Rendiconti del Seminario Matematico della Università di PadovaThe Mathematical Journal of the University of Padua

ISSN 0041-8994
e-ISSN: 2240-2926

Forthcoming Articles

(Last update: October 1, 2021)

The following articles will soon be published in this journal. Their publication status can be :

• online first – final version available;
• accepted – manuscript available.

Click on the title to view the paper.

Manuscripts of the accepted Papers

• Central algebraic geometry and seminormality

Abstract: We develop the theory of central ideals on commutative domains. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties. We provide a construction of the central seminormalization by a decomposition theorem in elementary central gluings. The existence of a central seminormalization is established in the affine case and for real schemes.

received 12.04.2021, accepted 23.09.2021 (48 pages)
• Degeneration of K3 surfaces with non-symplectic automorphisms

Abstract: We prove that a K3 surface with an automorphism acting on the global $2$-forms by a primitive $m$-th root of unity, $m \neq 1,2,3,4,6$, does not degenerate (assuming the existence of the so-called Kulikov models). A key result used to prove this is the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the $l$-adic cohomology groups of the surface.

received 15.04.2021, accepted 13.07.2021 (16 pages)
• A new approach to de Rham-Witt complexes, after Bhatt-Lurie-Mathew

Abstract: This is a report on [Bhatt, Bhargav; Lurie, Jacob; Mathew, Akhil. Revisiting the de Rham-Witt complex. Astérisque No. 424 (2021), 165 pp. arXiv:1805.05501v3 (2018)]. We review the main definitions and properties of the saturated de Rham-Witt complexes constructed there. A few complements are added, concerning the genesis of the notion, examples, and finiteness properties and open problems.

received 30.06.2021, accepted 11.07.2021 (43 pages)
• Classification of $p$-groups via their $2$-nilpotent multipliers

Abstract: For a $p$-group of order $p^n$, it is known that the order of $2$-nilpotent multiplier is equal to $|\mathcal{M}^{(2)}(G)|=p^{\frac12n(n-1)(n-2)+3-s_2(G)}$ for an integer $s_2(G)$. In this article, we characterize all of non abelian $p$-groups satisfying in $s_2(G)\in\{1,2,3\}$.

received 03.10.2020, accepted 18.06.2021 (11 pages)
• A formula for the minimal perimeter of clusters with density

Abstract: This paper deals with the isoperimetric problem for clusters in a Euclidean space with double density. In particular, we show that a limit of an isoperimetric minimizing sequence of clusters with volumes $\bf V$ is always isoperimetric for its own volumes (which may be smaller than $\bf V$). In particular, if it is strictly smaller, we provide an explicit formula.

received 05.03.2021, accepted 18.06.2021 (15 pages)
• Localizations and completions of stable $\infty$-categories

Abstract: We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $\infty$-category $\mathscr{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a homotopy commutative algebra in $\mathscr{M}$ we show that $E$-nilpotent completion, $E$-localization, and a suitable formal completion agree on bounded below objects when $E$ satisfies some reasonable conditions.

received 12.05.2021, accepted 12.06.2021 (67 pages)
• Note on Algebraic Irregular Riemann--Hilbert Correspondence

Abstract: The subject of this paper is an algebraic version of the irregular Riemann--Hilbert correspondence which was mentioned in [Y. Ito, $\mathbb{C}$-constructible enhanced Ind-Sheaves, Tsukuba J. Math., 44, no.1 (2020), pp. 155-201, arXiv:1910.09954]. In particular, we prove an equivalence of categories between the triangulated category of holonomic $\mathcal{D}$-modules on a smooth algebraic variety $X$ over $\mathbb{C}$ and the one of algebraic $\mathbb{C}$-constructible enhanced ind-sheaves on a bordered space $X^{an}_\infty$. Moreover we show that there exists a t-structure on the latter triangulated category whose heart is equivalent to the abelian category of holonomic $\mathcal{D}$-modules on $X$. Furthermore we shall consider simple objects of its heart and minimal extensions of objects of its heart.

