Proceedings of the Japan Academy, Series A, Mathematical Sciences Articles (Project Euclid)
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The latest articles from Proceedings of the Japan Academy, Series A, Mathematical Sciences on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 01 Jun 2011 09:15 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Analyticity and smoothing effect for the fifth order KdV type equation
http://projecteuclid.org/euclid.pja/1279719309
<strong>Kyoko Tomoeda</strong><p><strong>Source: </strong>Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 7, 101--106.</p><p><strong>Abstract:</strong><br/>
We consider the initial value problem for the reduced fifth order KdV type equation: $\partial_{t}u-\partial_{x}^{5}u-10\partial_{x}(u^{3})+5\partial_{x}(\partial_{x}u)^{2}=0$ which is obtained by removing the nonlinear term $10\partial_{x}(u\partial_{x}^{2} u)$ from the fifth order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables, if the initial data $\phi\in H^{s}(\mathbf{R})$ $(s>1/8)$ satisfies the condition
\begin{equation*}
∑_{k=0}^{∞}\frac{A_{0}^{k}}{k!}{\|}(x\partial_{x})^{k}φ{\|}_{H^{s}}<{∞},
\end{equation*}
for some constant $A_{0}(0<A_{0}<1)$. Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the argument used in [5].
</p>projecteuclid.org/euclid.pja/1279719309_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA $p$-analogue of Euler’s constant and congruence zeta functionshttps://projecteuclid.org/euclid.pja/1517454032<strong>Nobushige Kurokawa</strong>, <strong>Yuichiro Taguchi</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 2, 13--16.</p><p><strong>Abstract:</strong><br/>
A $p$-analogue of a formula of Euler on the Euler constant is given, and it is interpreted in terms of the absolute zeta functions of tori.
</p>projecteuclid.org/euclid.pja/1517454032_20180131220119Wed, 31 Jan 2018 22:01 ESTRational quotients of two linear forms in roots of a polynomialhttps://projecteuclid.org/euclid.pja/1517454034<strong>Artūras Dubickas</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 2, 17--20.</p><p><strong>Abstract:</strong><br/>
Let $f$ and $g$ be two linear forms with non-zero rational coefficients in $k$ and $\ell$ variables, respectively. We describe all separable polynomials $P$ with the property that for any choice of (not necessarily distinct) roots $\lambda_{1},\ldots,\lambda_{k+\ell}$ of $P$ the quotient between $f(\lambda_{1},\ldots,\lambda_{k})$ and $g(\lambda_{k+1},\ldots,\lambda_{k+\ell}) \ne 0$ belongs to $\mathbf{Q}$. It turns out that each such polynomial has all of its roots in a quadratic extension of $\mathbf{Q}$. This is a continuation of a recent work of Luca who considered the case when $k=\ell=2$, $f(x_{1},x_{2})$ and $g(x_{1},x_{2})$ are both $x_{1}-x_{2}$, solved it, and raised the above problem as an open question.
</p>projecteuclid.org/euclid.pja/1517454034_20180131220119Wed, 31 Jan 2018 22:01 ESTOn Koyama’s refinement of the prime geodesic theoremhttps://projecteuclid.org/euclid.pja/1519808413<strong>Muharem Avdispahić</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 3, 21--24.</p><p><strong>Abstract:</strong><br/>
We give a new proof of the best presently-known error term in the prime geodesic theorem for compact hyperbolic surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived, yielding to a further reduction of the error term outside a set of finite logarithmic measure.
</p>projecteuclid.org/euclid.pja/1519808413_20180228040034Wed, 28 Feb 2018 04:00 ESTComplete flat fronts as hypersurfaces in Euclidean spacehttps://projecteuclid.org/euclid.pja/1519808414<strong>Atsufumi Honda</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 3, 25--30.</p><p><strong>Abstract:</strong><br/>
By Hartman–Nirenberg’s theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. \textit{Flat fronts} are flat hypersurfaces with admissible singularities. Murata–Umehara gave a representation formula for complete flat fronts with non-empty singular set in Euclidean 3-space, and proved the four vertex type theorem. In this paper, we prove that, unlike the case of $n=2$, there do not exist any complete flat fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\geq 3)$.
