Kleisli Monoids Describing Approach Spaces

We study the functional ideal monad \(\mathbb {I} = (\mathsf {I}, m, e)\) on S e t and show that this monad is power-enriched. This leads us to the category \(\mathbb {I}\)- M o n of all \(\mathbb {I}\)-monoids with structure preserving maps. We show that this category is isomorphic to A p p, the category of approach spaces with contractions as morphisms. Through the concrete isomorphism, an \(\mathbb {I}\)-monoid (X,ν) corresponds to an approach space \((X, \mathfrak {A}),\) described in terms of its bounded local approach system. When I is extended to R e l using the Kleisli extension \(\check {\mathsf {I}},\) from the fact that \(\mathbb {I}\)- M o n and \((\mathbb {I},2)\)- C a t are isomorphic, we obtain the result that A p p can be isomorphically described in terms of convergence of functional ideals, based on the two axioms of relational algebras, reflexivity and transitivity. We compare these axioms to the ones put forward in Lowen (2015). Considering the submonad \(\mathbb {B}\) of all prime functional ideals, we show that it is both sup-dense and interpolating in \(\mathbb {I}\), from which we get that \((\mathbb {I},2)\)- C a t and \((\mathbb {B},2)\)- C a t are isomorphic. We present some simple axioms describing A p p in terms of prime functional ideal convergence.     

Conditional equations for quadratic functions

We consider quadratic functions f that satisfy the additional equation y² f(x) =  x² f(y) for the pairs \({ (x,y) \in \mathbb{R}^2}\) that fulfill the condition P(x, y) =  0 for some fixed polynomial P of two variables. If P(x, y) =  ax +  by +  c with \({ a , b , c \in \mathbb{R}}\) and \({(a^2 + b^2)c \neq 0}\) or P(x,y) =  x ⁿ ? y with a natural number \({n \geq 2}\), we prove that f(x) =  f(1) x² for all \({x \in \mathbb{R}}\). Some related problems, admitting quadratic functions generated by derivations, are considered as well.     

Equiconsistencies at subcompact cardinals

We present equiconsistency results at the level of subcompact cardinals. Assuming SBH_δ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □(δ) and □_δ fail, then δ is subcompact in a class inner model. If in addition □(δ⁺) fails, we prove that δ is \({\Pi_1^2}\) subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH_δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a \({\Pi_1^2}\) subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ⁺⁽ⁿ⁾ supercompact for all n < ω. We state some results at this level, and indicate how they are proved.     

A note on Jeśmanowicz’ conjecture

Je?manowicz [9] conjectured that, for positive integers m and n with m > n, gcd(m,n) = 1 and \({m\not\equiv n\pmod{2}}\), the exponential Diophantine equation \({(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z}\) has only the positive integer solution (x, y, z) = (2, 2, 2). We prove the conjecture for \({2 \| mn}\) and m + n has a prime factor p with \({p\not\equiv1\pmod{16}}\).     

O n ( m n,n, m n,m) relative differe nce sets with $$\text {gcd}( m,n)=1$$gcd( m,n)=1

There has been much research on \((p^{a},p^{b},p^{a},p^{a-b})\) relative difference sets with p a prime, while there are only a few results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\). The non-existence results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\) have only been obtained for the following five cases: (1) \(m=p,\ n=q,\ p>q\); (2) \(m=pq,\ n=3,\ p,q>3\); (3) \(m=4,\ n=p\); (4) \(m=2\) and (5) \(n=p\), where p, q are distinct odd primes. For the existence results, there are only four constructions of semi-regular relative difference sets in groups of size not a prime power with the forbidden subgroup having size larger than 2. In this paper, we present some more non-existence results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\). In particular, our result is a generalization of the main result of Hiramine’s work (J Comb Theory Ser A 117(7):996–1003, 2010). Meanwhile, we give a construction of non-abelian (16q, q, 16q, 16) relative difference sets, where q is a prime power with \(q\equiv 1\pmod {4}\) and \(q>4.2\times 10^{8}\). This is the third known infinite classes of non-abelian semi-regular relative difference sets.     

On a variant of Rado’s selection lemma and its equivalence with the Boolean prime ideal theorem

We establish that, in ZF (i.e., Zermelo–Fraenkel set theory minus the Axiom of Choice AC), the statement RLT: Given a set I and a non-empty set \({\mathcal{F}}\) of non-empty elementary closed subsets of 2 ^I satisfying the fip, if \({\mathcal{F}}\) has a choice function, then \({\bigcap\mathcal{F} \ne \emptyset}\) , which was introduced in Morillon (Arch Math Logic 51(7–8):739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem (see Sect. 1 for terminology). The result provides, on one hand, an affirmative answer to Morillon’s corresponding question in Morillon (2012) and, on the other hand, a negative answer—in the setting of ZFA (i.e., ZF with the axiom of extensionality weakened to permit the existence of atoms)—to the question in Morillon (2012) of whether RLT is equivalent to Rado’s selection lemma.     