received 14.07.2020, accepted 25.05.2021 (38 pages)
• Canonical universal locally finite groups

Abstract: We prove that for $\lambda = \beth_\omega$ or just $\lambda$ a strong limit singular cardinal of cofinality $\aleph_0$, if there is a universal member in the class $\mathbf K^{\rm lf}_\lambda$ of all locally finite groups of cardinality $\lambda$, then there is a canonical one. This is parallel to special models for elementary classes, which is the replacement of universal homogeneous ones and saturated ones in cardinals $\lambda = \lambda^{< \lambda}$. For showing the existence we rely on the existence of enough indecomposable such groups as was proved in [S. Shelah, Density of indecomposable locally finite groups, Rend. Semin. Mat. Univ. Padova 144 (2020), 253-270. MR 4186458]. More generally, we also deal with the existence of universal members in general classes for such cardinals.

received 25.06.2020, accepted 02.05.2021 (23 pages)
• An explicit self-dual construction of complete cotorsion pairs in the relative context

Abstract: Let $R\rightarrow A$ be a homomorphism of associative rings, and let $(\mathcal{F},\mathcal{C})$ be a hereditary complete cotorsion pair in $R{\operatorname{\mathsf{--Mod}}}$. Let $(\mathcal{F}_A,\mathcal{C}_A)$ be the cotorsion pair in $A{\operatorname{\mathsf{--Mod}}}$ in which $\mathcal{F}_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $\mathcal{F}$. Assuming that the $\mathcal{F}$-resolution dimension of every left $R$-module is finite and the class $\mathcal{F}$ is preserved by the coinduction functor $Hom_R(A,{-})$, we show that $\mathcal{C}_A$ is the class of all direct summands of left $A$-modules finitely (co)filtered by $A$-modules coinduced from $R$-modules from $\mathcal{C}$. Assuming that the class $\mathcal{F}$ is closed under countable products and preserved by the functor $Hom_R(A,{-})$, we prove that $\mathcal{C}_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $\mathcal{C}$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $\mathcal{F}$ have finite $\mathcal{F}$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $\omega+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra [L. Positselski, \textit{Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures}, Appendix C in collaboration with D. Rumynin, Appendix D in collaboration with S. Arkhipov. Monografie Matematyczne IMPAN, vol. 70, Birkhäuser/Springer Basel, 2010. xxiv+349 pp]. In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz-Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.

received 17.02.2021, accepted 02.05.2021 (71 pages)
• Refined Kolmogorov inequalities for the binomial distribution

Abstract: This paper presents refined versions of the well known Kolmogorov maximal inequality for the binomial distribution.

received 06.08.2020, accepted 12.04.2021 (20 pages)
• Algebraic curves admitting non-collinear Galois points

Abstract: This paper presents a criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves and describes its application via a novel example of a plane curve with non-collinear Galois points. In addition, this paper presents a new characterisation of the Fermat curve in terms of non-collinear Galois points.

received 02.12.2020, accepted 10.04.2021 (8 pages)
• A short proof of a non-vanishing result by Conca, Krattenthaler and Watanabe

Abstract: See the paper itself.

received 23.03.2021, accepted 08.04.2021 (3 pages)
• Height pairings of 1-motives

Abstract: The purpose of this work is to generalize, in the context of 1-motives, the $p$-adic height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we define a global pairing between the rational points of a 1-motive and its dual. We also provide a local pairing between disjoint zero-cycles of degree zero on a curve, which is done by considering the Picard and Albanese 1-motives associated to the curve.

received 16.09.2020, accepted 24.03.2021 (33 pages)
• Residual supersingular Iwasawa theory and signed Iwasawa invariants

