Diamond principles in Cichoń’s diagram

We present several models which satisfy CH and some -like principles while others fail, answering a question of Moore, Hruák and Damonja.     

Distributive proper forcing axiom and a left-right dichotomy of Cichoń’s diagram

In this paper, we study distributive proper forcing axiom (DPFA) and prove its consistency with a dichotomy of the Cichoń’s diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichoń’s diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.     

Dual Split Quaternions and Chasles’ Theorem in 3-Dimensional Minkowski Space {\mathbb{E}^{3}_{1}}

In 1831, Michel Chasles proved the existence of a fixed line under a general displacement in ${\mathbb{R}^3}$ . The fixed line called the screw axis of displacement was obtained by McCharthy in [10]. The purpose of this paper is to develop the method which is given for the pure rotation in [14], and thus to obtain the screw axis of spatial displacement in 3-dimensional Minkowski space. Firstly, we give a relation between dual vectors and lines in ${\mathbb{E}^{3}_{1}}$ , characterize the screw axis. Also, we discuss the dual split quaternion representation of a spatial displacement.     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.     

Certain Somos’s $$\varvec{}$$–$$\varvec{Q}$$Q type Dedekind $$\varvec{\eta }$$η-function identities

In this paper, we provide a new proof for the Dedekind \(\eta \)-function identities discovered by Somos. During this process, we found two new Dedekind \(\eta \)-function identities. Furthermore, we extract interesting partition identities from some of the \(\eta \)-function identities.     

Bäcklund’s Theorem for n Dimensional Lorentzian Submanifold in the Minkowski Space {\mathbb{E}_1^2n-1}

In this paper, Bäcklund’s Theorem is introduced on the Lorentzian n-submanifold of the Minkowski space \({\mathbb{E}_{1}^{2n-1}}\) by using the method of moving frames. Also, we prove the Integrability Theorem for the Lorentzian n-submanifold of the Minkowski space \({\mathbb{E}_{1}^{2n-1}}\).     

Fredholm and regularity theory of Douglis–Nirenberg elliptic systems on {\mathbb{R}^{N}}

We give a fairly complete exposition of the Fredholm properties of the Douglis–Nirenberg elliptic systems on ${\mathbb{R}^{N}}$ in the classical (unweighted) L ^p Sobolev spaces and under “minimal” assumptions about the coefficients. These assumptions rule out the use of classical pseudodifferential operator theory, although it is indirectly of assistance in places. After generalizing a necessary and sufficient condition for Fredholmness, already known in special cases, various invariance properties are established (index, null space, etc.), with respect to p and the Douglis–Nirenberg numbers. Among other things, this requires getting around the problem that the L ^p spaces are not ordered by inclusion. In turn, with some work, invariance leads to a regularity theory more general than what can be obtained by the method of differential quotients.     

Deitmar’s versus To?n-Vaquié’s schemes over {\mathbb{F}_{1}}

Deitmar introduced schemes over ${\mathbb {F}_{1}}$ , the so-called “field with one element”, as certain spaces with an attached sheaf of monoids, generalizing the definition of schemes as ringed spaces. On the other hand, To?n and Vaquié defined them as particular Zariski sheaves over the opposite category of monoids, generalizing the definition of schemes as functors of points. We show the equivalence between Deitmar’s and To?n-Vaquiés notions and establish an analog of the classical case of schemes over ${\mathbb {Z}}$ . This result has been assumed by the leading experts on ${\mathbb {F}_{1}}$ , but no proof was given. During the proof, we also conclude some new basic results on commutative algebra of monoids, such as a characterization of local flat epimorphisms and of flat epimorphisms of finite presentation. We also inspect the base-change functors from the category of schemes over ${\mathbb {F}_{1}}$ to the category of schemes over ${\mathbb {Z}}$ .     