</p>projecteuclid.org/euclid.pja/1519808414_20180228040034Wed, 28 Feb 2018 04:00 ESTSelf-similar measures for iterated function systems driven by weak contractionshttps://projecteuclid.org/euclid.pja/1522915216<strong>Kazuki Okamura</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 4, 31--35.</p><p><strong>Abstract:</strong><br/>
We show the existence and uniqueness for self-similar measures for iterated function systems driven by weak contractions. Our main idea is using the duality theorem of Kantorovich-Rubinstein and equivalent conditions for weak contractions established by Jachymski. We also show collage theorems for such iterated function systems.
</p>projecteuclid.org/euclid.pja/1522915216_20180405040037Thu, 05 Apr 2018 04:00 EDTInequalities for free multi-braid arrangementshttps://projecteuclid.org/euclid.pja/1522915218<strong>Michael Robert DiPasquale</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 4, 36--41.</p><p><strong>Abstract:</strong><br/>
Abe, Nuida, and Numata (2009) describe a large class of free multiplicities on the braid arrangement arising from signed-eliminable graphs. On a large cone in the multiplicity lattice, we prove that these are the only free multiplicities on the braid arrangement. We also give a conjecture on the structure of all free multiplicities on the braid arrangement.
</p>projecteuclid.org/euclid.pja/1522915218_20180405040037Thu, 05 Apr 2018 04:00 EDTOn a Galois group arising from an iterated maphttps://projecteuclid.org/euclid.pja/1524794414<strong>Masamitsu Shimakura</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 5, 43--48.</p><p><strong>Abstract:</strong><br/>
We study the irreducibility and the Galois group of the polynomial $f (a,x) = x^{8} +3ax^{6}+3a^{2}x^{4}+(a^{2}+1)ax^{2}+a^{2}+1$ over $\mathbf{Q}(a)$ and $\mathbf{Q}$. This polynomial is a factor of the 4-th dynatomic polynomial for the map $\sigma(x) = x^{3} + ax$.
</p>projecteuclid.org/euclid.pja/1524794414_20180426220030Thu, 26 Apr 2018 22:00 EDTNon-left-orderable surgeries on negatively twisted torus knotshttps://projecteuclid.org/euclid.pja/1524794415<strong>Kazuhiro Ichihara</strong>, <strong>Yuki Temma</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 5, 49--52.</p><p><strong>Abstract:</strong><br/>
We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non-left-orderable fundamental groups.
</p>projecteuclid.org/euclid.pja/1524794415_20180426220030Thu, 26 Apr 2018 22:00 EDTA note on the dimension of global sections of adjoint bundles for polarized 4-foldshttps://projecteuclid.org/euclid.pja/1524794416<strong>Yoshiaki Fukuma</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 5, 53--58.</p><p><strong>Abstract:</strong><br/>
Let $(X,L)$ be a polarized manifold defined over the field of complex numbers. In this paper, we consider the case where $\dim X=4$ and we prove that the second Hilbert coefficient $A_{2}(X,L)$ of $(X,L)$, which was defined in our previous paper, is non-negative. Furthermore we consider a question proposed by H. Tsuji for $\dim X=4$.
</p>projecteuclid.org/euclid.pja/1524794416_20180426220030Thu, 26 Apr 2018 22:00 EDTAutomorphism group of plane curve computed by Galois points, IIhttps://projecteuclid.org/euclid.pja/1527732017<strong>Takeshi Harui</strong>, <strong>Kei Miura</strong>, <strong>Akira Ohbuchi</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 6, 59--63.</p><p><strong>Abstract:</strong><br/>
Recently, the first author~[3] classified finite groups obtained as automorphism groups of smooth plane curves of degree $d \ge 4$ into five types. He gave an upper bound of the order of the automorphism group for each types. For one of them, the type (a-ii), that is given by $\max \{2d (d - 2), 60 d\}$. In this article, we shall construct typical examples of smooth plane curve $C$ by applying the method of Galois points, whose automorphism group has order $60d$. In fact, we determine the structure of the automorphism group of those curves.
</p>projecteuclid.org/euclid.pja/1527732017_20180530220044Wed, 30 May 2018 22:00 EDTAnalytic continuation of the multiple Fibonacci zeta functionshttps://projecteuclid.org/euclid.pja/1527732019<strong>Sudhansu Sekhar Rout</strong>, <strong>Nabin Kumar Meher</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 6, 64--69.</p><p><strong>Abstract:</strong><br/>
In this article, we prove the meromorphic continuation of the multiple Fibonacci zeta functions of depth 2: \begin{equation*} \sum_{0<n_{1}<n_{2}}\frac{1}{F_{n_{1}}^{s_{1}}F_{n_{2}}^{s_{2}}}, \end{equation*} where $F_{n}$ is the $n$-th Fibonacci number, $\mathop{\mathrm{Re}} (s_{1}) > 0$ and $\mathop{\mathrm{Re}} (s_{2}) > 0$. We compute a complete list of its poles and their residues. We also prove that multiple Fibonacci zeta values at negative integer arguments are rational.
