For n≥3, let r=r(n)≥3 be an integer. A hypergraph is r-uniform if each edge is a set of r vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear r-uniform hypergraphs on n→∞ vertices is determined asymptotically when the number of edges is m(n)=o(r−3n3/2). As one application, we find the probability of linearity for the independent-edge model of random r-uniform hypergraph when the expected number of edges is o(r−3n3/2). We also find the probability that a random r-uniform linear hypergraph with a given number of edges contains a given subhypergraph.

Square Heffter arrays are n×n arrays such that each row and each column contains k filled cells, each row and column sum is divisible by 2nk+1 and either x or −x appears in the array for each integer 1⩽x⩽nk. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face 2-colourable embedding of the complete graph K2nk+1 on an orientable surface, where for each colour, the faces give a k-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo 2nk+1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) k≡0(mod4); or (b) n≡1(mod4) and k≡3(mod4); or (c) n≡0(mod4), k≡3(mod4) and n≫k. As corollaries to the above we obtain pairs of orthogonal k-cycle decompositions of K2nk+1.

The fork is the tree obtained from the claw K1,3 by subdividing one of its edges once, and the antifork is its complement graph. We give a complete description of all graphs that do not contain the fork or antifork as induced subgraphs.

It is known that for each k≥4, there exists an irreducible sign pattern with minimum rank k that does not allow diagonalizability. However, it is shown in this paper that every square sign pattern A with minimum rank 2 that has no zero line allows diagonalizability with rank 2 and also with rank equal to the maximum rank of the sign pattern. In particular, every irreducible sign pattern with minimum rank 2 allows diagonalizability. On the other hand, an example is given to show the existence of a square sign pattern with minimum rank 3 and no zero line that does not allow diagonalizability; however, the case for irreducible sign patterns with minimum rank 3 remains open. In addition, for a sign pattern that allows diagonalizability, the possible ranks of the diagonalizable real matrices with the specified sign pattern are shown to be lengths of certain composite cycles. Some results on sign patterns with minimum rank 2 are extended to sign pattern matrices whose maximal zero submatrices are “strongly disjoint” (that is, their row index sets as well as their column index sets are pairwise disjoint).

A cyclic ordering of the points in a Mendelsohn triple system of order v (or MTS (v)) is called a sequencing. A sequencing D is ℓ-good if there does not exist a triple (x,y,z) in the MTS (v) such that 1. the three points x,y, and z occur (cyclically) in that order in D; and 2. {x,y,z} is a subset of ℓ cyclically consecutive points of D. In this paper, we prove some upper bounds on ℓ for MTS (v) having ℓ-good sequencings and we prove that any MTS (v) with v≥7 has a 3-good sequencing. We also determine the optimal sequencings of every MTS (v) with v≤10.

In this paper we investigate the geometric properties of the configuration consisting of a subspace Γ and a canonical subgeometry Σ in PG(n−1,qn), with Γ∩Σ=0̸. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of Σ from the vertex Γ. In particular we deal with the maximum scattered linear sets of the line PG(1,qn) found by Lunardon and Polverino (2001) and recently generalized by Sheekey (2016). Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajbók and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in PG(1,q6), yielding also to new examples of MRD-codes in Fq6×6 with left idealizer isomorphic to Fq6.

We generalize to sets with cardinality more than p a theorem of Rédei and Szőnyi on the number of directions determined by a subset U of the finite plane Fp2. A U-rich line is a line that meets U in at least #U∕p+1 points, while a U-poor line is one that meets U in at most #U∕p−1 points. The slopes of the U-rich and U-poor lines are called U-special directions. We show that either U is contained in the union of n=⌈#U∕p⌉ lines, or it determines “many” U-special directions. The core of our proof is a version of the polynomial method in which we study iterated partial derivatives of the Rédei polynomial to take into account the “multiplicity” of the directions determined by U.