Abstract: For an odd prime $p$ and a supersingular elliptic curve over a number field, this article introduces a multi-signed residual Selmer group, under certain hypotheses on the base field. This group depends purely on the residual representation at $p$, yet captures information about the Iwasawa theoretic invariants of the signed $p^\infty$-Selmer group that arise in supersingular Iwasawa theory. Working in this residual setting provides a natural framework for studying congruences modulo $p$ in Iwasawa theory.

received 16.07.2020, accepted 28.02.2021 (48 pages)
• Weakly S-semipermutable subgroups and p-nilpotency

Abstract: A subgroup $H$ of a finite group $G$ is said to be $S$-semipermutable in $G$ if $HG_{p}=G_{p}H$ for every Sylow subgroup $G_{p}$ of $G$ with $(|H|,p)=1$. A subgroup $H$ of $G$ is said to be Weakly $S$-semipermutable in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $S$-permutable and $H\cap T$ is $S$-semipermutable in $G$. In this paper we prove that for a finite group $G$, if some cyclic subgroups or maximal subgroups of $G$ are Weakly $S$-semipermutable in $G$, then $G$ is $p$-nilpotent.

received 02.12.2020, accepted 20.02.2021 (13 pages)
• Variational Hodge conjecture for complete intersections on hypersurfaces in projective space

Abstract: In this paper we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.

received 29.02.2020, accepted 26.12.2020 (10 pages)
• Fibrations in sextic del Pezzo surfaces with mild singularities

Abstract: We study sextic del Pezzo surface fibrations via root stacks.

received 14.10.2019, accepted 06.12.2020 (20 pages)
• A characterization of finite $\sigma$-soluble $P \sigma T$-groups

Abstract: Let $\mathfrak{F}$ be a non-empty class of groups, $G$ a finite group and $\mathcal{L}(G)$ be the lattice of all subgroups of $G$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_G(H/K))\in \mathfrak{F}$. Let $\mathcal{L}_{c\mathfrak{F}}(G)$ be the set of subgroups $A$ of $G$ such that every chief factor of $G$ between $A^G$ and $A_G$ is $\mathfrak{F}$-central in $G$; $\mathcal{L}_{\mathfrak{F}}(G)$ be the set of subgroups $A$ of $G$ such that $A^G/A_G\in \mathfrak{F}$. In this paper, we study the influence of $\mathcal{L}_{\mathfrak{F}}(G)$ and $\mathcal{L}_{c\mathfrak{F}}(G)$ on the structure of $G$, where $\mathfrak{F}$ is a normally hereditary saturated formation containing all $\sigma$-nilpotent groups and $D=G^{\mathfrak{F}}$ is $\sigma$-soluble. Moreover, we give a new characterization of a finite $\sigma$-soluble group to be a $P\sigma T$-group.

received 20.05.2020, accepted 25.11.2020 (10 pages)
• On Siegel's problem for E-functions

Abstract: Siegel defined in 1929 two classes of power series, the $E$-functions and $G$-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. He asked whether any $E$-function can be represented as a polynomial with algebraic coefficients in a finite number of $E$-functions of the form ${}_pF_q(\lambda z^{q-p+1})$, $q\ge p\ge 1$, with rational parameters. The case of $E$-functions of differential order less than or equal to 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring ${\bf G}$ of values taken by analytic continuations of $G$-functions at algebraic points must be a subring of the relatively small'' ring ${\bf H}$ generated by algebraic numbers, $1/\pi$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of ${\bf G}$ is a coefficient of the asymptotic expansion of a suitable $E$-function, which completes previous results of ours. We then prove (in two steps) that the coefficients of the asymptotic expansion of a hypergeometric $E$-function with rational parameters are in ${\bf H}$. Finally, we prove a similar result for $G$-functions.

received 07.11.2019, accepted 15.11.2020 (28 pages)
• Convolution identities of poly-Cauchy numbers with level $2$