</p>projecteuclid.org/euclid.pja/1527732019_20180530220044Wed, 30 May 2018 22:00 EDTOn the order of holomorphic curves with maximal deficiency sum, IIhttps://projecteuclid.org/euclid.pja/1531814465<strong>Nobushige Toda</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 7, 71--75.</p><p><strong>Abstract:</strong><br/>
In this paper we consider the order of holomorphic curves with maximal deficiency sum in the complex plane. The purpose of this paper is to weaken the condition treated in the paper [9]. As a special case we obtain the result in [9].
</p>projecteuclid.org/euclid.pja/1531814465_20180717040127Tue, 17 Jul 2018 04:01 EDTOn products of cyclic and abelian finite $p$-groups ($ p$ odd)https://projecteuclid.org/euclid.pja/1538186760<strong>Brendan McCann</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 8, 77--80.</p><p><strong>Abstract:</strong><br/>
For an odd prime $p$, it is shown that if $G = AB$ is a finite $p$-group, for subgroups $A$ and $B$ such that $A$ is cyclic and $B$ is abelian of exponent at most $p^{k}$, then $\Omega_{k}(A)B \unlhd G$, where $\Omega_{k}(A) = \langle g \in A \mid g^{ p^{k}} = 1 \rangle$.
</p>projecteuclid.org/euclid.pja/1538186760_20180928220642Fri, 28 Sep 2018 22:06 EDTCollapsing K3 surfaces and Moduli compactificationhttps://projecteuclid.org/euclid.pja/1538186761<strong>Yuji Odaka</strong>, <strong>Yoshiki Oshima</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 8, 81--86.</p><p><strong>Abstract:</strong><br/>
This note is a summary of our work [OO], which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kähler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not necessarily “maximally degenerating”. Our results also give a proof of Kontsevich-Soibelman [KS06,Conjecture 1] (cf., [GW00, Conjecture 6.2]) in the case of K3 surfaces as a byproduct.
</p>projecteuclid.org/euclid.pja/1538186761_20180928220642Fri, 28 Sep 2018 22:06 EDTThe cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomialshttps://projecteuclid.org/euclid.pja/1541059248<strong>Tatsuya Horiguchi</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 9, 87--92.</p><p><strong>Abstract:</strong><br/>
In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. We show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials.
</p>projecteuclid.org/euclid.pja/1541059248_20181101040213Thu, 01 Nov 2018 04:02 EDTThe signs of the Stieltjes constants associated with the Dedekind zeta functionhttps://projecteuclid.org/euclid.pja/1543201231<strong>Sumaia Saad Eddin</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 10, 93--96.</p><p><strong>Abstract:</strong><br/>
The Stieltjes constants $\gamma_{n}(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function $\zeta_{K}(s)$ at its pole $s=1$. In this paper, we establish a similar expression of $\gamma_{n}(K)$ as Stieltjes obtained in 1885 for $\gamma_{n}(\mathbf{Q})$. We also study the signs of $\gamma_{n}(K)$.
</p>projecteuclid.org/euclid.pja/1543201231_20181125220124Sun, 25 Nov 2018 22:01 ESTOn normalization of quasi-log canonical pairshttps://projecteuclid.org/euclid.pja/1543201232<strong>Osamu Fujino</strong>, <strong>Haidong Liu</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 94, Number 10, 97--101.</p><p><strong>Abstract:</strong><br/>
The normalization of an irreducible quasi-log canonical pair naturally becomes a quasi-log canonical pair.
</p>projecteuclid.org/euclid.pja/1543201232_20181125220124Sun, 25 Nov 2018 22:01 ESTThe number of orientable small covers over a product of simpliceshttps://projecteuclid.org/euclid.pja/1546830023<strong>Murat Altunbulak</strong>, <strong>Aslı Güçlükan İlhan</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 1, 1--5.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a formula for the number of orientable small covers over a product of simplices up to D-J equivalence. We also give an approximate value for the ratio between the number of small covers and the number of orientable small covers over a product of equidimensional simplices up to D-J equivalence.