A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-dicirculant is a bi-Cayley graph over a dicyclic group. In this paper, a classification of connected cubic bi-dicirculants is given. As byproducts, we show that every connected cubic vertex-transitive bi-dicirculant is Cayley, and up to isomorphism, there are two connected cubic edge-transitive bi-dicirculants, of which one has order 16, and the other has order 24.

In this paper, we give necessary and sufficient conditions on the compatibility of a kth-order homogeneous linear elliptic differential operator A and differential constraint C for solutions toAu=fsubject toCf=0 in Rn to satisfy the estimates‖Dk−ju‖Lnn−j(Rn)⩽c‖f‖L1(Rn) for j∈{1,…,min⁡{k,n−1}} and‖Dk−nu‖L∞(Rn)⩽c‖f‖L1(Rn) when k≥n.

We establish the continuity of a surface as a function of its first two fundamental forms for several Fréchet topologies, which include in particular those of the space Wloc1,p for the first fundamental form and of the space Llocp for the second fundamental form, for any p>2.

The conjugacy classes of groups and quasigroups form association schemes, in which the relation products are defined by collapsing group or quasigroup multiplications. In previous work, sharp transitivity was used to identify association schemes, such as certain Johnson schemes, which cannot appear as quasigroup schemes. Thus quasigroup schemes only constitute a fragment of the full set of all association schemes. Nevertheless, the current paper shows that every association scheme is in fact obtained by collapsing a quasigroup multiplication. In a second application of a similar technique, character quasigroups are constructed for each finite group, as analogues of the character groups of abelian groups, to encode the multiplicative structure of group characters. As infrastructure for these and related results, three key unifying concepts in compact closed categories are established: augmented comagmas, augmented magmas, and augmented quasigroups, the latter serving to capture such diverse structures as groups and Heyting algebras.

We prove that the von Neumann algebras of quantum permutation groups and quantum reflection groups have the Connes embedding property. We do this by establishing several new topological generation results for the quantum permutation groups SN+ and the quantum reflection groups HNs+. We use these results to show that these quantum groups admit sufficiently many “matrix models”. In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain “flat” matrix models for SN+ are inner faithful.

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$ . Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$ . The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

In this note it is shown that the class of all multipliers from the d-parameter Hardy space $H_{{\rm prod}}^1 ({\open T}^d)$ to L2 (𝕋d) is properly contained in the class of all multipliers from L logd/2L (𝕋d) to L2(𝕋d).

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$ -linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$ -coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

Subspace codes have attracted much attention in recent years due to their applications to error correction in random network coding. In this paper, we construct several kinds of large cyclic subspace codes via Sidon spaces and large subspace codes via unions of some Sidon spaces. Therefore, some known results are extended.

A digraph D=(V,A) has a good pair at a vertex r if D has a pair of arc-disjoint in- and out-branchings rooted at r. Let T be a digraph with t vertices u1,…,ut and let H1,…Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the composition Q=T[H1,…,Ht] is a digraph with vertex set {ui,ji∣1≤i≤t,1≤ji≤ni} and arc set A(Q)=∪i=1tA(Hi)∪{uijiupqp∣uiup∈A(T),1≤ji≤ni,1≤qp≤np}. If T is strongly connected, then Q is called a strong composition and if T is semicomplete, i.e., there is at least one arc between every pair of vertices, then Q is called a semicomplete composition. We obtain the following result: every strong digraph composition Q in which ni≥2 for every 1≤i≤t, has a good pair at every vertex of Q. The condition of ni≥2 in this result cannot be relaxed. We characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun and Wang's cyclic valley-hopping action as well as a formula of Brenti. Along the way, we give a new proof for the γ-positivity of the excedance number distribution over any fixed conjugacy class along with a combinatorial interpretation of the γ-coefficients.