S-forcing IIa: Adding diamonds and more applications: Coding sets,Arhangel’skii’s problem and ℒ[Q_1^{< \omega } ,Q_2^1 ]

We continue our earlier paper [20] by proving the equivalence, for regularκ>ω, of the existence of (κ, 1) morasses with built-in ♦ sequences and a strengthening, S_K◊ , of the forcing principle, S_K◊ of [20]. We obtain various applications of S_K◊, to wit: the existence of a stationary subset of [K₊]_<K with sup as coding function, the existence of a counterexample to Arhangel’skii’s conjecture (κ=ℵ₁) and compactness, axiomatizability and transfer properties for the Magidor-Malitz language ℒ (κ=ℵ₁). Research partially supported by NSF Grant MCS 8301042.     

An asymptotic version of Dumnicki’s algorithm for linear systems in $${\mathbb{CP}^2}$$

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .      

Vojta’s conjecture on blowups of $${\mathbb{P}^n}$$, greatest common divisors,and the abc conjecture

We will prove some cases of Vojta’s conjecture on blowups of \({\mathbb{P}^n}\), using Schmidt’s subspace theorem. The results can be stated as inequalities of greatest common divisors. Moreover, from Vojta’s conjecture on one further blowup at an infinitely near point, we derive a still-open special case of the abc-conjecture.     

Extension sets,affine designs,and Hamada’s conjecture

We introduce the notion of an extension set for an affine plane of order q to study affine designs \({\mathcal {D}}'\) with the same parameters as, but not isomorphic to, the classical affine design \({\mathcal {D}} = \mathrm {AG}_2(3,q)\) formed by the points and planes of the affine space \(\mathrm {AG}(3,q)\) which are very close to this geometric example in the following sense: there are blocks \(B'\) and B of \({\mathcal {D}'}\) and \({\mathcal {D}}\), respectively, such that the residual structures \({\mathcal {D}}'_{B'}\) and \({\mathcal {D}}_B\) induced on the points not in \(B'\) and B, respectively, agree. Moreover, the structure \({\mathcal {D}}'(B')\) induced on \(B'\) is the q-fold multiple of an affine plane \({\mathcal {A}}'\) which is determined by an extension set for the affine plane \(B \cong AG(2,q)\). In particular, this new approach will result in a purely theoretical construction of the two known counterexamples to Hamada’s conjecture for the case \(\mathrm {AG}_2(3,4)\), which were discovered by Harada et al. [7] as the result of a computer search; a recent alternative construction, again via a computer search, is in [23]. On the other hand, we also prove that extension sets cannot possibly give any further counterexamples to Hamada’s conjecture for the case of affine designs with the parameters of some \(\mathrm {AG}_2(3,q)\); thus the two counterexamples for \(q=4\) might be truly sporadic. This seems to be the first result which establishes the validity of Hamada’s conjecture for some infinite class of affine designs of a special type. Nevertheless, affine designs which are that close to the classical geometric examples are of interest in themselves, and we provide both theoretical and computational results for some particular types of extension sets. Specifically, we obtain a theoretical construction for one of the two affine designs with the parameters of \(\mathrm {AG}_2(3,3)\) and 3-rank 11 and for an affine design with the parameters of \(\mathrm {AG}_2(3,4)\) and 2-rank 17 (in both cases, just one more than the rank of the classical example).     

Periods and Young’s diagram of Fermat–Euler’s geometrical progressions of residues

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.     

On commuting polynomial automorphisms of {\mathbb{C}}^{k} , k ≥ 3

We characterize the polynomial automorphisms of ${\mathbb{C}}^3We characterize the polynomial automorphisms of , which commute with a regular automorphism. We use their meromorphic extension to and consider their dynamics on the hyperplane at infinity. We conjecture the additional hypothesis under which the same characterization is true in all dimensions. We give a partial answer to a question of S. Smale that in our context can be formulated as follows: can any polynomial automorphism of be the uniform limit on compact sets of polynomial automorphisms with trivial centralizer (i.e. )? Partially supported by Progetto MURST di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse. Supported by Istituto Nazionale Alta Matematica, “F. Severi”, Roma and G.N.S.A.G.A., Roma.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

Classification of solutions to the higher order Liouville’s equation on {\mathbb{R}^{2m}}

We classify the solutions to the equation (−Δ) ^m u = (2m − 1)!e ^2mu on giving rise to a metric with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric at infinity, and we observe that the pull-back of this metric to S ^2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.