</p>projecteuclid.org/euclid.pja/1546830023_20190106220117Sun, 06 Jan 2019 22:01 ESTLeft-orderability for surgeries on twisted torus knotshttps://projecteuclid.org/euclid.pja/1546830024<strong>Anh Tuan Tran</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 1, 6--10.</p><p><strong>Abstract:</strong><br/>
We show that the fundamental group of the 3-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if $\frac{p}{q}$ is sufficiently close to 0.
</p>projecteuclid.org/euclid.pja/1546830024_20190106220117Sun, 06 Jan 2019 22:01 ESTClassification of irreducible symmetric spaces which admit standard compact Clifford–Klein formshttps://projecteuclid.org/euclid.pja/1548990026<strong>Koichi Tojo</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 2, 11--15.</p><p><strong>Abstract:</strong><br/>
We give a classification of irreducible symmetric spaces which admit standard compact Clifford–Klein forms. The method uses representation theory over the real number field and the criterion for properness and cocompactness of the action on homogeneous spaces due to T. Kobayashi.
</p>projecteuclid.org/euclid.pja/1548990026_20190131220134Thu, 31 Jan 2019 22:01 ESTHypertranscendence of the multiple sine function for a complex periodhttps://projecteuclid.org/euclid.pja/1548990027<strong>Masaki Kato</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 2, 16--19.</p><p><strong>Abstract:</strong><br/>
It is known that the multiple sine function for a “rational” period satisfies an algebraic differential equation. However, for a non-“rational” period, the differential algebraicity of the multiple sine function is obscure. In this paper, we prove that, if there exists a non-real element in the set $\{\omega_{j}/\omega_{i}|1\leq i<j\leq r\}$, the multiple sine function $\text{Sin}_{r}(x,(\omega_{1},\cdots,\omega_{r}))$ does not satisfy any algebraic differential equation.
</p>projecteuclid.org/euclid.pja/1548990027_20190131220134Thu, 31 Jan 2019 22:01 ESTExamples of isometric immersions of $\mathbf{R}^{2}$ into $\mathbf{R}^{4}$ with vanishing normal curvaturehttps://projecteuclid.org/euclid.pja/1551344432<strong>Hiroshi Mori</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 3, 21--23.</p><p><strong>Abstract:</strong><br/>
We construct a family of isometric immersions of $\mathbf{R}^{2}$ into $\mathbf{R}^{4}$ with vanishing normal curvature.
</p>projecteuclid.org/euclid.pja/1551344432_20190228040115Thu, 28 Feb 2019 04:01 ESTA note on the non-vanishing of Poincaré series for the Fricke group $\Gamma_{0}^{+}(p)$https://projecteuclid.org/euclid.pja/1551344433<strong>SoYoung Choi</strong>, <strong>Bo-Hae Im</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 3, 24--29.</p><p><strong>Abstract:</strong><br/>
For each prime $p$, we give an upper bound in $m$ for Poincaré series $P_{k}^{+}(z,m)$ of weight $k$ for $\Gamma_{0}^{+}(p)$ to be non-vanishing.
</p>projecteuclid.org/euclid.pja/1551344433_20190228040115Thu, 28 Feb 2019 04:01 ESTModular forms of weight $3m$ and elliptic modular surfaceshttps://projecteuclid.org/euclid.pja/1554084020<strong>Shouhei Ma</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 4, 31--36.</p><p><strong>Abstract:</strong><br/>
We prove that the graded ring of modular forms of weight divisible by 3 is naturally isomorphic to a certain log canonical ring of the associated elliptic modular surface. This extends the Shioda correspondence between weight 3 cusp forms and holomorphic 2-forms.
</p>projecteuclid.org/euclid.pja/1554084020_20190331220107Sun, 31 Mar 2019 22:01 EDTHitting times to spheres of Brownian motions with drifts starting from the originhttps://projecteuclid.org/euclid.pja/1554084023<strong>Yuji Hamana</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 4, 37--39.</p><p><strong>Abstract:</strong><br/>
We investigate the first hitting times to spheres of Brownian motions with constant drifts. In the case when the Brownian motion starts from a point in $\mathbf{R}^{d}$ except for the origin, an explicit formula for the density function of the hitting time has been obtained. When the starting point is the origin, we represent the density function by means of the density of the hitting time of the Brownian motion without the drift.
