Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666212020Existence and stability of stationary solutions to the Allen-Cahn equation discretized in space and time197210ENAmy Poh Ai LingDivision of Mathematics and Physics, Graduate School of Natural Science and Technology, Okayama UniversityMasaharuTaniguchiResearch Institute for Interdisciplinary Science, Okayama University The existence and stability of the Allen–Cahn equation discretized in space and time are studied in a finite spatial interval. If a parameter is less than or equals to a critical value, the zero solution is the only stationary solution. If the parameter is larger than the critical value, one has a positive stationary solution and this positive stationary solution is asymptotically stable.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666212020Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space179195ENShoichiFujimoriDepartment of Mathematics, Hiroshima UniversityYuKawakamiGraduate School of Natural Science and Technology, Kanazawa UniversityMasatoshiKokubuDepartment of Mathematics, School of Engineering, Tokyo Denki UniversityWayneRossmanDepartment of Mathematics, Faculty of Science, Kobe UniversityMasaakiUmeharaDepartment of Mathematical and Computing Sciences, Tokyo Institute of TechnologyKotaroYamadaDepartment of Mathematics, Tokyo Institute of Technology Catenoids in de Sitter 3-space S<sup>3</sup><sub>1</sub> belong to a certain class of
space-like constant mean curvature one surfaces. In a previous work, the authors
classified such catenoids, and found that two different classes of countably many exceptional elliptic catenoids are not realized as closed subsets in S<sup>3</sup><sub>1</sub> . Here we show that such exceptional catenoids have closed analytic extensions in S<sup>3</sup><sub>1</sub> with interesting properties.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666212020Crystal interpretation of a formula on the branching rule of types Bn, Cn, and Dn87178ENToyaHiroshimaDepartment of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka UniversityThe branching coefficients of the tensor product of finite-dimensional irreducible Uq(g)-modules, where g is so(2n + 1, C) (Bn-type), sp(2n,C) (Cn-type), and so(2n,C) (Dn-type), are expressed in
terms of Littlewood-Richardson (LR) coefficients in the stable region. We give an interpretation of this relation by Kashiwarafs crystal theory by providing an explicit surjection from the LR crystal of type Cn to the disjoint union of Cartesian product of LR crystals of An|1-type and by proving that LR crystals of types Bn and Dn are identical to the corresponding LR crystal of type Cn in the stable region.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666212020Unstable higher Toda brackets2786ENHideakiOshimaIbaraki UniversityKatsumiOshimaNo potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666212020A representation for algebraic K-theory of quasi-coherent modules over affine spectral schemes125ENMarikoOharaDepartment of Mathematical Sciences Shinshu University In this paper, we study K-theory of spectral schemes by using locally free sheaves. Let us regard the K-theory as a functor K on affine spectral schemes. Then, we prove that the group completion
ΩB<sup>G</sup>(B<sup>G</sup>GL) represents the sheafification of K with respect to Zariski (resp. Nisnevich) topology G, where B<sup>G</sup>GL is a classifying space of a colimit of affine spectral schemes GLn.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019Terwilliger Algebras of Some Group Association Schemes199204ENNurHamidFaculty of Mathematics and Physics, Kanazawa UniversityManabuOuraFaculty of Mathematics and Physics, Kanazawa University The Terwilliger algebra plays an important role in the theory of association schemes. The present paper gives the explicit structures of the Terwilliger algebras of the group association schemes of the finite groups PSL(2, 7), A6, and S6.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019Passage of property (Bw) from two operators to their tensor product187198ENM.H.M.RashidDepartment of Mathematics& Statistics Faculty of Science P.O.Box(7) Muftah University A Banach space operator satisfies property (Bw) if the complement of its B-Weyl spectrum in its the spectrum is the set of finite multiplicity isolated eigenvalues of the operator. Property (Bw) does not transfer from operators T and S to their tensor product T ⊗ S. We give necessary and /or sufficient conditions ensuring the passage of property (Bw) from T and S to T ⊗ S. Perturbations by Riesz operators are considered.