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Let S be a multiplicative subset of a ring R. A right ideal A of R is referred to as S - principal if there exist an element s 2 S and a principal right ideal aR of R such that As ⊆ aR ⊆ A. A ring is referred to as an S -principal right ideal ring (S -PRIR) if every right ideal is S -principal. This paper examines S -PRIRs, which extend the notion of principal right ideal rings. Various examples, including several extensions of S -PRIRs are investigated, and some practical results are proven. A noncommutative S -PRIR that is not a principal right ideal ring is found, and the S -variants of the Eakin-Nagata-Eisenbud theorem and Cohen's theorem for S -PRIRs are proven. [ABSTRACT FROM AUTHOR]

We prove the following two basis theorems for ∑12-sets of reals: (1) Every nonthin ∑12-set has a perfect A12-subset if and only if it has a nonthin A12-subset, and this is equivalent to the statement that there is a nonconstructible real. (2) Every uncountable ∑12-set has an uncountable A12-subset if and only if either every real is constructible or ωL2 is countable. We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect Π12-set with no nonempty Π12-thin subset, strengthening a result of Harrington [4]. [ABSTRACT FROM AUTHOR]

For a given set $S\subset \mathbb{N}$ , $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s. Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$ , then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$. This solves a problem of Chen and Lev ['Integer sets with identical representation functions', Integers 16 (2016), A36] under the condition $m>r$. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$ , let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$. Kiss et al. ['Groups, partitions and representation functions', Publ. Math. Debrecen 85 (3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$ , then $|G|=2|A|$ , and (b) if $h$ is even and $|G|=2|A|$ , then $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$. We prove that $R_{G\setminus A,h}(g)-(-1)^{h}R_{A,h}(g)$ does not depend on $g$. In particular, if $h$ is even and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for some $g\in G$ , then $|G|=2|A|$. If $h>1$ is odd and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$ , then $R_{A,h}(g)=\frac{1}{2}|A|^{h-1}$ for all $g\in G$. If $h>1$ is odd and $|G|$ is even, then there exists a subset $A$ of $G$ with $|A|=\frac{1}{2}|G|$ such that $R_{A,h}(g)\not =R_{G\setminus A,h}(g)$ for all $g\in G$. [ABSTRACT FROM AUTHOR]

This paper presents the following definition which is a natural combination of the definitions of asymptotically equivalence, I-convergence, statistical limit, lacunary sequence, and Wijsman convergence of weight g; where g:N→[0,∞) is a function satisfying limn→∞ g(n)=∞ and n/g(n) →0 as n →∞ for sequence of sets. Let (X,ρ) be a metric space, θ ={kr} be a lacunary sequence and I ⊆ 2N be an admissible ideal. For any non-empty closed subsets Ak, Bk ⊆ X such that d (x, Ak > 0 and d(x, Bk) > 0 for each x ∊ X, we say that the sequences {Ak} and {Bk} are Wijsman I-asymptotically lacunary statistical equivalent of multiple L of weight g if for every ∊>0,δ>0 and for each x∊X, {r∊N:1/g(hr)∣{k∊Ir:∣ d(x,Ak)/d(x,Bk)-L∣≥∊}∣≥δ}∊I (denoted by AksLθ(IW)~g Bk). We mainly investigate their relationship and also make some observations about these classes. [ABSTRACT FROM AUTHOR]

In this paper, we study the shared values and uniqueness of meromorphic functions on annulus, and obtain one theorem about meromorphic functions on annulus sharing some distinct values, and this result is an improvement of some theorems given by Cao, Yi [4, 5], Kondratyuk and Laine[9]. [ABSTRACT FROM AUTHOR]

Abstract We describe a unified approach to estimating the dimension of f − 1 (A) for any G -equivariant map f : X → Y and any closed G -invariant subset A ⊆ Y in terms of connectivity of X and dimension of Y , where G is either a cyclic group of order p k , a p -torus (p a prime), or a torus. [ABSTRACT FROM AUTHOR]

Abstract We introduce the notion of completely ω -balanced semitopological group and prove that this property is preserved under subgroups and topological products. We also show that a regular semitopological group G admits a homeomorphic embedding as a subgroup into a product of strongly metrizable semitopological groups if and only if G is completely ω -balanced and I r (G) ≤ ω. [ABSTRACT FROM AUTHOR]