</p>projecteuclid.org/euclid.pja/1554084023_20190331220107Sun, 31 Mar 2019 22:01 EDTAbsolute multiple sine functionshttps://projecteuclid.org/euclid.pja/1556092942<strong>Nobushige Kurokawa</strong>, <strong>Hidekazu Tanaka</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 5, 41--46.</p><p><strong>Abstract:</strong><br/>
In this paper we formulate a unified theory of multiple sine functions by using a view point of absolute zeta functions and absolute automorphic forms.
</p>projecteuclid.org/euclid.pja/1556092942_20190424040255Wed, 24 Apr 2019 04:02 EDTTermination of extremal rays of divisorial type for the power of étale endomorphismshttps://projecteuclid.org/euclid.pja/1556092943<strong>Yoshio Fujimoto</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 5, 47--51.</p><p><strong>Abstract:</strong><br/>
Let $f\colon X\to X$ be a non-isomorphic étale endomorphism of a smooth projective variety $X$. Suppose that there exists a $K_{X}$-negative extremal ray $R'\subset \overline{\mathrm{NE}}(X)$ of fiber type. Then we give a sufficient condition for a $K_{X}$-negative extremal ray $R\subset \overline{\mathrm{NE}}(X)$ of divisorial type to terminate under a suitable power $f^{k}$ of $k > 0$.
</p>projecteuclid.org/euclid.pja/1556092943_20190424040255Wed, 24 Apr 2019 04:02 EDTInfinitely many elliptic curves of rank exactly two IIhttps://projecteuclid.org/euclid.pja/1559268170<strong>Keunyoung Jeong</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 6, 53--57.</p><p><strong>Abstract:</strong><br/>
Under the parity conjecture, an infinite family of elliptic curves of rank 2 with a torsion subgroup of order 2 or 3 is constructed.
</p>projecteuclid.org/euclid.pja/1559268170_20190530220340Thu, 30 May 2019 22:03 EDTErdősian functions and an identity of Gausshttps://projecteuclid.org/euclid.pja/1559268171<strong>Tapas Chatterjee</strong>, <strong>Suraj Singh Khurana</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 6, 58--63.</p><p><strong>Abstract:</strong><br/>
A famous identity of Gauss gives a closed form expression for the values of the digamma function $\psi(x)$ at rational arguments $x$ in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdős which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for $n$th Catalan number in terms of these functions.
</p>projecteuclid.org/euclid.pja/1559268171_20190530220340Thu, 30 May 2019 22:03 EDTWeierstrass points on hyperelliptic modular curveshttps://projecteuclid.org/euclid.pja/1563955329<strong>Daeyeol Jeon</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 7, 65--69.</p><p><strong>Abstract:</strong><br/>
In this paper, we find all Weierstrass points on the hyperelliptic modular curves $X_{0}(N)$ whose hyperelliptic involutions are non-exceptional, i.e., induced by matrices in $\mathrm{GL}_{2}(\mathbf{R})$.
</p>projecteuclid.org/euclid.pja/1563955329_20190724040301Wed, 24 Jul 2019 04:03 EDTLipschitz characterization for exponentially weighted Bergman spaces of the unit ballhttps://projecteuclid.org/euclid.pja/1563955331<strong>Hong Rae Cho</strong>, <strong>Soohyun Park</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 7, 70--74.</p><p><strong>Abstract:</strong><br/>
The paper concerns the weighted Bergman spaces of the complex unit ball with exponential weights. We characterize the space with respect to Lipschitz type conditions using norm equivalence lemma.
</p>projecteuclid.org/euclid.pja/1563955331_20190724040301Wed, 24 Jul 2019 04:03 EDTRelationship between orbit decomposition on the flag varieties and multiplicities of induced representationshttps://projecteuclid.org/euclid.pja/1563955332<strong>Taito Tauchi</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 7, 75--79.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a real reductive Lie group and $H$ a closed subgroup. T. Kobayashi and T. Oshima established a finiteness criterion of multiplicities of irreducible $G$-modules occurring in the regular representation $C^{\infty}(G/H)$ by a geometric condition, referred to as \textit{real sphericity}, namely, $H$ has an open orbit on the real flag variety $G/P$. This note discusses a refinement of their theorem by replacing a minimal parabolic subgroup $P$ with a general parabolic subgroup $Q$ of $G$, where a careful analysis is required because the finiteness of the number of $H$-orbits on the partial flag variety $G/Q$ is not equivalent to the existence of $H$-open orbit on $G/Q$.
