An asymptotic formula for the primitive of Hardy’s function

Let Z(t) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term O(T ^1/6log?T) is given for the integral of Z(t) over the interval [0,T], with special attention paid to the critical cases when the fractional part of \(\sqrt{T/2\pi }\) is close to \(\frac{1}{4}\) or \(\frac{3}{4}\).     

Definable relations in finite-dimensional subspace lattices with involution

For a large class of finite dimensional inner product spaces V, over division \(*\)-rings F, we consider definable relations on the subspace lattice \(\mathsf{L}(V)\) of V, endowed with the operation of taking orthogonals. In particular, we establish translations between the relevant first order languages, in order to associate these relations with definable and invariant relations on F—focussing on the quantification type of defining formulas. As an intermediate structure we consider the \(*\)-ring \(\mathsf{R}(V)\) of endomorphisms of V, thereby identifying \(\mathsf{L}(V)\) with the lattice of right ideals of \(\mathsf{R}(V)\), with the induced involution. As an application, model completeness of F is shown to imply that of \(\mathsf{R}(V)\) and \(\mathsf{L}(V)\).     

Nonequilibrium statistical mechanics of a solid immersed in a continuum

We study actions of the symmetric group S₄ on K3 surfaces X that satisfy the following condition: there exists an equivariant birational contraction \(\bar r:X \to \bar X\) to a K3 surface \(\bar X\) with ADE singularities such that the quotient space \(\bar X\)/S₄ is isomorphic to P². We prove that up to smooth equivariant deformations there exist exactly 15 such actions of the group S₄ on K3 surfaces, and that these actions are realized as actions of the Galois groups on the Galoisations \(\bar X\) of the dualizing coverings of the plane which are associated with plane rational quartics without A₄, A₆, and E₆ singularities as their singular points.     

Dirac flow on the 3-sphere

We illustrate some well-known facts about the evolution of the 3-sphere (S³, g) generated by the Ricci flow. We define the Dirac flow and study the properties of the metric \(\bar g = dt^2 + g(t)\), where g(t) is a solution of the Dirac flow. In the case of a metric g conformally equivalent to the round metric on S³ the metric \(\bar g\) is of constant curvature. We study the properties of solutions in the case when g depends on two functional parameters. The flow on differential 1-forms whose solution generates the Eguchi–Hanson metric was written down. In particular cases we study the singularities developed by these flows.     

A characterization of the Riemann extension in terms of harmonicity

If (M,?) is a manifold with a symmetric linear connection, then T*M can be endowed with the natural Riemann extension \(\bar g\) (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to \(\bar g\) initiated by C. L.Bejan and O.Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure \(\mathcal{P}\) on (T*M, \(\bar g\)) and prove that \(\mathcal{P}\) is harmonic (in the sense of E.Garciá-Río, L.Vanhecke and M. E.Vázquez-Abal (1997)) if and only if \(\bar g\) reduces to the classical Riemann extension introduced by E.M. Patterson and A.G. Walker (1952).     

Pointfree Functorial Polar Functions

C denotes the category of compact regular frames with frame homomorphisms. A function \(\mathcal {X}\), which assigns to each C-object F a subalgebra of \(\mathcal {P}(F)\) that contains the complemented elements of F is said to be a polar function. An essential extension H of F is a \(\mathcal {X}\)-splitting frame of F if whenever \(p \in \mathcal {X}(F)\), then the polar generated by p in H is complemented. For F∈ C we examine the least \(\mathcal {X}\)-splitting extension and prove that every invariant polar function generates a C-hull class of frames. In addition, we define the concept of a functorial polar function and prove that each functorial polar function generates an epireflective subcategory of the category compact regular frames with skeletal maps.     

Revisiting Fryszkowski’s problem

Professor Andrzej Fryszkowski formulated, at the 2nd Symposium on Nonlinear Analysis in Toruń, September 13–17, 1999, the following problem: given \(\alpha \in (0,1)\), an arbitrary non-empty set \(\Omega \) and a set-valued mapping \(F:\Omega \rightarrow 2^{\Omega }\), find necessary and (or) sufficient conditions for the existence of a (complete) metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d. Com?neci (Stud. Univ. Babe?-Bolyai Math. 62:537–542, 2017) provided necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d, in case that \(\alpha \in (0,\frac{1}{2})\) and there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) . We improve Com?neci’s result by allowing \(\alpha \) to belong to the interval (0, 1). In addition, we provide necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) such that F is a Nadler set-valued \(\alpha \)-similarity with respect to d, in case that \(\alpha \in (0,1)\), there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) and F is non-overlapping.     