Abstract Motivated by results of Juhász and van Mill in [13] , we define the cardinal invariant w t (X) , the weak tightness of a topological space X , and show that | X | ≤ 2 L (X) w t (X) ψ (X) for any Hausdorff space X (Theorem 2.8). As w t (X) ≤ t (X) for any space X , this generalizes the well-known cardinal inequality | X | ≤ 2 L (X) t (X) ψ (X) for Hausdorff spaces (Arhangel′skiĭ [1] , Šapirovskiĭ [17]) in a new direction. Theorem 2.8 is generalized further using covers by G κ -sets, where κ is a cardinal, to show that if X is a power homogeneous compactum with a countable cover of dense, countably tight subspaces then | X | ≤ c , the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting. [ABSTRACT FROM AUTHOR]

Abstract The famous Rosenthal–Lacey theorem asserts that for each infinite compact set K the Banach space C (K) admits a quotient which is either an isomorphic copy of c or ℓ 2. What is the case when the uniform topology of C (K) is replaced by the pointwise topology? Is it true that C p (X) always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space X the function space C p (X) has an infinite-dimensional metrizable quotient if X either contains an infinite discrete C ⁎ -embedded subspace or else X has a sequence (K n) n ∈ N of infinite compact subsets such that for every n the space K n contains two disjoint topological copies of K n + 1. Applying the latter result, we show that under ◊ there exists a zero-dimensional Efimov space K whose function space C p (K) has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Ka̧kol and Śliwa on infinite-dimensional separable quotients of C p -spaces. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce the concept of a Geraghty's cyclic contraction which is a generalized cyclic contractive mapping and provide best proximity point theorems for such a mapping in complete metric space and uniformly convex Banach space. Moreover, we also give some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

Let C = { C 1 , C 2 , … } , K = { K 1 , K 2 , … } be countable families of subsets of the Euclidean plane R 2 whose diameters tend to zero and whose closures are continua such that cl ( C i ) ∩ cl ( C j ) = ∅ and cl ( K i ) ∩ cl ( K j ) = ∅ , for i ≠ j , i , j ∈ N . If all sets from both families C and K are ambiently homeomorphic to each other via orientation preserving automorphisms of R 2 , and the sets ⋃ C , ⋃ K are dense in R 2 , then there exists an automorphism f : R 2 → R 2 of the plane such that { f [ C ] : C ∈ C } = K . [ABSTRACT FROM AUTHOR]

We show that for every finite nonempty subset of ℕ ≥ 2 {\mathbb{N}_{\geq 2}} there are a numerical monoid H and a squarefree element a ∈ H {a\in H} whose set of lengths 𝖫 ⁢ (a) {\mathsf{L}(a)} is equal to L. [ABSTRACT FROM AUTHOR]

It is shown that certain lower semi-continuous maps from a paracompact space to the family of closed subsets of the bundle space of a Banach bundle admit continuous selections. This generalization of the theorem of Douady, dal Soglio-Herault, and Hofmann on the fullness of Banach bundles has applications to establishing conditions under which the induced maps between the spaces of sections of Banach bundles are onto. Another application is to a generalization of the theorem of Bartle and Graves [3] for Banach bundle maps that are onto their images. Other applications of the selection theorem are to the study of the M-ideals of the space of bounded sections begun in [4] and continued in [9] . A class of Banach bundles that generalizes the class of trivial Banach bundles is introduced and some properties of these Banach bundles are discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive the following bound on the size of a k‐wise L‐intersecting family (resp. cross L‐intersecting families) modulo a prime number: (i) Let p be a prime, l 0 < p, and n ≡ l 0 ( mod p ). Let L = { l 1 , l 2 , … , l s } and K = { k 1 , k 2 , … , k r } be two disjoint subsets of { 0 , 1 , … , p − 1 } such that l 0 < k i < l 0 + p + 2 r − s − 1, or k i < min { l 0 + 1 , l 0 + 2 r − s − 1 }. Suppose that F = { F 1 , F 2 , … , F m } is a family of subsets of [n] such that | F i | ( mod p ) ∈ K for every 1 ⩽ i ⩽ m and | F i 1 ∩ F i 2 ∩ ⋯ ∩ F i k | ( mod p ) ∈ L for every collection { F i 1 , F i 2 , … , F i k } of k distinct subsets in F , k ⩾ 2. Then, m ⩽ ( k − 1 ) n − 1 s + n − 1 s − 1 + ⋯ + n − 1 s − 2 r + 1 . This result may be considered as a modular version of Theorem 1.10 in [J. Q. Liu, S. G. Zhang, S. C. Li, H. H. Zhang, Eur. J. Combin. 58 (2016), 166‐180]. (ii)Let p be a prime, l 0 < p, and n ≡ l 0 ( mod p ). Let L = { l 1 , l 2 , … , l s } and K = { k 1 , k 2 , … , k r } be two subsets of { 0 , 1 , … , p − 1 } such that min K > s − r, or l 0 < k i < l 0 + p + r − s, or k i < min { l 0 + 1 , l 0 + r − s }. Suppose that A = { A 1 , A 2 , … , A m } and B = { B 1 , B 2 , … , B m } are two families of subsets of [n] such that (1) | A i ∩ B j | ( mod p ) ∈ L for every pair i ≠ j; (2) | A i ∩ B i | ( mod p ) ∉ L for every 1 ⩽ i ⩽ m; (3) | B i | ( mod p ) ∈ K for every 1 ⩽ i ⩽ m. Then, m ⩽ n s + n s − 1 + ⋯ + n s − r + 1 . This result extends the well‐known Alon–Babai–Suzuki theorem to two cross L‐intersecting families. [ABSTRACT FROM AUTHOR]