</p>projecteuclid.org/euclid.pja/1563955332_20190724040301Wed, 24 Jul 2019 04:03 EDTCorrigendum to “Real abelian fields satisfying the Hilbert-Speiser condition for some small primes $p$”https://projecteuclid.org/euclid.pja/1563955333<strong>Humio Ichimura</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 7, 80--82.</p><p><strong>Abstract:</strong><br/>
We correct and change Proposition 1 and the proof of Proposition 2 of the previous paper [7].
</p>projecteuclid.org/euclid.pja/1563955333_20190724040301Wed, 24 Jul 2019 04:03 EDTLimiting cases of Sobolev inequalities on stratified groupshttps://projecteuclid.org/euclid.pja/1570003225<strong>Michael Ruzhansky</strong>, <strong>Nurgissa Yessirkegenov</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 8, 83--87.</p><p><strong>Abstract:</strong><br/>
In this paper we present critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities concerning the limiting cases of the embedding theorems for Sobolev spaces on stratified groups. Moreover, using the critical Gagliardo-Nirenberg inequality the existence of least energy solutions of the nonlinear Schrödinger type equations can be obtained. We also express the best constant in the critical Gagliardo-Nirenberg inequality in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations.
</p>projecteuclid.org/euclid.pja/1570003225_20191002040125Wed, 02 Oct 2019 04:01 EDTSymplectic structures on free nilpotent Lie algebrashttps://projecteuclid.org/euclid.pja/1570003226<strong>Viviana del Barco</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 8, 88--90.</p><p><strong>Abstract:</strong><br/>
In this short note we show a necessary and sufficient condition for the existence of symplectic structures on free nilpotent Lie algebras and their one-dimensional trivial extensions.
</p>projecteuclid.org/euclid.pja/1570003226_20191002040125Wed, 02 Oct 2019 04:01 EDTOn the mod 2 cohomology of the classifying space of the exceptional Lie group $E_{6}$https://projecteuclid.org/euclid.pja/1572595221<strong>Masaki Kameko</strong>, <strong>Masaki Nakagawa</strong>, <strong>Tetsu Nishimoto</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 9, 91--96.</p><p><strong>Abstract:</strong><br/>
We determine the mod 2 cohomology ring of the classifying space of the exceptional Lie group $E_{6}$ and the action of the Steenrod algebra on it.
</p>projecteuclid.org/euclid.pja/1572595221_20191101040042Fri, 01 Nov 2019 04:00 EDTSpace-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-spacehttps://projecteuclid.org/euclid.pja/1572595222<strong>Shintaro Akamine</strong>, <strong>Masaaki Umehara</strong>, <strong>Kotaro Yamada</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 9, 97--102.</p><p><strong>Abstract:</strong><br/>
Consider a surface $S$ immersed in the Lorentz-Minkowski 3-space $\mathbf{R}^{3}_{1}$. A complete light-like line in $\mathbf{R}^{3}_{1}$ is called an \textit{entire null line} on the surface $S$ in $\mathbf{R}^{3}_{1}$ if it lies on $S$ and consists of only null points with respect to the induced metric. In this paper, we show the existence of embedded space-like maximal graphs containing entire null lines. If such a graph is defined on a convex domain in $\mathbf{R}^{2}$, then it must be contained in a light-like plane (cf. Remark~3.3). Our example is critical in the sense that it is defined on a certain non-convex domain.
</p>projecteuclid.org/euclid.pja/1572595222_20191101040042Fri, 01 Nov 2019 04:00 EDTA note on flatness of some fiber type contractionshttps://projecteuclid.org/euclid.pja/1572595223<strong>Eleonora Anna Romano</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 9, 103--106.</p><p><strong>Abstract:</strong><br/>
We discuss the flatness property of some fiber type contractions of complex smooth projective varieties of arbitrary dimensions. We relate the flatness of some morphisms having one-dimensional fibers with their conic bundles structures, also in the general case in which some mild singularities of the varieties are admitted.
</p>projecteuclid.org/euclid.pja/1572595223_20191101040042Fri, 01 Nov 2019 04:00 EDTGraph equivariant cohomological rigidity for GKM graphshttps://projecteuclid.org/euclid.pja/1575255621<strong>Matthias Franz</strong>, <strong>Hitoshi Yamanaka</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 10, 107--110.</p><p><strong>Abstract:</strong><br/>
We formulate the notion of an isomorphism of GKM graphs. We then show that two GKM graphs have isomorphic graph equivariant cohomology algebras if and only if the graphs are isomorphic.