Extremes of Gaussian random fields with regularly varying dependence structure

Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ ²(?) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t ₀. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t ₀. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t ₀. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.     

On the Stability of the Zero Solution of a Second-Order Differential Equation under a Periodic Perturbation of the Center

Small periodic perturbations of the oscillator \(\ddot x + {x^{2n}}\) sgn x = Y(t, x, \(\dot x\)) are considered, where n < 1 is a positive integer and the right-hand side is a small perturbation periodic in t, which is an analytic function in \(\dot x\) and x in a neighborhood of the origin. New Lyapunov-type periodic functions are introduced and used to investigate the stability of the equilibrium position of the given equation. Sufficient conditions for asymptotic stability and instability are given.     

Witten’s perturbation on strata with general adapted metrics

Let M be a stratum of a compact stratified space A. It is equipped with a general adapted metric g, which is slightly more general than the adapted metrics of Nagase and Brasselet–Hector–Saralegi. In particular, g has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then, g is called good. We consider the maximum/minimum ideal boundary condition, \(d_{\mathrm{max/min}}\), of the compactly supported de Rham complex on M, in the sense of Brüning–Lesch. Let \(H^*_{\mathrm{max/min}}(M)\) and \(\Delta _{\mathrm{max/min}}\) denote the cohomology and Laplacian of \(d_{\mathrm{max/min}}\). The first main theorem states that \(\Delta _{\mathrm{max/min}}\) has a discrete spectrum satisfying a weak form of the Weyl’s asymptotic formula. The second main theorem is a version of Morse inequalities using \(H_{\mathrm{max/min}}^*(M)\) and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for \(d_{\mathrm{max/min}}\) of the Witten’s perturbation of the de Rham complex. Another ingredient is certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory. The condition on g to be good is general enough in the following sense. Assume that A is a stratified pseudomanifold, and consider its intersection homology \(I^{\bar{p}}H_*(A)\) with perversity \(\bar{p}\); in particular, the lower and upper middle perversities are denoted by \(\bar{m}\) and \(\bar{n}\), respectively. Then, for any perversity \(\bar{p}\le \bar{m}\), there is an associated good adapted metric on M satisfying the Nagase isomorphism \(H^r_{\mathrm{max}}(M)\cong I^{\bar{p}}H_r(A)^*\) (\(r\in \mathbb {N}\)). If M is oriented and \(\bar{p}\ge \bar{n}\), we also get \(H^r_{\mathrm{min}}(M)\cong I^{\bar{p}}H_r(A)\). Thus our version of the Morse inequalities can be described in terms of \(I^{\bar{p}}H_*(A)\).     

Joint universality for Dirichlet L-functions and zeros of their linear combinations in the $$\chi $$-aspect

We establish a joint universality theorem for Dirichlet L-functions in the character aspect. This is an extension of a result obtained by Bagchi and Gonek independently, and is an analogue of the joint universality for Dirichlet L-functions in the t-aspect. Zeros of linear combinations of Dirichlet L-functions in the t-aspect have been investigated by various authors. Using our joint universality theorem, we investigate zeros of such combinations from a new viewpoint. More precisely, we show that for any region \(\Omega \) in the strip \(1/2< \mathrm {Re}\,s <1\), any non-zero meromorphic functions \(H_1 (s), \dots , H_r(s)\) on \(\Omega \) with \(r \ge 2\) and any positive integer N, there exist a positive integer m and Dirichlet characters \(\varphi _1, \dots , \varphi _r \bmod m\) such that \(\sum _{j=1}^r H_j (s) L(s, \varphi _r)\) has at least N distinct zeros in \(\Omega \).     

Sets with even partition functions and 2-adic integers

For P ? \(\mathbb{F}_2 \)[z] with _P(0) = 1 and deg(_P) ≥ 1, let \(\mathcal{A}\) = \(\mathcal{A}\)(P) (cf. [4], [5], [13]) be the unique subset of ? such that Σ _n≥0 p(\(\mathcal{A}\), n)z ⁿ ≡ P(z) (mod 2), where p(\(\mathcal{A}\), n) is the number of partitions of n with parts in \(\mathcal{A}\). Let p be an odd prime and P ? \(\mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + z ^p in \(\mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \(\mathcal{A}\) = \(\mathcal{A}\)(P) of the form 2 ^k m are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Saïd and J.-L. Nicolas [6] to all primes p.     