Let (X, T) be a topological dynamical system, and ℱ be a family of subsets of ℤ+. (X, T) is strongly ℱ-sensitive if there is δ > 0 such that for each non-empty open subset U there are x, y ∈ U with {n ∈ ℤ+ : d(Tnx, Tny) > δ} ∈ ℱ. Let ℱt (resp. ℱip, ℱfip) consist of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets). The following Auslander-Yorke’s type dichotomy theorems are obtained: (1) a minimal system is either strongly ℱip-sensitive or an almost one-to-one extension of its ∞-step nilfactor; (2) a minimal system is either strongly ℱip-sensitive or an almost one-to-one extension of its maximal distal factor; (3) a minimal system is either strongly ℱt-sensitive or a proximal extension of its maximal distal factor. [ABSTRACT FROM AUTHOR]

Abstract: In the realm of Lindelöf metric spaces the following results are obtained in ZF: (i) If X = ( X , d ) is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice CAC, the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of R has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space X = ( X , d ) is quasi totally bounded implies CAC ω. (iv) The proposition “every quasi totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem CUC and CAC. Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between CAC ( R ) and CAC. [ABSTRACT FROM AUTHOR]

It is shown that the Bolzano property characterizes the covering dimension. Some relationships between the Steinhaus chain property and a dimension are discussed. Finally a new dimension for arbitrary topological spaces is defined. [ABSTRACT FROM AUTHOR]

In this paper, we study the Heisenberg uniqueness pairs corresponding to a finite number of parallel lines Γ. We give a necessary condition and a sufficient condition for a subset Λ of R 2 so that ( Γ , Λ ) becomes a HUP. [ABSTRACT FROM AUTHOR]

It is given a simplified and self-contained proof of the classical Michael's finite-dimensional selection theorem. The proof is based on approximate selections constructed stepwise over skeletons of nerves of covers. The method is also applied to simplify the proof of the Schepin-Brodsky's generalisation of this theorem. [ABSTRACT FROM AUTHOR]

In set theory without the axiom of choice [ABSTRACT FROM AUTHOR]

A set x is Dedekind infinite if there is an injection from ω into x; otherwise x is Dedekind finite. A set x is power Dedekind infinite if [ABSTRACT FROM AUTHOR]

Given a topological Ramsey space [ABSTRACT FROM AUTHOR]

We define the notion of Euler characteristic for definable quotients in an arbitrary o-minimal structure and prove some fundamental properties. [ABSTRACT FROM AUTHOR]

We show that for every finite nonempty subset of ℕ≥2 there are a numerical monoid H and a squarefree element a ∈ H whose set of lengths L(a) is equal to L. [ABSTRACT FROM AUTHOR]

Let Ω be a Lebesgue-measurable set in Rn of finite positive Lebesgue measure. In this note we calculate the lack of equi-measurability of the set Kc(Ω), c > 0, of all Lebesgue-measurable functions f : Ω → R such that 0 ≤ ƒ ≤ c, a.e. on Ω. From our result we repair a gap in the Example 2.3 of the paper [APPELL, J.--DE PASCALE, E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. B (6) 3 (1984), 497-515]. [ABSTRACT FROM AUTHOR]