</p>projecteuclid.org/euclid.pja/1575255621_20191201220043Sun, 01 Dec 2019 22:00 ESTA generalization of the Tutte polynomialshttps://projecteuclid.org/euclid.pja/1575255622<strong>Tsuyoshi Miezaki</strong>, <strong>Manabu Oura</strong>, <strong>Tadashi Sakuma</strong>, <strong>Hidehiro Shinohara</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 95, Number 10, 111--113.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.
</p>projecteuclid.org/euclid.pja/1575255622_20191201220043Sun, 01 Dec 2019 22:00 ESTQuasi traveling waves with quenching in a reaction-diffusion equation in the presence of negative powers nonlinearityhttps://projecteuclid.org/euclid.pja/1577264416<strong>Yu Ichida</strong>, <strong>Takashi Okuda Sakamoto</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 1, 1--6.</p><p><strong>Abstract:</strong><br/>
The quasi traveling waves with quenching of $u_{t} = u_{xx} + (1-u)^{-\alpha}$ for $\alpha \in 2 \mathbf{N}$ are considered. The existence of quasi traveling waves with quenching and their quenching rates are studied by applying the Poincaré compactification.
</p>projecteuclid.org/euclid.pja/1577264416_20191225040034Wed, 25 Dec 2019 04:00 ESTThe second moment for counting prime geodesicshttps://projecteuclid.org/euclid.pja/1577264417<strong>Ikuya Kaneko</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 1, 7--12.</p><p><strong>Abstract:</strong><br/>
A brighter light has freshly been shed upon the second moment of the Prime Geodesic Theorem. We work with such moments in the two and three dimensional hyperbolic spaces. Letting $E_{\Gamma}(X)$ be the error term arising from counting prime geodesics associated to $\Gamma = \mathrm{PSL}_{2}(\mathbf{Z}[i])$, the bound $E_{\Gamma}(X) \ll X^{3/2+\epsilon}$ is proved in a square mean sense. Our second moment bound is the pure counterpart of the work of Balog \textit{et al.} for $\Gamma = \mathrm{PSL}_{2}(\mathbf{Z})$, and the main innovation entails the delicate analysis of sums of Kloosterman sums. We also infer pointwise bounds from the standpoint of the second moment. Finally, we announce the pointwise bound $E_{\Gamma}(X) \ll X^{67/42+\epsilon}$ for $\Gamma = \mathrm{PSL}_{2}(\mathbf{Z}[i])$ by an application of the Weyl-type subconvexity.
</p>projecteuclid.org/euclid.pja/1577264417_20191225040034Wed, 25 Dec 2019 04:00 ESTContact loci, motivic Milnor fibers of nondegenerate singularitieshttps://projecteuclid.org/euclid.pja/1580785219<strong>Quy Thuong Lê</strong>, <strong>Tat Thang Nguyen</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 2, 13--17.</p><p><strong>Abstract:</strong><br/>
Inspired by Denef-Loeser’s identity of the Euler characteristic with compact supports of the contact loci with the Lefschetz numbers of a complex singularity, we study sheaf cohomology groups of contact loci of complex nondegenerate singularities. Moreover, also for these singularities, we obtain a motivic analogue of Lê Dũng Tráng’s work on a monodromy relation of a complex singularity and its restriction to a generic hyperplane.
</p>projecteuclid.org/euclid.pja/1580785219_20200203220037Mon, 03 Feb 2020 22:00 ESTZariski tuples for a smooth cubic and its tangent lineshttps://projecteuclid.org/euclid.pja/1580785220<strong>Shinzo Bannai</strong>, <strong>Hiro-o Tokunaga</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 2, 18--21.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the geometry of two-torsion points of elliptic curves in order to distinguish the embedded topology of reducible plane curves consisting of a smooth cubic and its tangent lines. As a result, we obtain a new family of Zariski tuples consisting of such curves.
</p>projecteuclid.org/euclid.pja/1580785220_20200203220037Mon, 03 Feb 2020 22:00 ESTThe Gibbs phenomenon for Stromberg waveletshttps://projecteuclid.org/euclid.pja/1583204420<strong>Vazgen Gagik Mikayelyan</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 3, 23--27.</p><p><strong>Abstract:</strong><br/>
The Gibbs phenomenon for Stromberg wavelets is studied. It is proved that the Gibbs phenomenon for partial sums of Fourier-Stromberg series occurs for almost all points of $\mathbf{R}$.