A spectral-like decomposition for transitive Anosov flows in dimension three

Given a (transitive or non-transitive) Anosov vector field X on a closed three dimensional manifold M, one may try to decompose (M, X) by cutting M along tori and Klein bottles transverse to X. We prove that one can find a finite collection \(\{S_1,\dots ,S_n\}\) of pairwise disjoint, pairwise non-parallel tori and Klein bottles transverse to X, such that the maximal invariant sets \(\Lambda _1,\dots ,\Lambda _m\) of the connected components \(V_1,\dots ,V_m\) of \(M-(S_1\cup \dots \cup S_n)\) satisfy the following properties:

• each \(\Lambda _i\) is a compact invariant locally maximal transitive set for X;
• the collection \(\{\Lambda _1,\dots ,\Lambda _m\}\) is canonically attached to the pair (M, X) (i.e. it can be defined independently of the collection of tori and Klein bottles \(\{S_1,\dots ,S_n\}\));
• the \(\Lambda _i\)’s are the smallest possible: for every (possibly infinite) collection \(\{S_i\}_{i\in I}\) of tori and Klein bottles transverse to X, the \(\Lambda _i\)’s are contained in the maximal invariant set of \(M-\cup _i S_i\).

To a certain extent, the sets \(\Lambda _1,\dots ,\Lambda _m\) are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition \(V_1,\dots ,V_m\), equipped with the restriction of the Anosov vector field X, are “almost unique up to topological equivalence”.

     

On needed reals

Given a binary relationR, we call a subsetA of the range ofR R-adequate if for everyx in the domain there is someyεA such that (x, y)εR. Following Blass [4], we call a realη ”needed” forR if in everyR-adequate set we find an element from whichη is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets,Cof(\(\mathcal{N}\)), is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get the related notion of being “weakly needed”. We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].     

On the approximation of continuous functions of two variables

It is shown that if P _m ^α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ _mn ^α,β (x, y)} _m,n=0 ^r = {P _m ^α,β (x)P _n ^α,β (y)} _{m, n=0} ^r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω _N×N = ?ub;(x _i , y _i ) _i,j=0 ^N , where x _i and y _i are the zeros of the Jacobi polynomial P _n ^α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]², we construct the discrete partial Fourier-Jacobi sums of the rectangular type S _{m, n, N} ^α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S _{m, n, N} ^α,β ∥ of the discrete sums S _{m, n, N} ^α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) ^{q + 1/2}), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S _{m, n, N} ^α,β (f; x, y) are considered.     

Involution words II: braid relations and atomic structures

Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori–Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted involutions x, y in a Coxeter group W with automorphism \(*\), we associate a set of involution words \(\hat{\mathcal {R}}_*(x,y)\). This set is the disjoint union of the reduced words of a set of group elements \(\mathcal {A}_*(x,y)\), which we call the atoms of y relative to x. The atoms, in turn, are contained in a larger set \(\mathcal {B}_*(x,y) \subset W\) with a similar definition, whose elements are referred to as Hecke atoms. Our main results concern some interesting properties of the sets \(\hat{\mathcal {R}}_*(x,y)\) and \(\mathcal {A}_*(x,y) \subset \mathcal {B}_*(x,y)\). For finite Coxeter groups, we prove that \(\mathcal {A}_*(1,y)\) consists of exactly the minimal-length elements \(w \in W\) such that \(w^* y \le w\) in Bruhat order, and we conjecture a more general property for arbitrary Coxeter groups. In type A, we describe a simple set of conditions characterizing the sets \(\mathcal {A}_*(x,y)\) for all involutions \(x,y \in S_n\), giving a common generalization of three recent theorems of Can et al. We show that the atoms of a fixed involution in the symmetric group (relative to \(x=1\)) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the “Chinese relation” studied by Cassaigne et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of “braid relations” spanning the involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto’s theorem for involution words in arbitrary Coxeter groups.     

Analogues of some Tauberian theorems for bounded variation

In this paper definitions for “bounded variation”, “subsequences”, “Pringsheim limit points”, and “stretchings” of a double sequence are presented. Using these definitions and the notion of regularity for four dimensional matrices, the following two questions will be answered. First, if there exists a four dimensional regular matrix A such that Ay = Σ _k,l=1,1 ^∞∞ a _m,n,k,l y _k,l is of bounded variation (BV) for every subsequence y of x, does it necessarily follow that x ∈ BV? Second, if there exists a four dimensional regular matrix A such that Ay ∈ BV for all stretchings y of x, does it necessarily follow that x ∈ BV? Also some natural implications and variations of the two Tauberian questions above will be presented.     