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019On the classification of ruled minimal surfaces in pseudo-Euclidean space173186ENYuichiroSatoDepartment of Mathematical Sciences Tokyo Metropolitan University This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a counter-example on the problem of Bernstein type.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019The Factorization of 2 and 3 in Cyclic Quartic Fields167172ENStephen C.BrownDepartment of Computer Science, Mathematics, Physics and Statistics I.K. Barber School of Arts and Sciences University of British ColumbiaChad T.DavisDepartment of Computer Science, Mathematics, Physics and Statistics I.K. Barber School of Arts and Sciences University of British Columbia Due to a theorem of Dedekind, factoring ideals generated by prime numbers in number fields is easily done given that said prime number does not divide the index of the field. In this paper, we determine the prime ideal factorizations of both 2 and 3 in cyclic quartic fields whose index is divisible by one of or both of these primes.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019The number of simple modules in a block with Klein four hyperfocal subgroup159166ENFuminoriTasakaNational Institute of Technology Tsuruoka College A 2-block of a finite group having a Klein four hyperfocal subgroup has the same number of irreducible Brauer characters as the corresponding 2-block of the normalizer of the hyperfocal subgroup.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019Cesaro Orlicz sequence spaces and their Kothe-Toeplitz duals141158ENKuldipRajSchool of Mathematics Shri Mata Vaishno Devi UniversityRenuAnandSchool of Mathematics Shri Mata Vaishno Devi UniversitySuruchiPandohSchool of Mathematics Shri Mata Vaishno Devi UniversityThe present paper focus on introducing certain classes of Cesàro Orlicz sequences over n-normed spaces. We study some topological and algebraic properties of these spaces. Further, we examine relevant relations among the classes of these sequences. We show that these spaces are made n-BK-spaces under certain conditions. Finally, we compute the Köthe-Toeplitz duals of these spaces.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems129139ENNobukazuShimenoSchool of Science and Technology Kwansei Gakuin University We prove that the radial part of the class one Whittaker function on a split semisimple Lie group can be obtained as an appropriate limit of the Heckman-Opdam hypergeometric function.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019Complex interpolation of smoothness Triebel-Lizorkin-Morrey spaces99128ENDenny IvanalHakimDepartment of Mathematics and Information Sciences, Tokyo Metropolitan UniversityToruNogayamaDepartment of Mathematics and Information Sciences, Tokyo Metropolitan UniversityYoshihiroSawanoDepartment of Mathematics and Information Sciences, Tokyo Metropolitan University This paper extends the result in [8] to Triebel-Lizorkin-Morrey spaces which contains 4 parameters p, q, r, s. This paper reinforces our earlier paper [8] by Nakamura, the first and the third authors in two different directions. First, we include the smoothness parameter s and the second smoothness parameter r. In [8] we assumed s = 0 and r = 2. Here we relax the conditions on s and r to s ¸ R and 1 < r ≤ . Second, we apply a formula obtained by Bergh in 1978 to prove our main theorem without using the underlying sequence spaces.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019On the structure of the profile of finite connected quandles8598ENTaisukeWatanabe We verify some cases of a conjecture by C. Hayashi on the structure of the profile of a finite connected quandle.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019On the Diophantine equation in the form that a sum of cubes equals a sum of quintics7584ENFarzaliIzadiMehdi Baghalaghdam Department of Mathematics Faculty of Science Azarbaijan Shahid Madani UniversityMehdiBaghalaghdamMehdi Baghalaghdam Department of Mathematics Faculty of Science Azarbaijan Shahid Madani UniversityNo potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019Reconstruction of inertia groups associated to log divisors from a configuration space group equipped with its collection of log-full subgroups3773ENKazumiHigashiyamaResearch Institute for Mathematical Sciences Kyoto University In the present paper, we study configuration space groups. The goal of this paper is to reconstruct group-theoretically the inertia groups associated to various types of log divisors of a log configuration space of a smooth log curve from the associated configuration space group equipped with its collection of log-full subgroups.