-continuous setting by introducing two suitable Lyapunov-functions. Afterwards, we introduce a time-discrete variant of this model by applying the implicit Eulerian time-stepping scheme and demonstrate that it can be recasted in an explicit manner. As our main contribution, we show that the same Lyapunov-functions of the time-continuous model transfer to the time-discrete case to establish global asymptotic stability of both equilibria where even non-equidistant time grids can be used. Finally, we compare our suggested non-standard finite-diffence-method with the classical explicit Eulerian time-stepping method, one second-order Runge-Kutta time-stepping scheme and an explicit-implicit non-standard finite-difference-method. We see that only both non-standard finite-difference-methods conserve non-negativity for arbitrary time-step sizes and that they behave similarly for small time-step sizes.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01285-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147561524","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 conditions determining formal automorphisms of integro-differential operators","authors":"Alberto Lastra,&nbsp;Sławomir Michalik,&nbsp;Maria Suwińska","doi":"10.1007/s00010-026-01281-x","DOIUrl":"10.1007/s00010-026-01281-x","url":null,"abstract":"<div><p>Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems in different settings. The main achievements provide a three-statement result not only in the framework of general formal power series, but also on subspaces appearing in applications such as Gevrey settings and moment differential operators.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01281-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147561501","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":"Graceful Antimagic Injection via Combinatorial Nullstellensatz","authors":"R. Y. Wulandari,&nbsp;R. Simanjuntak,&nbsp;S. W. Saputro","doi":"10.1007/s00010-026-01267-9","DOIUrl":"10.1007/s00010-026-01267-9","url":null,"abstract":"&lt;div&gt;&lt;p&gt;For a simple graph &lt;i&gt;G&lt;/i&gt;, a &lt;i&gt;graceful antimagic injection&lt;/i&gt; &lt;i&gt;f&lt;/i&gt; in [0, &lt;i&gt;M&lt;/i&gt;] is a map &lt;span&gt;(f: V(G)rightarrow [0, M])&lt;/span&gt; such that the induced edge labels are different for all distinct edges and the vertex weights are different for all distinct vertices, where the induced edge label of &lt;i&gt;uv&lt;/i&gt; is &lt;span&gt;(f^*(uv)=|f(u)-f(v)|)&lt;/span&gt; and the vertex weight of &lt;i&gt;v&lt;/i&gt; is &lt;span&gt;(omega (v)=sum _{uvin E(G)}f^*(uv))&lt;/span&gt;. If &lt;span&gt;(mathcal {Y})&lt;/span&gt; is a class of graphs of order &lt;i&gt;n&lt;/i&gt; and size &lt;i&gt;m&lt;/i&gt; not containing &lt;span&gt;(bigcup K_2)&lt;/span&gt;, &lt;span&gt;(GA(mathcal {Y},n,m))&lt;/span&gt; denotes the smallest integer &lt;i&gt;M&lt;/i&gt; such that every &lt;span&gt;(Gin mathcal {Y})&lt;/span&gt; admits a graceful antimagic injection in [0, &lt;i&gt;M&lt;/i&gt;]. For &lt;span&gt;(mathcal {G}_Delta subset mathcal {Y})&lt;/span&gt; with a maximum degree &lt;span&gt;(Delta )&lt;/span&gt;, we utilize Alon’s Combinatorial Nullstellensatz to find an upper bound for &lt;span&gt;(GA(mathcal {G}_Delta ,n,m))&lt;/span&gt;, that is, &lt;span&gt;(GA(mathcal {G}_Delta ,n,m))&lt;/span&gt; &lt;span&gt;(le n+p(m-p)+Delta (2n-4))&lt;/span&gt;, where &lt;span&gt;(p=min {Delta ,lfloor frac{n}{2}rfloor })&lt;/span&gt;. We also prove that if &lt;span&gt;(mathcal {T}_q subset mathcal {Y})&lt;/span&gt; is a class of trees whose base tree has &lt;i&gt;q&lt;/i&gt; inner vertices, then &lt;span&gt;(GA(mathcal {T}_q,n,n-1) le 