In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra Aˆ of Lie algebra A=gl∞⊕gl∞ with respect to certain automorphism groups. We then prove that restricted modules of level ℓ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras VAˆ(ℓ,0) (or VAˆ(2ℓ,0)). Furthermore, we determine irreducible modules and equivariant quasi modules for the simple vertex algebra LAˆ(ℓ,0) with ℓ a positive integer. In particular, we prove that every quasi-finite unitary highest weight (irreducible) module of level ℓ for type A trigonometric Lie algebra gives rise to an irreducible equivariant quasi LAˆ(ℓ,0)-module.

We investigate the hyperspace GH(Rn) of the isometry classes of all non-empty compact subsets of a Euclidean space in the Gromov-Hausdorff metric. It is proved that for any n≥1, GH(Rn) is homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of a Euclidean space Rn equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This is further applied to prove that 2Rn/E(n) is homeomorphic to the open cone OCone(Ch(Bn)/O(n)), where Ch(Bn) stands for the set of all A∈2Rn for which the closed Euclidean unit ball Bn is the least circumscribed ball (the Chebyshev ball). These results lead to determine the complete topological structure of GH(Rn) for n≤2, namely, we prove that GH(Rn) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n≤2, GH(Bn) is homeomorphic to the Hilbert cube.

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset \mathbb{R}$ is a $(\delta ,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$.

In this paper, we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are $(-\Delta )^{-\alpha /2}$, and we develop basic analogous properties with respect to meromorphic continuation, residues, Fourier transforms, and relations to conformal geometry and representations of the conformal group.

A general smooth curve of genus six lies on a quintic del Pezzo surface. Artebani and Kondō [ 4] construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne–Mumford stack ${\mathfrak{P}}_0$ parametrizing certain stable surface-curve pairs, which essentially resolves this map. Moreover, we give an explicit description of pairs in ${\mathfrak{P}}_0$ containing special curves.

We study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting.

The property of non-catastrophicity of multidimensional convolutional codes is studied. In particular, an algebraic and system-theoretic characterization of non-catastrophicity is offered in the multidimensional setting, and the Massey–Sain classical criterion is extended to this setting.

An RE-m-coloring of a graph G is a vertex m-coloring of G, which is relaxed (every vertex shares the same color with at most one neighbor) and equitable (the sizes of all color classes differ by at most one). In this article, we prove that every planar graph with minimum degree at least 2 and girth at least 8 has an RE-m-coloring for each integer m≥4. We use the discharging method and Hall’s Theorem to simply the structures of counterexamples.

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers, are Hausdorff moment sequences. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.

An upper bound on the diameter of the Stable Matching (Stable Marriage) polytope is known to be ⌊n2⌋ where n is the number of men (or women) involved in the matching. The current work complements that result by providing a lower bound and an algorithm computing it. It also presents a class of Stable Matching instances for which the lower bound coincides with the above-mentioned upper bound.

Let G=(V,E) be a graph and Γ an Abelian group both of order n. A Γ-distance magic labeling of G is a bijection ℓ:V→Γ for which there exists μ∈Γ such that ∑x∈N(v)ℓ(x)=μ for all v∈V, where N(v) is the neighborhood of v. Froncek showed that the Cartesian product Cm□Cn, m,n≥3 is a Zmn-distance magic graph if and only if mn is even. It is also known that if mn is even then Cm□Cn has Zα×A-magic labeling for any α≡0(modlcm(m,n)) and any Abelian group A of order mn∕α. However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution of this problem by proving some necessary conditions. We further prove that for n even the graph Cn□Cn has a Γ-distance magic labeling for any Abelian group Γ of order n2. Moreover we show that if m≠n, then there does not exist a (Z2)m+n-distance magic labeling of the Cartesian product C2m□C2n. We also give a necessary and sufficient condition for Cm□Cn with gcd(m,n)=1 to be Γ-distance magic.