Abstract: Poly-Cauchy numbers with level $2$ are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we introduce the Stirling numbers of the first kind with level $2$ in order to establish some relations with poly-Cauchy numbers with level $2$. Then, we show several convolution identities of poly-Cauchy numbers with level $2$. In particular, that of three poly-Cauchy numbers with level $2$ can be expressed as a simple form. \\ {\bf Keywords:} Poly-Cauchy numbers, hyperbolic functions, inverse hyperbolic functions, convolutions, Stirling numbers of the first kind

received 19.09.2020, accepted 11/11/2020 (19 pages)
• On the extension of even families of non-congruent numbers

Abstract: A method that extends existing families of even non-congruent numbers to produce new families of non-congruent numbers with arbitrarily many distinct prime factors is presented. We show that infinitely many new non-congruent numbers can be generated by appending a suitable collection of primes onto any even non-congruent number whose corresponding congruent number elliptic curve has 2-Selmer rank of zero. Our method relies upon Monsky's formula for computing the 2-Selmer rank of the congruent number elliptic curve. Even non-congruent numbers constructed according to our result have an unlimited number of prime factors in each odd congruence class modulo eight, and have congruent number elliptic curves with 2-Selmer rank equal to zero.

received 15.07.2019, accepted 02.10.2020 (29 pages)
• G-equivariance of formal models of flag varieties

Abstract: Let ${\mathbb{G}}$ be a split connected reductive group scheme over the ring of integers $\mathfrak{o}$ of a finite extension $L|{\mathbb{Q}}_p$ and $\lambda\in X({\mathbb{T}})$ an algebraic character of a split maximal torus ${\mathbb{T}}\subseteq\mathbb{G}$. Let us also consider $X^{\text{rig}}$ the rigid analytic flag variety of $\mathbb{G}$ and $G=\mathbb{G}(L)$. In the first part of this paper, we introduce a family of $\lambda$-twisted differential operators on a formal model $\mathcal{Y}$ of $X^{\text{rig}}$. We compute their global sections and we prove coherence together with several cohomological properties. In the second part, we define the category of coadmissible $G$-equivariant arithmetic $\mathcal{D}(\lambda)$-modules over the family of formal models of the rigid flag variety $X^{\text{rig}}$. We show that if $\lambda$ is such that $\lambda + \rho$ is dominant and regular ($\rho$ being the Weyl character), then the preceding category is anti-equivalent to the category of admissible locally analytic $G$-representations, with central character $\lambda$. In particular, we generalize the results in [C. HUYGHE – D. PATEL – T. SCHMIDT – M. STRAUCH “$\mathcal{D}^\dagger$-affinity of formal models of flag varieties”. arXiv:1501.05837v2 (to appear in Mathematical Research Letters). Preprint 2017] for algebraic characters.

received 25.02.2020, accepted 22.09.2020 (78 pages)
• Huppert’s conjecture and almost simple groups

Abstract: Let $G$ be a finite group and $cd(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is $H_{0}=PSL(2,q)$ with $q=2^{f}$ ($f$ prime) such that $cd(G) =cd(H)$, then there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. In view of Huppert's conjecture (2000), the main result of this paper gives rise to some examples that $G$ is not necessarily a direct product of $A$ and $H$, and consequently, we cannot extend this conjecture to almost simple groups.

received 23.02.2020, accepted 08.09.2020 (13 pages)
• Finite groups with $H_{\cal L}$-embedded subgroups

Abstract: Let $G$ be a finite soluble group and let $\mathfrak{F}$ be a class of groups. A chief factor $H/K$ of $G$ is said to be $\mathfrak{F}$-central (in $G$) if $(H/K)\rtimes (G/C_{G}(H/K)) \in \mathfrak{F}$; we write ${\cal L}_{c\mathfrak{F}}(G)$ to denote the set of all subgroups $A$ of $G$ such that every chief factor $H/K$ of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$ central in $G$. Let $\cal L$ be a set of subgroups of $G$. We say that a subgroup $A$ of $G$ is $H_{\cal L}$-embedded in $G$ provided $A$ is a Hall subgroup of some subgroup $E\in {\cal L}$. In this paper, we study the structure of $G$ under the condition that every subgroup of $G$ is $H_{\cal L}$-embedded in $G$, where ${\cal L}={\cal L}_{c\mathfrak{F}}(G)$ for some hereditary saturated formation $\mathfrak{F}$. Some known results are generalized.