2n+q-3)&lt;/span&gt;. We introduce a variant of the previous notion of injection, that is, the graceful &lt;span&gt;(mathbb {D})&lt;/span&gt;-antimagic injection. Let &lt;i&gt;G&lt;/i&gt; be a simple graph with diameter &lt;span&gt;({textrm{diam}}(G))&lt;/span&gt;. Suppose that &lt;span&gt;(mathbb {D})&lt;/span&gt; is a nonempty subset of the distance set &lt;span&gt;({0,1,)&lt;/span&gt; &lt;span&gt;(ldots , {textrm{diam}}(G)})&lt;/span&gt; and the &lt;span&gt;(mathbb {D})&lt;/span&gt;-&lt;i&gt;neighborhood&lt;/i&gt; of a vertex &lt;i&gt;v&lt;/i&gt;, &lt;span&gt;(N_{mathbb {D}}(v))&lt;/span&gt;, is the set of all vertices &lt;i&gt;u&lt;/i&gt; with &lt;span&gt;(d(u,v)in mathbb {D})&lt;/span&gt;. &lt;i&gt;A graceful&lt;/i&gt; &lt;span&gt;(mathbb {D})&lt;/span&gt;-&lt;i&gt;antimagic injection&lt;/i&gt; &lt;i&gt;f&lt;/i&gt; in [0, &lt;i&gt;N&lt;/i&gt;] is a map &lt;span&gt;(f: V(G)rightarrow [0,N])&lt;/span&gt; such that all induced edge labels are distinct and all &lt;span&gt;(mathbb {D})&lt;/span&gt;-vertex weights are also distinct, where the &lt;span&gt;(mathbb {D})&lt;/span&gt;-vertex weight of &lt;i&gt;v&lt;/i&gt; is the sum of the labels of the vertices in &lt;span&gt;(N_{mathbb {D}}(v))&lt;/span&gt;. If &lt;span&gt;(mathcal {X})&lt;/span&gt; is a class of graphs of order &lt;i&gt;n&lt;/i&gt; and size &lt;i&gt;m&lt;/i&gt; such that no two vertices have the same &lt;span&gt;(mathbb {D})&lt;/span&gt;-neighborhood, the smallest possible &lt;i&gt;N&lt;/i&gt; such that every graph in &lt;span&gt;(mathcal {X})&lt;/span&gt; has a graceful &lt;span&gt;(mathbb {D})&lt;/span&gt;-antimagic injection is denoted by &lt;span&gt;(Gmathbb {D}A(mathcal {X},n,m))&lt;/span&gt;. Let the &lt;span&gt;(mathbb {D})&lt;/span&gt;-&lt;i&gt;degree&lt;/i&gt; of a vertex &lt;i&gt;v&lt;/i&gt; be the cardinality of &lt;span&gt;(N_{mathbb {D}}(v))&lt;/span&gt;. For &lt;span&gt;(mathcal {G}_{Delta ,Delta _{mathbb {D}}} subset mathcal {X})&lt;/span&gt; with a maximum degree &lt;span&gt;(Delta )&lt;/span&gt; and a maximum &lt;span&gt;(mathbb {D})&lt;/span&gt;-degree &lt;span&gt;(Delta _{mathbb {D}})&lt;/span&gt;, we prove that &lt;span&gt;(Gmathbb {D}A(mathca","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147559871","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":"A hexagonal functional equation","authors":"Bruce Ebanks","doi":"10.1007/s00010-026-01271-z","DOIUrl":"10.1007/s00010-026-01271-z","url":null,"abstract":"<div><p>Our main objective is to solve the hexagonal functional equation </p><div><div><span>$$begin{aligned} F(x_1&amp;rho (y_1), x_2 sigma (y_2)) - F(x_1 rho (y_1), x_2) - F(x_1, x_2 sigma (y_2))&amp;= F(x_1 y_1, x_2 y_2) - F(x_1 y_1, x_2) - F(x_1, x_2 y_2) end{aligned}$$</span></div></div><p>for an unknown <span>(F: S_1times S_2 rightarrow H)</span>, where <span>(S_1,S_2)</span> are monoids with respective involutions <span>(rho )</span> and <span>(sigma )</span>, and <i>H</i> is an Abelian group. Our method uses results about the solutions <span>(f_1,f_2,f_3:S rightarrow H)</span> of </p><div><div><span>$$begin{aligned} f_1(xy)+ f_2(xsigma (y)) = f_3(x), quad x,y in S, end{aligned}$$</span></div></div><p>which is a Pexiderized version of the extended Jensen equation on semigroups. We are motivated by some earlier results about similar functional equations on groups. In those results the equation is considered for <span>(S_1 = S_2 = G)</span>, a group, and with <span>(H = {mathbb {C}})</span>. Chung <i>et al.</i> (J. Korean Math. Soc. <b>38</b>, 37-47 (2001)) solved the main equation under the assumptions that <i>G</i> is a 2-divisible group and <span>(rho (x) = sigma (x) = x^{-1})</span> for all <span>(x in G)</span>. Later Hunt <i>et al.