</p>projecteuclid.org/euclid.pja/1583204420_20200302220041Mon, 02 Mar 2020 22:00 ESTLocal comparisons of homological and homotopical mixed Hodge polynomialshttps://projecteuclid.org/euclid.pja/1583204421<strong>Shoji Yokura</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 3, 28--31.</p><p><strong>Abstract:</strong><br/>
For a simply connected complex algebraic variety $X$, by the mixed Hodge structures $(W_{\bullet}, F^{\bullet})$ and $(\tilde{W}_{\bullet}, \tilde{F}^{\bullet})$ of the homology group $H_{*}(X;\mathbf{Q})$ and the homotopy groups $\pi_{*}(X)\otimes \mathbf{Q}$ respectively, we have the following mixed Hodge polynomials \begin{equation*} \mathit{MH}_{X}(t,u,v):= ∑_{k,p,q} \dim (\mathit{Gr}_{F_{•}}^{p} \mathit{Gr}^{W_{•}}_{p+q} H_{k} (X;\mathbf{C})) t^{k} u^{-p} v^{-q}, \end{equation*} \begin{equation*} \mathit{MH}^{π}_{X}(t,u,v):= ∑_{k,p,q} \dim (\mathit{Gr}_{\tilde{F}_{•}}^{p} \mathit{Gr}^{\tilde{W}_{•}}_{p+q} (π_{k}(X) øtimes \mathbf{C})) t^{k}u^{-p} v^{-q}, \end{equation*} which are respectively called \textit{the homological mixed Hodge polynomial} and \textit{the homotopical mixed Hodge polynomial}. In this paper we discuss some inequalities concerning these two mixed Hodge polynomials.
</p>projecteuclid.org/euclid.pja/1583204421_20200302220041Mon, 02 Mar 2020 22:00 ESTOn a Diophantine equation involving powers of Fibonacci numbershttps://projecteuclid.org/euclid.pja/1585728123<strong>Krisztián Gueth</strong>, <strong>Florian Luca</strong>, <strong>László Szalay</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 4, 33--37.</p><p><strong>Abstract:</strong><br/>
This paper deals with the diophantine equation $F_{1}^{p}+2F_{2}^{p}+\cdots +kF_{k}^{p}=F_{n}^{q}$, an equation on the weighted power terms of Fibonacci sequence. For the exponents $p,q\in\{1,2\}$ the problem has already been solved in ad hoc ways using the properties of the summatory identities appear on the left-hand side of the equation. Here we suggest a uniform treatment for arbitrary positive integers $p$ and $q$ which works, in practice, for small values. We obtained all the solutions for $p,q\le 10$ by testing the new approach.
</p>projecteuclid.org/euclid.pja/1585728123_20200401040219Wed, 01 Apr 2020 04:02 EDTCharacterizations of weaving $K$-frameshttps://projecteuclid.org/euclid.pja/1588212028<strong>Animesh Bhandari</strong>, <strong>Debajit Borah</strong>, <strong>Saikat Mukherjee</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 5, 39--43.</p><p><strong>Abstract:</strong><br/>
In distributed signal processing frames play significant role as redundant building blocks. Bemrose \textit{et al.} were motivated from this concept, as a result they introduced weaving frames in Hilbert space. Weaving frames have useful applications in sensor networks, likewise weaving $K$-frames have been proved to be useful during signal reconstructions from the range of a bounded linear operator $K$. This article focuses on study, characterization of weaving $K$-frames in different spaces. Paley-Wiener type perturbations and conditions on erasure of frame components have been assembled to scrutinize woven-ness of $K$-frames.
</p>projecteuclid.org/euclid.pja/1588212028_20200429220040Wed, 29 Apr 2020 22:00 EDTNew results on slowly varying functions in the Zygmund sensehttps://projecteuclid.org/euclid.pja/1590652896<strong>Edward Omey</strong>, <strong>Meitner Cadena</strong>. <p><strong>Source: </strong>Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 96, Number 6, 45--49.</p><p><strong>Abstract:</strong><br/>
Very recently Seneta [15] has provided a characterization of slowly varying functions $L$ in the Zygmund sense by using the condition, for each $y>0$, \begin{equation} x\left(\frac{L(x+y)}{L(x)}-1\right)\to0 \text{as} x\to∞. \label{Lb1} \end{equation} We extend this result by considering a wider class of functions and a more general condition than (\ref{Lb1}). Further, a representation theorem for this wider class is provided.
</p>projecteuclid.org/euclid.pja/1590652896_20200528040152Thu, 28 May 2020 04:01 EDT