received 13.10.2019, accepted 07.09.2020 (11 pages)
• Filtration relative, l'idéal de Bernstein et ses pentes

Abstract: Let $f_i:X\rightarrow{\bf C}$, for $i$ integer between $1$ and $p$, be analytic functions defined on a complex analytic variety $X$. Let us consider ${\cal D}_X$ the ring of linear differential operators and ${\cal D}_X [s_1, \ldots, s_p] = {\bf C}_X [s_1, \ldots, s_p] \otimes_ {\bf C} {\cal D}_X$. Let $m$ be a section of a holonomic ${\cal D}_X$-Module. We denote ${\cal B}(m, x_0, f_1, \ldots, f_p)$ the ideal of ${ \bf C} [s_1, \ldots, s_p]$ constituted by the polynomials $b$ satisfying in the neighborhood of $x_0\in X$ : $$B (s_1, \ldots, s_p) m f_1^{ s_1} \ldots f_p ^{s_p} \in {\cal D}_X [s_1, \ldots, s_p] \, m f_1^{s_1 + 1} \ldots f_p^{s_p + 1} \; .$$ This ideal is called Bernstein's ideal. C. Sabbah shows the existence for every $x_0\in X$ of a finite set ${\cal H}$ of linear forms with coefficients in ${\bf N}$, such that: $$\prod_{H \in {\cal H}} \prod_{i \in I_{\cal H}} (H (s_1, \ldots, s_p) + \alpha_{H , i}) \in {\cal B} (m, x_0, f_1, \ldots, f_p) \;,$$ where $\alpha_{H,i}$ are complex numbers. The purpose of this article is to show in particular the existence of a minimal set ${\cal H}$. In addition, when $m$ is a section of a holonomic regular ${\cal D}_X$-Module, we will precise geometrically this set from the characteristic variety of ${\cal D}_X$-Module generated by $m$. We introduce and study especially the relative characteristic variety of the ${\cal D}_X[s_1, \ldots, s_p]$-Modules related to our problem. This allows to specify the structure of the Bernstein's ideals.

received 07.04.2020, accepted 04.08.2020 (51 pages)
• Ding modules and dimensions over formal triangular matrix rings

Abstract: Let $T=\biggl(\begin{matrix} A&0\\U&B \end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_{A}$ and $_{B}U$ have finite flat dimensions, then a left $T$-module $\biggl(\begin{matrix} M_{1}\\ M_{2}\end{matrix}\biggr)_{\varphi^{M}}$ is Ding projective if and only if $M_{1}$ and $M_{2}/\mathrm{im}(\varphi^{M})$ are Ding projective and the morphism $\varphi^{M}$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a right $T$-module $(W_{1}, W_{2})_{\varphi_{W}}$ is Ding injective if and only if $W_{1}$ and $\ker(\widetilde{{\varphi_{W}}})$ are Ding injective and the morphism $\widetilde{{\varphi_{W}}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.

received 25.05.2019, accepted 03.08.2020 (22 pages)
• Markov triples with two Fibonacci components

Abstract: In this paper, we prove that there are at most finitely many pairs of Fibonacci numbers $(x,y)=(F_m,F_n)$ with the property that $m\le n$ and the pair $(m,n)\not\in \{(1,2r-1),~(1,2),~(2,2r+1),~(2r+1,2r+3): r\ge 1\}$ such that $(x,y,z)$ is a Markov triple for some integer $z$.

received 30.06.2020, accepted 26.07.2020 (33 pages)
• Inequalities involving pi(x)