</i> (Aequat. Math. <b>90</b>, 87-96 (2016)) studied the special case <span>(rho = sigma )</span> of our equation on a general group, and Lin <i>et al.</i> (Aequat. Math. <b>97</b>, 639-648 (2023)) studied our equation on a general group. The authors of the latter two articles claim that they solved the respective equations without assuming that the group <i>S</i> is 2-divisible, but some solutions are missing. We correct the latter two results and extend all three results to the case that <i>S</i> is a monoid.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147559872","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":"A Note on Torsion Confinement and Sharpness for a Reflected Jensen Functional Equation on Groups","authors":"Đặng Võ Phúc","doi":"10.1007/s00010-026-01279-5","DOIUrl":"10.1007/s00010-026-01279-5","url":null,"abstract":"<div><p>Let <i>G</i> be a group with identity <i>e</i>, <i>H</i> an abelian group written additively, and <span>(varphi :Grightarrow G)</span> an endomorphism. We study solutions <span>(f:Grightarrow H)</span> of the reflected Jensen equation, </p><div><div><span>$$ fbig (x,varphi (y)big )+fbig (varphi (y)^{-1},xbig )=2f(x), qquad f(e)=0, qquad (forall x,y in G). $$</span></div></div><p>Writing <span>(u=varphi (y)in textrm{Im},varphi )</span>, this becomes <span>(f(xu)+f(u^{-1}x)=2f(x))</span> for all <span>(xin G)</span> and <span>(uin textrm{Im},varphi )</span>. In contrast to the classical Jensen equation <span>(f(xy)+f(xy-1) = 2f(x))</span>, the reflected endomorphism variant considered here exhibits a strong torsion-connement phenomenon. Our main result identifies a precise torsion obstruction in a purely elementary way. If </p><div><div><span>$$ K:=langle textrm{Im},varphi cap J(G)rangle , qquad H[4]:={hin H:4h=0}, $$</span></div></div><p>then every solution satisfies the stronger right <i>K</i>-invariance modulo <i>H</i>[4]: </p><div><div><span>$$ f(x)-f(xk)in H[4]qquad (forall xin G, forall kin K). $$</span></div></div><p>In particular, taking <span>(x=e)</span> yields the torsion confinement <span>(f(K)subseteq H[4])</span>. Consequently, if <span>(varphi )</span> is surjective, <span>(G=langle J(G)rangle )</span> (e.g. Coxeter and dihedral groups), and <i>H</i> is 2-torsion-free, then <span>(H[4]={0})</span> and hence <span>(fequiv 0)</span> on <i>G</i>. Finally, we give an explicit example on the dihedral group of order 8 with values in <span>(mathbb {Z}_4)</span> showing that the exponent 4 is sharp: the confinement cannot, in general, be improved to <i>H</i>[2].</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147559873","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":"The best Ulam constant of a first-order linear differential operator","authors":"Alina Ramona Baias,&nbsp;Alexandra Paicu,&nbsp;Dorian Popa","doi":"10.1007/s00010-026-01272-y","DOIUrl":"10.1007/s00010-026-01272-y","url":null,"abstract":"<div><p>The aim of this paper is to give an Ulam stability result for the linear differential operator <span>(D:C^{1}(mathbb {R}, X)rightarrow C(mathbb {R}, X))</span> defined by <span>(Dy=y^{prime }+fcdot y,)</span> where <i>X</i> is a real or complex Banach space. Moreover, if there exists <span>(inf limits _{xin mathbb {R}}|Re f(x)|=m)</span> and <span>(mne 0,)</span> we prove that <span>(K=frac{1}{m})</span> is the best Ulam constant of the operator. As applications, we provide some stability results for Hill’s operator and for the n-th order linear differential operator.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147559863","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}