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019Berezin-Weyl quantization of Heisenberg motion groups1935ENBenjaminCahenDLepartement de mathLematiques UniversitLe de Lorraine We introduce a SchrNodinger model for the generic representations of a Heisenberg motion group and we construct adapted Weyl correspondences for these representations by adapting the method introduced in [ B. Cahen, Weyl quantization for semidirect products, Differential Geom. Appl. 25 (2007), 177-190].No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666112019On the existence of non-finite coverings of stable curves over complete discrete valuation rings118ENYuYangResearch Institute for Mathematical Sciences Kyoto UniversityLet R be a complete discrete valuation ring with algebraically residue field of characteristic p > 0 and X a stable curve over R. In the present paper, we study the geometry of coverings of X. Under certain assumptions, we prove that, by replacing R by a finite extension of R, there exists a morphism of stable curves f : Y ¨ X over R such that the morphism fĊ : YĊ ¨ XĊ induced by f on generic fibers is finite étale and the morphism fs : Ys ¨ Xs induced by f on special fibers is non-finite.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018On the profinite abelian Beckmann-Black problem233240ENNourGhaziUniversity of Damascus, Faculty of Sciences, Department of MathematicsThe main topic of this paper is to generalize the problem of Beckmann-Black for pro�nite groups. We introduce the Beckmann-Black problem for complete systems of �finite groups and for unramified extensions. We prove that every Galois extension of profi�nite abelian group over a Ġ-free fi�eld is the specialization of some tower of regular Galois extensions of the same group.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018A binomial-coefficient identity arising from the middle discrete series of SU(2,2)221231ENTakahiroHayataGraduate School of Science and Engineering, Yamagata UniversityMasaoIshikawaGraduate School of Natural Science and Technology, Okayama UniversityThe aim of this paper is to answer the question in Remark 8.2 of Takahiro Hayata, Harutaka Koseki, and Takayuki Oda, Matrix coefficients of the middle discrete series of SU(2; 2), J. Funct. Anal. 185 (2001), 297{341, by giving an elementary proof of certain identities on binomials.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Necessary and sufficient Tauberian conditions for the A^r method of summability209219ENÖzerTaloDepartment of Mathematics Faculty of Science and Letters Manisa Celal Bayar UniversityFeyziBas̨arİnönü UniversityMóricz and Rhoades determined the necessary and sufficient Tauberian conditions for certain weighted mean methods of summability in [Acta. Math. Hungar. 102(4) (2004), 279{285]. In the present paper, we deal with the necessary and sufficient Tauberian conditions for the Ar method which was introduced by Bas̨ar in [Fırat Üniv. Fen & Müh. Bil. Dergisi 5(1)(1993), 113{117].No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Indecomposability of various profinite groups arising from hyperbolic curves175208ENArataMinamideResearch Institute for Mathematical Sciences Kyoto UniversityIn this paper, we prove that the etale fundamental group of a hyperbolic curve over an arithmetic field [e.g., a finite extension field of Q or Qp] or an algebraically closed field is indecomposable [i.e., cannot be decomposed into the direct product of nontrivial profinite groups]. Moreover, in the case of characteristic zero, we also prove that the etale fundamental group of the configuration space of a curve of the above type is indecomposable. Finally, we consider the topic of indecomposability in the context of the comparison of the absolute Galois group of Q with the Grothendieck-Teichmuller group GT and pose the question: Is GT indecomposable? We give an affirmative answer to a pro-l version of this questionNo potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018An alternative proof of some results on the framed bordism classes of low rank simple Lie groups165173ENHaruoMinamiNara University of EducationWe present a uni�ed proof of some known results on the framed bordism classes of low rank simple Lie groups.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Arithmetic of positive integers having prime sums of complementary divisors155164ENKenichiShimizuWe study a class of integers called SP numbers (Sum Prime numbers). An SP number is by de�nition a positive integer d that gives rise to a prime number (a + b)=gcd(4; 1 + d) from every factorization d = ab. We also discuss properties of SP numbers in relations with arithmetic of imaginary quadratic �elds (least split primes, exponents of ideal class groups). Further we point out that special cases of SP numbers provide the problems of distribution of prime numbers (twin primes, Sophi-Germain primes, quadratic progressions). Finally, we consider the problem whether there exist in�nitely many SP numbers.