Abstract: We present several inequalities involving the prime-counting function $\pi(x)$. Here, we give two examples of our results. We show that $$\frac{16}{9} \pi(x)\pi(y)\leq \pi^2(x+y)$$ is valid for all integers $x,y\geq 2$. The constant factor $16/9$ is best possible. The special case $x=y$ leads to $$\frac{4}{3}\leq \frac{\pi(2x)}{\pi(x)} \quad (x=2,3,...),$$ where the lower bound $4/3$ is sharp. This complements Landau's well-known inequality $$\frac{\pi(2x)}{\pi(x)}\leq 2 \quad (x=2,3,...).$$ Moreover, we prove that the inequality $$\Bigl(2\frac{\pi(x+y)}{x+y}\Bigr)^s \leq \Bigl( \frac{\pi(x)}{x}\Bigr)^s + \Bigl(\frac{\pi(y)}{y}\Bigr)^s \quad (0< s\in \mathbb{R})$$ holds for all integers $x,y\geq 2$ if and only if $s\leq s_0=0.94745...$. Here, $s_0$ is the only positive solution of $$\Bigl( \frac{16}{7}\Bigr)^t -\Bigl( \frac{6}{5}\Bigr)^t=1.$$

received 04.02.2020, accepted 20.07.2020 (15 pages)
• Quasi-variational Structure Approach to Systems with Degenerate Diffusions

Abstract: We consider the following system consisting of one strongly nonlinear partial differential inclusion (PDI in short) with one linear PDE and one ODE, which describes a tumor invasion phenomenon with a haptotaxis effect and was originally proposed in [Chaplain and Anderson, Mathematical modelling of tissue invasion, Cancer modelling and simulation'', 269-297, Chapman & Hall/CRC Math. Biol. Med. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2003]: \begin{equation*} \left\{ \begin{array}{l} u_t-\nabla \cdot (d_u (v) \nabla \beta (v\,;w)-u \nabla \lambda (v))+\beta (v\,;u) \ni 0,\\ v_t=-avw,\\ w_t=d_w \Delta w -bw+cu. \end{array} \right. \end{equation*} This system has two interesting characteristics. One is that the diffusion coefficient $d_u$ for the unknown function $u$ in the partial differential inclusion depends on the function $v$, which is also unknown in this system. The other is that the diffusion flux $\nabla \beta (v\,;u)$ of $u$ also depends on $v$. Moreover, we are especially interested in the case that $\beta (v\,;u)$ is nonsmooth and degenerate in general under suitable assumptions. These facts make it difficult for us to treat this system mathematically. In order to overcome these mathematical difficulties, we apply the theory of evolution inclusions on the real Hilbert space $V^*$, the dual space of $V$, with a quasi-variational structure for the inner products, which is established in [A. Ito, Evolution inclusion on a real Hilbert space with quasi-variational structure for inner products, J. Convex Analysis, 26 (2019), 1185-1252], and show the existence of time global solutions to the initial-boundary value problem for this system.

received 05.03.2020, accepted 11.07.2020 (64 pages)
• On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents

Abstract: We consider a differential system $x\frac{d}{dx} Y=GY$, where $G$ is a $m\times m$ matrix whose coefficients are power series which converge and are bounded on the open unit disc $D(0,1^-)$. Assume that $G(0)$ is a diagonal matrix with $p$-adic integer coefficients. Then there exists a solution matrix of the form $Y=F \exp(G(0)\log x)$ at $x=0$ if all exponent differences are $p$-adically non-Liouville numbers. We give an example where $F$ is analytic on the $p$-adic open unit disc and has log-growth grater than $m$. Under some conditions, we prove that if a solution matrix at a generic point has log-growth $\delta$, then $F$ has log-growth $\delta$.

received 11.12.2019, accepted 02.07.2020 (6 pages)
• Une construction d'extensions faiblement non ramifiées d'un anneau de valuation