A Minimum Principle for Potentials with Application to Chebyshev Constants

For “Riesz-like” kernels K(x,y) = f(|x?y|) on A×A, where A is a compact d-regular set \(A\subset \mathbb {R}^{p}\), we prove a minimum principle for potentials \(U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)\), where μ is a Borel measure supported on A. Setting \(P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)\), the K-polarization of μ, the principle is used to show that if {ν _N } is a sequence of measures on A that converges in the weak-star sense to the measure ν, then P _K (ν _N )→P _K (ν) as \(N\to \infty \). The continuous Chebyshev (polarization) problem concerns maximizing P _K (μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes P _K (μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {ν _N } is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of ν _N as \(N \to \infty \) is a solution to the continuous problem.     

Maximum scattered linear sets and MRD-codes

The rank of a scattered \({\mathbb F}_q\)-linear set of \({{\mathrm{{PG}}}}(r-1,q^n)\), rn even, is at most rn / 2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r, n, q (rn even) for scattered \({\mathbb F}_q\)-linear sets of rank rn / 2. In this paper, we prove that the bound rn / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered \({\mathbb F}_q\)-linear sets of \({{\mathrm{{PG}}}}(1,q^n)\) of maximum rank n yield \({\mathbb F}_q\)-linear MRD-codes with dimension 2n and minimum distance \(n-1\). We generalize this result and show that scattered \({\mathbb F}_q\)-linear sets of \({{\mathrm{{PG}}}}(r-1,q^n)\) of maximum rank rn / 2 yield \({\mathbb F}_q\)-linear MRD-codes with dimension rn and minimum distance \(n-1\).     

Rigidity of unary algebras and its application to the $${\mathcal {HS} = \mathcal {SH}}$$ problem

H. P. Gumm and T. Schröder stated a conjecture that the preservation of preimages by a functor T for which |T1| = 1 is equivalent to the satisfaction of the class equality \({{\mathcal {HS}}({\sf K}) = {\mathcal {SH}}({\sf K})}\) for any class K of T-coalgebras. Although T. Brengos and V. Trnková gave a positive answer to this problem for a wide class of Set-endofunctors, they were unable to find the full solution. Using a construction of a rigid unary algebra we prove \({{\mathcal {HS}} \neq {\mathcal {SH}}}\) for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously.     

Improvements of the Weil bound for Artin–Schreier curves

For the Artin–Schreier curve y ^q ? y = f(x) defined over a finite field \({{\mathbb F}_q}\) of q elements, the celebrated Weil bound for the number of \({{\mathbb F}_{q^r}}\)-rational points can be sharp, especially in super-singular cases and when r is divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz’s work on ?-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra \({\sqrt{q}}\) factor in the error term.     

On the 1/3–2/3 Conjecture

Let (P, ≤) be a finite poset (partially ordered set), where P has cardinality n. Consider linear extensions of P as permutations x₁x₂?x_n in one-line notation. For distinct elements x, y ∈ P, we define ?(x ? y) to be the proportion of linear extensions of P in which x comes before y. For \(0\leq \alpha \leq \frac {1}{2}\), we say (x, y) is an α-balanced pair if α ≤ ?(x ? y) ≤?1 ? α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. We make progress on this conjecture by showing that it holds for certain families of posets. These include lattices such as the Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. We also consider various posets which satisfy the stronger condition of having a 1/2-balanced pair. For example, this happens when the poset has an automorphism with a cycle of length 2. Various questions for future research are posed.     

Erratum to: The generalized strong recurrence for non-zero rational parameters

In the present paper, we prove that self-approximation of \({\log \zeta (s)}\) with d = 0 is equivalent to the Riemann Hypothesis. Next, we show self-approximation of \({\log \zeta (s)}\) with respect to all nonzero real numbers d. Moreover, we partially filled a gap existing in “The strong recurrence for non-zero rational parameters” and prove self-approximation of \({\zeta(s)}\) for 0 ≠ d = a/b with |a?b| ≠ 1 and gcd(a,b) = 1.     

Classification of affine symmetry groups of orbit polytopes

Let G be a finite group acting linearly on a vector space V. We consider the linear symmetry groups \({\text {GL}}(Gv)\) of orbits \(Gv\subseteq V\), where the linear symmetry group \({\text {GL}}(S)\) of a subset \(S\subseteq V\) is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V, which is Zariski open in V, and show that the groups \({\text {GL}}(Gv)\) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group \({\text {GL}}(Gw)\) such that V is the linear span of Gw. If the underlying characteristic is zero, “isomorphic” can be replaced by “conjugate in \({\text {GL}}(V)\).” Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (Geom Dedicata 6(3):331–337, 1977.  https://doi.org/10.1007/BF02429904).     

On the Monge–Ampère equation with boundary blow-up: existence,uniqueness and asymptotics

We consider the Monge–Ampère equation det D ² u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ?Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) is positive in Ω and non-negative on ?Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in \({\mathbb R}^N\) with N ≥ 2. We give asymptotic estimates of the behaviour of such solutions near ?Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ?Ω and \(b\equiv 0\) on ?Ω.