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018A non-symmetric diffusion process on the Wiener space137153ENIchiroShigekawaDepartment of Mathematics Graduate School of Science Kyoto UniversityWe discuss a non-symmetric diffusion process on the Wiener space. The process we consider is generated by A = L + b, L being the Ornstein-Uhlenbeck operator and b being a vector �eld. Under suitable integrability condition for b, we show the existence of associated diffusion process. We also investigate the domain of the generator. Further we consider a similar problem in the �nite dimensional Euclidean space.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018A remark on a central limit theorem for non-symmetric random walks on crystal lattices109135ENRyuyaNambaGraduate School of Natural Sciences, Okayama UniversityRecently, Ishiwata, Kawabi and Kotani [4] proved two kinds of central limit theorems for non-symmetric random walks on crystal lattices from the view point of discrete geometric analysis developed by Kotani and Sunada. In the present paper, we establish yet another kind of the central limit theorem for them. Our argument is based on a measure-change technique due to Alexopoulos [1].No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Primary decompositions in abelian R-categories91108ENKenichiSatoGraduate School of Natural Science and Technology Okayama UniversityYujiYoshinoGraduate School of Natural Science and Technology Okayama UniversityWe shall generalize the theory of primary decomposition and associated prime ideals of �nitely generated modules over a noetherian ring to general objects in an abelian R-category where R is a noetherian commutative ring.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Stable splittings of the complex connective K-theory of BSO(2n+1)7389ENTsung-HsuanWuDepartment of Mathematics National Tsing Hua UniversityWe give the stable splittings of the complex connective K-theory of the classifying space BSO(2n + 1), n≥1.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Absolute continuity of the representing measures of the transmutation operators attached to the root system of type BC25972ENKhalifaTrimẻcheDepartment of Mathematics Faculty of sciences of Tunis UniversityWe prove in this paper the absolute continuity of the representing measures of the transmutation operators Vk, tVk and VkW, tVkW associated respectively to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type BC2.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Tomita-Takesaki theory and its application to the structure theory of factors of type III3758ENToshihikoMasudaGraduate School of Mathematics, Kyushu UniversityWe give a survey of Tomita-Takesaki theory and the development of analysis of structure of type III factors, which started from Tomita-Takesaki theory.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15666012018Review on higher homotopies in the theory of H-spaces136ENYutakaHemmiDepartment of Mathematics Faculty of Science and Technology Kochi UniversityHigher homotopy in the theory of H-spaces started from the works by Sugawara in the 1950th. In this paper we review the development of the theory of H-spaces associated with it. Mainly there are two types of higher homotopies, homotopy associativity and homotopy commutativity. We give explanations of the polytopes used as the parameter spaces of those higher forms.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017Blowup and global existence of a solution to a semilinear reaction-diffusion system with the fractional Laplacian175218ENTomoyukiKakehiDepartment of Mathematics, Okayama UniversityYoshihitoOshitaDepartment of Mathematics, Okayama UniversityIn this paper, we deal with the semilinear reaction diffusion system with the fractional Laplacian.<br>
<img src="http://www.lib.okayama-u.ac.jp/www/mjou/mjou_59_175.png"><br>
where <i>p,q</i> > 1 and 0 < <i>α</i> < 1. We study the existence of a global in time solution, the blowup of a solution, and the life span of the blowup solution to the above reaction-diffusion system for sufficiently small initial data.