We consider the Horn-Tarski condition for the extension of (signed) measures (resp., non-negative measures) in the setup of field-valued assignments. For a finite collection 𝓒 of subsets of Ω, we find that the extension from 𝓒 over the collection exp Ω of all subsets of Ω is implied by, and indeed equivalent to, a certain type of Frobenius theorem (resp. a certain type of Farkas lemma). This links classical notions of linear algebra with a generalized version of Horn-Tarski condition on extensions of measures. We also observe that for a general (infinite) 𝓒 the Horn-Tarski condition guarantees the extension of signed measures (here the standard Zorn lemma applies). However, we find out that the extensions for non-negative ordered-field-valued measures are generally not available. [ABSTRACT FROM AUTHOR]

It is shown that a subspace of a product of countably many Lašnev spaces is Pytkeev if and only if it has countable tightness. In particular, this implies a result [6] that for an ultrafilter p on ω , the space { p } ∪ ω cannot be embedded in any product of countably many Lašnev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we derive a best possible k -wise extension to the well-known Snevily theorem on set systems (Snevily, 2003) which strengthens the well-known theorem by Füredi and Sudakov (2004). We also provide a conjecture which gives a common generalization to all existing non-modular L -intersection theorems. [ABSTRACT FROM AUTHOR]

The k -restricted arc connectivity of digraphs is a common generalization of the arc connectivity and the restricted arc connectivity. An arc subset S of a strong digraph D is a k -restricted arc cut if D − S has a strong component D ′ with order at least k such that D − V ( D ′ ) contains a connected subdigraph with order at least k . The k -restricted arc connectivity λ k ( D ) of a digraph D is the minimum cardinality over all k -restricted arc cuts of D . Let D be a strong digraph with order n ≥ 6 and minimum degree δ ( D ) . In this paper, we first show that λ 3 ( D ) exists if δ ( D ) ≥ 3 and, furthermore, λ 3 ( D ) ≤ ξ 3 ( D ) if δ ( D ) ≥ 4 , where ξ 3 ( D ) is the minimum 3-degree of D . Next, we prove that λ 3 ( D ) = ξ 3 ( D ) if δ ( D ) ≥ n + 3 2 . Finally, we give examples showing that these results are best possible in some sense. [ABSTRACT FROM AUTHOR]

The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matoušek and Wagner, and of Dotterrer, Kaufman and Wagner). We also discuss a simple generalization in which the vertices are weighted according to some probability distribution. This allows to use von Neumann's minimax theorem to deduce a dual statement. [ABSTRACT FROM AUTHOR]

In 1940 Fisher famously showed that if there exists a non-trivial ( v,k,λ)-design, then λ( v-1)⩾ k( k-1). Subsequently Bose gave an elegant alternative proof of Fisher's result. Here, we show that the idea behind Bose's proof can be generalised to obtain new bounds on the number of blocks in ( v,k,λ)-coverings and -packings with λ( v-1)< k( k-1). [ABSTRACT FROM AUTHOR]

For metrizable continua, there exists the well-known notion of a Whitney map. If X is a nonempty, compact, and metric space, then any Whitney map for any closed subset of 2X can be extended to a Whitney map for 2X [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem. [ABSTRACT FROM AUTHOR]

The inverse nodal problem for the Sturm-Liouville operator L(q, a) arisen from acoustic scattering problems was considered. The authors showed that q(x) on the interval [0, 1] can be uniquely determined by the interior twin-dense nodal subset WT([0, 1]), or WT([a0, 1]), respectively. Using the above results, two uniqueness theorems of the spherically symmetric speed of sound η(r) for acoustic scattering problems are obtained. [ABSTRACT FROM AUTHOR]

Completely regular semigroups $$\mathcal {C}\mathcal {R}$$ are unions of their subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by $$\mathcal {L}(\mathcal {C}\mathcal {R})$$ . The Polák theorem concerns the computation of joins in $$\mathcal {L}(\mathcal {C}\mathcal {R})$$ . The $$\mathbf {B}$$ -relation on $$\mathcal {L}(\mathcal {C}\mathcal {R})$$ identifies varieties with the same bands. We elaborate upon two nontrivial conditions in Polák's theorem applied to certain subsets of $$\mathcal {C}\mathcal {R}$$ which amounts to solving particular equations in $$\mathcal {L}(\mathcal {C}\mathcal {R})$$ . [ABSTRACT FROM AUTHOR]