A subset C of the vertex set of a graph Γ is called a perfect code in Γ if every vertex of Γ is at distance no more than 1 to exactly one vertex of C. A subset C of a group G is called a perfect code of G if C is a perfect code in some Cayley graph of G. In this paper we give sufficient and necessary conditions for a subgroup H of a finite group G to be a perfect code of G. Based on this, we determine the finite groups that have no nontrivial subgroup as a perfect code, which answers a question by Ma, Walls, Wang and Zhou.

Let q be a prime-power, and n≥3 an odd integer. We determine the structure of the Weierstrass semigroup H(P) where P is an arbitrary Fq2-rational point of GK2,n where GK2,n stands for the Fq2n-maximal curve of Beelen and Montanucci. We prove that these points are Weierstrass points, and we compute the Frobenius dimension of GK2,n. Using these results, we also show that GK2,n is isomorphic to the Güneri–Garcìa–Stichtenoth only for n=3. Furthermore, AG codes and AG quantum codes from the GK2,n are constructed and discussed. In some cases, they have better parameters compared with those of the known linear codes.

We show that the codimension sequence of the algebra of k×k matrices over the Grassmann algebra, cn(Mk(E)), is asymptotic to αn1−k22(2k2)n, where α is an undetermined constant.

Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the Plücker relations in Clifford's geometric algebra. We discover that the Plücker relations can be fully characterized in terms of the geometric product, without the need for a confusing hodgepodge of many different formalisms and mathematical traditions found in the literature.

We study uniform measures in the first Heisenberg group H equipped with the Korányi metric dH. We prove that 1-uniform measures are proportional to the spherical 1-Hausdorff measure restricted to an affine horizontal line, while 2-uniform measures are proportional to spherical 2-Hausdorff measure restricted to an affine vertical line. We also show that each 3-uniform measure which is supported on a vertically ruled surface is proportional to the restriction of spherical 3-Hausdorff measure to an affine vertical plane, and that no quadratic x3-graph can be the support of a 3-uniform measure. According to a result of Merlo, every 3-uniform measure is supported on a quadratic variety; in conjunction with our results, this shows that all 3-uniform measures are proportional to spherical 3-Hausdorff measure restricted to an affine vertical plane. We establish our conclusions by deriving asymptotic formulas for the measures of small extrinsic balls in (H,dH) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in H.

We show that real and imaginary parts of equivariant spherical harmonics on ${{\mathbb{S}}}^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left (\frac{N^2 - m^2}{2} + N\right ) $. Hence, if $\frac{m}{N}= c $ for fixed $0 < c < 1$, then the genus has order $N^3$.

Boolean functions used in symmetric-key cryptosystems must have high second-order nonlinearity to withstand several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since the maximal second-order nonlinearity of all Boolean functions in n variables equals the covering radius of the Reed–Muller code with length 2n and order r. It is well-known that providing a tight lower bound on the second-order nonlinearity of a general Boolean function with high algebraic degree is a hard task, excepting a few special classes of Boolean functions. In this paper, we improve the lower bounds on the second-order nonlinearity of three classes of Boolean functions of the form fi(x)=Tr1n(xdi) in n variables for i=1,2 and 3, where Tr1n denotes the absolute trace mapping from F2n to F2 and di’s are of the form (1) d1=2m+1+3 and n=2m, (2) d2=2m+2m+12+1, n=2m with odd m, and (3) d3=22r+2r+1+1 and n=4r with even r.

The strong fractional choice number of a graph G is the infimum of those real numbers r such that G is (⌈rm⌉,m)-choosable for every positive integer m. The strong fractional choice number of a family G of graphs is the supremum of the strong fractional choice number of graphs in G. We denote by Qk the class of series–parallel graphs with girth at least k. This paper proves that the strong fractional choice number of Qk equals 2+1⌊(k+1)∕4⌋.

This paper proves that if G is a planar graph without 4-cycles and l-cycles for some l∈{5,6,7}, then there exists a matching M such that AT(G−M)≤3. This implies that every planar graph without 4-cycles and l-cycles for some l∈{5,6,7} is 1-defective 3-paintable.