Subgraph Transversal of Graphs

Given graphs F and G, denote by \({\tau_F}(G)\) the cardinality of a smallest subset \(T {\subseteq}V(G)\) that meets every maximal F-free subgraph of G. Erdös, Gallai and Tuza [9] considered the question of bounding \(\tau_{\overline{K_2}}(G)\) by a constant fraction of |G|. In this paper, we will give a complete answer to the following question: for which F, is τ _F (G) bounded by a constant fraction of |G|?In addition, for those graphs F for which \({\tau_F}(G)\) is not bounded by any fraction of |G|, we prove that \(\tau_F(G)\le|G|-\frac{1}{2}\sqrt{|G|}+\frac{1}{2}\), provided F is not K _k or \(\overline{K_k}\).     

A characterization of the normal distribution using stationary max-stable processes

Consider a max-stable process of the form \(\eta (t) = \max _{i\in \mathbb {N}} U_{i} \mathrm {e}^{\langle X_{i}, t\rangle - \kappa (t)}\), \(t\in \mathbb {R}^{d}\), where \(\{U_{i}, i\in \mathbb {N}\}\) are points of the Poisson process with intensity u ^?2du on (0,∞), X _i , \(i\in \mathbb {N}\), are independent copies of a random d-variate vector X (that are independent of the Poisson process), and \(\kappa :\mathbb {R}^{d} \to \mathbb {R}\) is a function. We show that the process η is stationary if and only if X has multivariate normal distribution and κ(t)?κ(0) is the cumulant generating function of X. In this case, η is a max-stable process introduced by R. L. Smith.     

Triviality of Iwasawa module associated to some abelian fields of prime conductors

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.     

Generalized Kloosterman sum with primes

The work is devoted to generalized Kloosterman sums modulo a prime, i.e., trigonometric sums of the form \(\sum\nolimits_{p \leqslant x} {\exp \left\{ {2\pi i\left( {a\bar p + {F_k}\left( p \right)} \right)/q} \right\}} \) and \(\sum\nolimits_{n \leqslant x} {\mu \left( n \right)\exp \left\{ {2\pi i\left( {a\bar n + {F_k}\left( n \right)} \right)/q} \right\}} \), where q is a prime number, \(\left( {a,q} \right) = 1,m\bar m \equiv 1\left( {\bmod {\kern 1pt} q} \right)\), F _k (u) is a polynomial of degree k ≥ 2 with integer coefficients, and p runs over prime numbers. An upper estimate with a power saving is obtained for the absolute values of such sums for x ≥ q^1/2+ε.     

Levi Extension Theorems for Meromorphic Functions of Weak Type in Infinite Dimension

The aim of paper is to give some results, that prepare for studying the problem on cross theorems for separately \((\cdot , W)\)-meromorphic functions. Some general versions of extension theorem of Levi type are extended to the classes of meromorphic functions f on \(D \times (\Delta _r {\setminus } \overline{\Delta })\) with values in a locally convex space F. Here, the function f is assumed that, for each \(z \in D_*,\) the function \(f_z = f(z, \cdot )\) has a (F, W)-meromorphic extension to \(\Delta _r,\) where F is either a locally (or sequentially) complete locally convex space or a Fréchet space, the space \(W \subseteq F'\) is separating (or determines boundedness), \(\Delta _r = \{\lambda \in {\mathbb C}: |\lambda | < r\}\) with \(r > 1, \Delta = \Delta _1\) and D is either a domain in \({\mathbb C}^n\) or a balanced domain in a Fréchet space containing a non-pluripolar balanced convex compact subset, \(D_*\) is dense in D.     

Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $$\mathbb {C}^2$$

Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).

     

Torsion and divisibility in finitely generated commutative semirings

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties—torsion and almost-divisibility (an element \(a\in S\) is called almost-divisible in S if there is \(b\in \mathbb {N}\cdot a\) such that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements \(f\in F(X)\) such that for every epimorphism \(\pi \) of F(X) torsion and almost-divisibility of \(\pi (f)\) are equivalent in \(\pi (F(X))\).