This paper focuses on the relationship between an L-subset and the system of level-elements induced by it, where the underlying lattice L is a complete residuated lattice and the domain set of L-subset is an L-partially ordered set (X,P). Firstly, we obtain the sufficient and necessary condition that an L-subset is represented by its system of level-elements. Then, a new representation theorem of intersection-preserving L-subsets is shown by using union-preserving system of elements. At last, another representation theorem of compatible intersection-preserving L-subsets is obtained by means of compatible union-preserving system of elements. [ABSTRACT FROM AUTHOR]

The generalized closure operator cl* induces a topology t* . In this paper, we study the topology t* on lower bounded Alexandroff spaces. We prove that ( X, t* ) is a submaximal T0 Alexandroff space. We get some new results about the relation between t* and tα . Then we prove that a subset A in a lower bounded space is g - closed set if and only if A is α - open in the dual space. [ABSTRACT FROM AUTHOR]

A set of sets is called a family. Two families A and B are said to be cross-t-intersecting if each set in A intersects each set in B in at least t elements. An active problem in extremal set theory is to determine the maximum product of sizes of cross-t-intersecting subfamilies of a given family. This incorporates the classical Erdõs-Ko-Rado (EKR) problem. We prove a cross-t-intersection theorem for weighted subsets of a set by means of a new subfamily alteration method, and use the result to provide solutions for three natural families. For r ∊ [n] = {1, 2, . . . , n}, let ([n] r) be the family of r-element subsets of [n], and let ([n] ⩽r ) be the family of subsets of [n] that have at most r elements. Let Fn,r,t be the family of sets in ([n] ⩽r ) that contain [t]. We show that if g : ([m] ⩽r ) → R+ and h : ([n] ⩽ s ) → R+ are functions that obey certain conditions, A ⊆ ([m] ⩽ r ) , B ⊆ ([n] ⩽ s ) , and A and B are cross-t-intersecting, then and equality holds if A = Fm,r,t and B = Fn,s,t. We prove this in a more general setting and characterize the cases of equality. We use the result to show that the maximum product of sizes of two cross-t-intersecting families A ⊆ ([m] r ) and B ⊆ ([n] s ) is (m-t/r-t )(n-t/s-t ) for min{m, n} ⩾ n0(r, s, t), where n0(r, s, t) is close to best possible. We obtain analogous results for families of integer sequences and for families of multisets. The results yield generalizations for k ⩾ 2 cross-t-intersecting families, and EKR-type results. [ABSTRACT FROM AUTHOR]

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper, we consider the analogous multiplicative setting of the cyclic group (ℤ/qℤ)x and prove a similar result. For all suitably large primes q we define Pη to be the set of primes less than ηq, viewed naturally as a subset of (ℤ/qℤ)x. Considering the k-fold product set P(kη = {p1p1 ... pk : pi ∈ Pη}, we show that, for η » q-1/4+ε, there exists a constant k depending only on ε such that P(kη = (ℤ/qℤ)x. Erdős conjectured that, for η = 1, the value k = 2 should sufiice: although we have not been able to prove this conjecture, we do establish that P(2)1 has density at least 1/64 v(1 + o(1)). We also formulate a similar theorem in almost-primes, improving on existing results. [ABSTRACT FROM AUTHOR]

We study the class of overconvergent subanalytic subsets of a k-affinoid space X when k is a non-archimedean field. These are the images along the projection X x Bn → X of subsets defined by inequalities between functions on X x Bn which are overconvergent in the variables of Bn. In particular, we study the local nature, with respect to X, of overconvergent subanalytic subsets. We show that they behave well with respect to the Berkovich topology, but not the G-topology. This gives counterexamples to previous results on the subject, and a way to correct them. Moreover, we study the case dim (X) = 2, for which a simpler characterisation of overconvergent subanalytic subsets is proven. [ABSTRACT FROM AUTHOR]

The article discusses the study about the existence of two-colorings of vertex set in the property B of a uniform hypergraphs. Topics discussed include the least number m(n) of edges of an n-uniform hypergraph without property B, generalizations of the Erdos-Hajnal problem and the auxiliary result which is based on a hypergraph to have a property Bk.