The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a possibility by defining and studying an analogy of the Tamari lattice on Motzkin paths. While our generalization is not a lattice, each of its connected components is isomorphic to an interval in the classical Tamari lattice. With this structural result, we proceed to the enumeration of components and intervals in the poset of Motzkin paths we defined. We also extend the structural and enumerative results to Schröder paths. We conclude by a discussion on the relation between our work and that of Baril and Pallo (2014).

Inspired by the Kruskal–Katona theorem a minimization problem is studied, where the role of the shadow is replaced by the image of the action of a certain subset of the monoid of increasing functions. One of our main results shows that compressed sets are a solution to this problem. Several applications to simplicial complexes are discussed.

Pairs of complementary sequences such as Golay pairs have zero sum autocorrelation at all non-trivial phases. Several generalizations are known where conditions on either the autocorrelation function, or the entries of the sequences are altered. We aim to unify most of these ideas by introducing autocorrelation functions that apply to any sequences with entries in a set equipped with a ring-like structure which is closed under multiplication and contains multiplicative inverses. Depending on the elements of the chosen set, the resulting complementary pairs may be used to construct a variety of combinatorial structures such as Hadamard matrices, complex generalized weighing matrices, and signed group weighing matrices. We may also construct quasi-cyclic and quasi-constacyclic linear codes which over finite fields of order less than 5 are also Hermitian self-orthogonal. As the literature on binary and ternary Golay sequences is already quite deep, one intention of this paper is to survey and assimilate work on more general pairs of complementary sequences and related constructions of combinatorial objects, and to combine the ideas into a single theoretical framework.

In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.

We give a short new computation of the quantum cohomology of an arbitrary smooth (semiprojective) toric variety $X$, by showing directly that the Kodaira–Spencer map of Fukaya–Oh–Ohta–Ono defines an isomorphism onto a suitable Jacobian ring. In contrast to previous results of this kind, $X$ need not be compact. The proof is based on the purely algebraic fact that a class of generalized Jacobian rings associated to $X$ are free as modules over the Novikov ring. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.

The mass transference principle, proved by Beresnevich and Velani in 2006, is a strong result that gives lower bounds for the Hausdorff dimension of limsup sets of balls. We present a version for limsup sets of open sets of arbitrary shape.

We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [ 27] on traces of singular moduli.

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results.(1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules.(2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators.(3) Theorem III gives several characterizations of an analytic Toeplitz operator.(4) Theorem IV characterizes asymptotic Toeplitz operators.(5) Theorem V is a commutant lifting theorem.(6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

Given a double vector bundle $D\to M$, we define a bigraded bundle of algebras $W(D)\to M$ called the “Weil algebra bundle”. The space ${\mathcal{W}}(D)$ of sections of this algebra bundle ”realizes” the algebra of functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebra bundles of $D$ and those of the double vector bundles $D^{\prime},\ D^{\prime\prime}$ obtained from $D$ by duality operations. We show that ${\mathcal{V}\mathcal{B}}$-algebroid structures on $D$ are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the 3rd. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ”classical” version of Voronov’s result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that ${\mathcal{W}}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad–Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multi-vector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author’s previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. Here we apply this strategy to spaces with lower curvature bound.

We study the derivative nonlinear wave equation $- \partial _{tt} u + \Delta u = |\nabla u|^2$ on $\mathbb{R}^{1 +3}$. The deterministic theory is determined by the Lorentz-critical regularity $s_L = 2$, and both local well-posedness above $s_L$ as well as ill-posedness below $s_L$ are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities $s\geqslant 1.984$. In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.

Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

We consider the adjoint double layer potential (Neumann–Poincaré (NP)) operator appearing in 3-dimensional elasticity. We show that the recent result about the polynomial compactness of this operator for the case of a homogeneous media follows without additional calculations from previous considerations by Agranovich et al., based upon pseudodifferential operators. Further on, we define the NP operator for the case of a nonhomogeneous isotropic media and show that its properties depend crucially on the character of nonhomogeneity. If the Lamé parameters are constant along the boundary, the NP operator is still polynomially compact. On the other hand, if these parameters are not constant, two or more intervals of continuous spectrum may appear, so the NP operator ceases to be polynomially compact. However, after a certain modification, it becomes polynomially compact again. Finally, we evaluate the rate of convergence of discrete eigenvalues of the NP operator to the tips of the essential spectrum.

Keevash and Mycroft [ 19] developed a geometric theory for hypergraph matchings and characterized the dense simplicial complexes that contain a perfect matching. Their proof uses the hypergraph regularity method and the hypergraph blow-up lemma recently developed by Keevash. In this note we give a new proof of their results, which avoids these complex tools. In particular, our proof uses the lattice-based absorbing method developed by the author and a recent probabilistic argument of Kohayakawa, Person, and the author.

We consider the topological classification of finitely determined map germs $f:(\mathbb{R}^n,0)\to (\mathbb{R}^p,0)$ with $f^{-1}(0)\neq \{0\}$. Associated with $f$ we have a link diagram, which is well defined up to topological equivalence. We prove that $f$ is topologically $\mathcal{A}$-equivalent to the generalized cone of its link diagram.

In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over $\mathbb{F}_{q}$ is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by $q$ .

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k≥5) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size.

Recently, Beck posed two conjectures on the difference between the number of (respectively, distinct) parts in the odd partitions of n and the number of (respectively, distinct) parts in the distinct partitions of n. These two conjectures were first confirmed by Andrews using generating functions, and then generalized by Fu and Tang, and Yang in different ways. Motivated by Yang’s work, we present two more generalized results and prove them both analytically and combinatorially.

Let a torus T act in a Hamiltonian fashion on a compact symplectic manifold (M,ω). The assignment ring AT(M) is an extension of the equivariant cohomology ring HT(M); it is modeled on the GKM description of the equivariant cohomology of a GKM space. We show that AT(M) is a finitely generated S(t⁎)-module, and give a criterion guaranteeing that a given set of assignments generates (alternatively, is a basis for) this module. We define two new types of assignments, delta classes and bridge classes, and show that if the torus T is 2-dimensional, then all assignments of sufficiently high degree are generated by cohomological, delta, and bridge classes. In particular, if M is 6-dimensional, then we can find a basis of such classes.

The cluster of a crossing in a graph drawing in the plane is the set of the four end-vertices of its two crossed edges. Two crossings are independent if their clusters do not intersect. In this paper, we prove that every plane graph with independent crossings has an equitable partition into m induced forests for any m≥8. Moreover, we decrease this lower bound 8 for m to 6, 5, 4 and 3 if we additionally assume that the girth of the considering graph is at least 4, 5, 6 and 26, respectively.

In 1996, Bang-Jensen, Gutin, and Li proposed the following conjecture: If D is a strong digraph of order n where n≥2 with the property that d(x)+d(y)≥2n−1 for every pair of dominated non-adjacent vertices {x,y}, then D is hamiltonian. In this paper, we give an infinite family of counterexamples to this conjecture. In the same paper, they showed that for the above x,y, if they satisfy the condition either d(x)≥n, d(y)≥n−1 or d(x)≥n−1, d(y)≥n, then D is hamiltonian. It is natural to ask if there is an integer k≥1 such that every strong digraph of order n satisfying either d(x)≥n+k, d(y)≥n−1−k, or d(x)≥n−1−k, d(y)≥n+k, for every pair of dominated non-adjacent vertices {x,y}, is hamiltonian. In this paper, we show that k must be at most n−5 and prove that every strong digraph with k=n−4 satisfying the above condition is hamiltonian, except for one digraph on 5 vertices.

We give a bijection between the set of ordinary partitions and that of self-conjugate partitions with some restrictions. Also, we show the relationship between hook lengths of a self-conjugate partition and its corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.