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017Scattering and semi-classical asymptotics for periodic Schrödinger operators with oscillating decaying potential149174ENMouezDimassiUniversitLe Bordeaux I, Institut de MathLematiques de BordeauxAnh Tuan DuongDepartment of Mathematics, Hanoi National University of EducationIn the semi-classical regime (i.e., <i>h</i> ↘ 0), we study the effect of an oscillating decaying potential <i>V</i> (<i>hy, y</i>) on the periodic Schrödinger operator <i>H</i>. The potential <i>V</i> (<i>x, y</i>) is assumed to be smooth, periodic with respect to <i>y</i> and tends to zero as |<i>x</i>| → ∞. We prove the existence of <i>O</i>(<i>h<sup>|n</sup></i>) eigenvalues in each gap of the operator <i>H</i> + <i>V</i> (<i>hy, y</i>). We also establish a Weyl type asymptotics formula of the counting function of eigenvalues with optimal remainder estimate. We give a weak and pointwise asymptotic expansions in powers of <i>h</i> of the spectral shift function corresponding to the pair (<i>H</i> + <i>V</i> (<i>hy, y</i>),<i>H</i>). Finally, under some analytic assumption on the potential V we prove the existence of shape resonances, and we give their asymptotic expansions in powers of <i>h<sup>1/2</sup></i>. All our results depend on the Floquet eigenvalues corresponding to the periodic Schrödinger operator <i>H</i> +<i>V</i> (<i>x, y</i>), (here <i>x</i> is a parameter).No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017On the (1 | C<sub>2</sub>) condition141147ENLe Van AnDepartment of Natural Education, Ha Tinh UniversityNguyen Thi Hai AnhDepartment of Natural Education, Ha Tinh UniversityNgo Sy TungDepartment of Mathematics, Vinh UniversityIn this paper, we give some results on (1 | C<sub>2</sub>)|modules and 1|continuous modules.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017Higher-dimensional absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras131140ENMitsuyasuHashimotoDepartment of Mathematics Faculty of Science, Okayama UniversityIn this paper, we define and discuss higher-dimensional and absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras. In particular, we compare these with the relative notions defined by Scheja and Storch. We also prove the validity of codimension two-argument for modules over a coherent sheaf of algebras with a 2-canonical module, generalizing a result of the author.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017A note on balance equations for doubly periodic minimal surfaces117130ENPeterConnorDepartment of Mathematical Sciences, Indiana University South BendMost known examples of doubly periodic minimal surfaces in R<sup>3</sup> with parallel ends limit as a foliation of R<sup>3</sup> by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a corresponding three-parameter family of doubly periodic minimal surfaces. In this note we derive a differential equation that is equivalent to the balance equations for doubly periodic minimal surfaces. This allows for the generation of many more solutions to the balance equations, enabling the construction of increasingly complicated surfaces.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017A remark on the Lavallee-Spearman-Williams-Yang family of quadratic fields113116ENKwang-SeobKimSchool of Mathematics, Korea Institute for Advanced StudyYasuhiroKishiDepartment of Mathematics, Aichi University of EducationIn [4], M. J. Lavallee, B. K. Spearman, K. S. Williams and Q. Yang introduced a certain parametric D5-quintic polynomial and studied its splitting field. The present paper gives an infinite family of quadratic fields with class number divisible by 5 by using properties of its polynomial.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017Gauss maps of cuspidal edges in hyperbolic 3-space, with application to flat fronts93111ENYutaOgataDepartment of Mathematics, Graduate School of Science, Kobe UniversityKeisukeTeramotoDepartment of Mathematics, Graduate School of Science, Kobe UniversityWe study singularities of de Sitter Gauss map images of cuspidal edges in hyperbolic 3-space. We show relations between singularities of de Sitter Gauss map images and differential geometric properties of cuspidal edges. Moreover, we apply this result to flat fronts in hyperbolic 3-space.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017An arithmetic function arising from the Dedekind ψ function8192ENColinDefantDepartment of Mathematics, University of FloridaWe define ψ‾ to be the multiplicative arithmetic function that satisfies<br>
<img src="http://www.lib.okayama-u.ac.jp/www/mjou/mjou_59_81.png"><br>
for all primes <i>p</i> and positive integers α. Let <i>λ(n)</i> be the number of iterations of the function <i>ψ‾</i> needed for <i>n</i> to reach 2. It follows from a theorem due to White that <i>λ</i> is additive. Following Shapiro's work on the iterated <i>φ</i> function, we determine bounds for <i>λ</i>. We also use the function <i>λ</i> to partition the set of positive integers into three sets <i>S<sub>1</sub>, S<sub>2</sub>, S<sub>3</sub></i> and determine some properties of these sets.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665912017Evaluation of convolution sums and some remarks on cusp forms of weight 4 and level 127179ENB.RamakrishhanHarish-Chandra Research InstituteBrundabanSahuSchool of Mathematical Sciences National Institut