Let n , p , r be positive integers with n ≥ p ≥ r . A rank- r ‾ subset of n by p matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to r . A classical theorem of Flanders states that the dimension of a rank- r ‾ linear subspace must be less than or equal to nr , and it characterizes the spaces with the critical dimension nr . Linear subspaces with dimension close to the critical one were later studied by Atkinson, Lloyd and Beasley over fields with large cardinality; their results were recently extended to all fields [18] . Using a new method, we obtain a classification of rank- r ‾ affine subspaces with large dimension, over all fields. This classification is then used to double the range of (large) dimensions for which the structure of rank- r ‾ linear subspaces is known for all fields. [ABSTRACT FROM AUTHOR]

We investigate the definable subsets of Stone algebras, introduce the notion of pseudo o-minimality of partially ordered structures, and prove that the completion of the theory of Stone algebras is pseudo o-minimal. We also give the decomposition theorem of Stone algebras. [ABSTRACT FROM AUTHOR]

Let Γ be a discrete group. Following Linnell and Schick one can define a continuous ring c(Γ) associated with Γ. They proved that if the Atiyah Conjecture holds for a torsion-free group Γ, then c(Γ) is a skew field. Also, if Γ has torsion and the Strong Atiyah Conjecture holds for Γ, then c(Γ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = ℤ2≀ℤ. It is known that ℂ(ℤ2≀ℤ) does not even have a classical ring of quotients. Our main result is that if H is amenable, then c(ℤ2≀ H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930s. [ABSTRACT FROM AUTHOR]

We determine lower and exact estimates of Kolmogorov, Gelfand and linear n-widths of unit balls in Sobolev norms in Lp-spaces on compact Riemannian manifolds. As it was shown by us previously these lower estimates are exact asymptotically in the case of compact homogeneous manifolds. The proofs rely on two-sides estimates for the near-diagonal localization of kernels of functions of elliptic operators. [ABSTRACT FROM AUTHOR]

Let K be a non-empty closed subset of a Banach space X endowed with a graph G. We obtain fixed point theorems for nonself G-contractions of Chatterjea type. Our new results complement and extend recent related results [Berinde, V., Păcurar, M., The contraction principle for nonself mappings on Banach spaces endowed with a graph, J. Nonlinear Convex Anal. 16 (2015), no. 9, 1925-1936; Balog, L., Berinde, V., Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph, Carpathian J. Math. 32 (2016), no. 3 (in press)] and thus provide more general and flexible tools for studying nonlinear functional equations. [ABSTRACT FROM AUTHOR]

The Bogomolov conjecture claims that a closed subvariety containing a dense subset of small points is a special kind of subvariety. In the arithmetic setting over number ι elds, the Bogomolov conjecture for abelian varieties has already been established as a theorem of Ullmo and Zhang, but in the geometric setting over function ι elds, it has not yet been solved completely. There are only some partial results known such as the totally degenerate case due to Gubler and our recent work generalizing Gubler's result. The key in establishing the previous results on the Bogomolov conjecture is the equidistribution method due to Szpiro, Ullmo and Zhang with respect to the canonical measures. In this paper we exhibit the limits of this method, making an important contribution to the geometric version of the conjecture. In fact, by the crucial investigation of the support of the canonical measure on a subvariety, we show that the conjecture in full generality holds if the conjecture holds for abelian varieties which have anywhere good reduction. As a consequence, we establish a partial answer that generalizes our previous result. [ABSTRACT FROM AUTHOR]

We show that Friedman's finite adjacent Ramsey theorem is unprovable in Peano Arithmetic, and give a new proof of the unprovability of the Paris--Harrington theorem in PA. We also determine the status of these theorems for each dimension. It is to be noted that the finite adjacent Ramsey theorem for dimension d is equivalent to the Paris--Harrington theorem for dimension d + 1 . [ABSTRACT FROM AUTHOR]

We show that Friedman's finite adjacent Ramsey theorem is unprovable in Peano Arithmetic, and give a new proof of the unprovability of the Paris--Harrington theorem in PA. We also determine the status of these theorems for each dimension. It is to be noted that the finite adjacent Ramsey theorem for dimension d is equivalent to the Paris--Harrington theorem for dimension d + 1 . [ABSTRACT FROM AUTHOR]