Abstract: Given a valuation ring $V$, with residue field $F$ and value group $\Gamma$, we give a sufficient condition for a local ring dominating $V$ to be a valuation ring with the same value group. When $V$ contains a field $k$, we apply this result to the problem of constructing a valuation ring $W$ containing $V$ and a prescribed extension $k'$ of $k$, with value group $\Gamma$ and residue field generated by $k'$ and $F$; this is possible in particular when either $k'$ or $F$ is separable over $k$.

received 16.03.2020, accepted 18.06.2020 (14 pages)
• Rational motivic path spaces and Kim’s relative unipotent section conjecture

Abstract: We develop the foundations of commutative algebra objects in the category of motives, which we call motivic dga's''. Work of White and of Cisinski-D\'eglise provides us with a suitable model structure. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga, and provides us with a factorization of Kim's relative unipotent section conjecture into several smaller conjectures with a homotopical flavor.

received 03.02.2020, accepted 06.06.2020 (53 pages)
• Nonexistence of Minimizers for a Nonlocal Perimeter with a Riesz and a Background Potential

Abstract: We consider the nonexistence of minimizers for the energy containing a nonlocal perimeter with a general kernel $K$, a Riesz potential, and a background potential in $\mathbb{R}^N$ with $N\geq2$ under the volume constraint. We show that the energy has no minimizer for a sufficiently large volume under suitable assumptions on $K$. The proof is based on the partition of a minimizer and the comparison of the sum of the energy for each part with the energy for the original configuration.

received 14.04.2020, accepted 03.06.2020 (27 pages)
• Bloch's Theorem for Heat Mappings

Abstract: In this paper we give a proof via the contraction mapping principle of a Bloch-type theorem for (normalised) heat Bochner-Takahashi $K$-mappings, that is, mappings that are solutions to the heat equation, and which also satisfy a weak form of $K$-quasiregularity. We also provide estimates from below for the radius of the univalent balls covered by this family of functions.

received 05.02.2020, accepted 16.05.2020 (20 pages)
• Torsors under abelian schemes via Picard schemes

Abstract: Given a torsor $P$ under a principally polarised abelian scheme $J$, projective over a noetherian base, we use the Picard scheme of $P$ to write down an explicit extension of $\mathbb{Z}$ by $J$ giving the class of $P$. As an application we give a version of Bhatt's period-index result valid over an arbitrary noetherian base.

received 02.03.2020, accepted 07.04.2020 (10 pages)
• Analytic General Solutions of nonlinear second-order $q$-Difference Equations with a double characteristic value

Abstract: As far as the author knows it seems that an existence theorem of a solution of a general nonlinear $q$-difference equation is not known. In this paper we will investigate a nonlinear second order $q$-difference equation whose characteristic equation has only one solution and will show analytic general solutions of such an equation. Further we will show an example.

received 17.02.2020, accepted 26.03.2020 (23 pages)
• On join irreducible J-trivial semigroups

Abstract: A pseudovariety of semigroups is join irreducible if whenever it is contained in the complete join of some pseudovarieties, then it is contained in one of the pseudovarieties. A finite semigroup is join irreducible if it generates a join irreducible pseudovariety. New finite $\mathscr{J}$-trivial semigroups $\mathcal{C}_n$ ($n \geq 2$) are exhibited with the property that while each $\mathcal{C}_n$ is not join irreducible, the monoid $\mathcal{C}_n^I$ is join irreducible. The monoids $\mathcal{C}_n^I$ are the first examples of join irreducible $\mathscr{J}$-trivial semigroups that generate pseudovarieties that are not self-dual. Several sufficient conditions are also established under which a finite semigroup is not join irreducible. Based on these results, join irreducible pseudovarieties generated by a $\mathscr{J}$-trivial semigroup of order up to six are completely described. It turns out that besides known examples and those generated by $\mathcal{C}_2^I$ and its dual monoid, there are no further examples.

received 18.07.2019, accepted 24.03.2020 (26 pages)
• A generalization of the total mean curvature

Abstract: A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is proved.

received 30.08.2019, accepted 18.03.2020 (6 pages)