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In 1943, Hadwiger conjectured that every graph with no K t + 1 minor is t-colorable for every t ≥ 0 . While Hadwiger's conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980 s, Kostochka and Thomason independently proved that every graph with no K t minor has average degree O ( t log ⁡ t ) and thus is O ( t log ⁡ t ) -list-colorable. Recently, the authors and Song proved that every graph with no K t minor is O ( t ( log ⁡ t ) β ) -colorable for every β > 1 4 . Here, we build on that result to show that every graph with no K t minor is O ( t ( log ⁡ t ) β ) -list-colorable for every β > 1 4 . Our main new tool is an upper bound on the number of vertices in highly connected K t -minor-free graphs: We prove that for every β > 1 4 , every Ω ( t ( log ⁡ t ) β ) -connected graph with no K t minor has O ( t ( log ⁡ t ) 7 / 4 ) vertices.

In this paper, we study the null controllability of a nonlinear age, space and two-sex structured population dynamics model. This model is such that the nonlinearity and the couplage are at birth level. We consider a population with males and females and we are dealing with two cases of null controllability problems. The first problem is related to the total extinction, which means that, we estimate a time \begin{document}$ T $\end{document} to bring the male and female subpopulation density to zero. The second case concerns null controllability of male or female subpopulation. Since the absence of males or females in the population stops births; so, if we have the total extinction of the females at time \begin{document}$ T, $\end{document} and if \begin{document}$ A $\end{document} is the life span of the individuals, at time \begin{document}$ T+A $\end{document} one will get certainly the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after the Schauder's fixed point theorem.

The number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2 n / 17700000 . On the other hand, for infinitely many n, there exists a triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is 2 15 n / log 2 ⁡ ( n ) .

The submonoid of the 3-strand braid group B 3 generated by σ 1 and σ 1 σ 2 is known to yield an exotic Garside structure on B 3 . We introduce and study an infinite family ( M n ) n ≥ 1 of Garside monoids generalizing this exotic Garside structure, i.e., such that M 2 is isomorphic to the above monoid. The corresponding Garside group G ( M n ) is isomorphic to the ( n , n + 1 ) -torus knot group–which is isomorphic to B 3 for n = 2 and to the braid group of the exceptional complex reflection group G 12 for n = 3 . This yields a new Garside structure on ( n , n + 1 ) -torus knot groups, which already admit several distinct Garside structures. The ( n , n + 1 ) -torus knot group is an extension of B n + 1 , and the Garside monoid M n surjects onto the submonoid Σ n of B n + 1 generated by σ 1 , σ 1 σ 2 , … , σ 1 σ 2 ⋯ σ n , which is not a Garside monoid when n > 2 . Using a new presentation of B n + 1 that is similar to the presentation of G ( M n ) , we nevertheless check that Σ n is an Ore monoid with group of fractions isomorphic to B n + 1 , and give a conjectural presentation of it, similar to the defining presentation of M n . This partially answers a question of Dehornoy–Digne–Godelle–Krammer–Michel.

This paper introduces two fundamental types of ordinal sum constructions for partially ordered monoids that are determined by two specific partial orderings on the disjoint union of the partially ordered monoids. Both ordinal sums of partially ordered monoids are generalized with the help of operators on posets, which combine, in some sense, the properties of interior and closure operators on posets. The proposed approach provides a unified view on several known constructions of ordinal sums of t-norms and t-conorms on posets (lattices) and introduces generalized ordinal sums of uninorms on posets (lattices).

This paper proves the conjectures (previously proved in the case n = 0 ) that for any Coxeter system ( W , S ) and any natural number n, the set of n-small roots for W is bipodal and the set of n-low elements of W is a Garside shadow. The proof here uses special cases ( k = 3 ) of properties (notably, their existence) of certain maximal rank k reflection subgroups of W, for non-negative integers k.

In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed graph admits a homomorphism, of graphs with underlying graph of the form P n □ P m , of Cartesian products of signed paths, of Cartesian products of signed complete graphs and of Cartesian products of signed cycles.

Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is at least $2\sqrt{p-1}$. Along the way we prove a uniform lower bound for the number of $p$-regular classes in a finite simple group of Lie type in terms of its rank and size of the underlying field., 46 pages

Several combinatorial actions of the affine Weyl group of type C ˜ n on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these permutation representations are multiplicity-free. The proof uses a general construction of Gelfand subgroups in the affine Weyl groups of types C ˜ n and B ˜ n .

Garside theory emerged from the study of Artin groups and their generalizations. Finite-type Artin groups admit two types of interval Garside structures corresponding to their standard and dual presentations. Concerning affine Artin groups, Digne established interval Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to interval Garside structures. In this paper, shifting attention from dual presentations to other nice presentations for the affine Artin group of type A ˜ discovered by Shi and Corran–Lee–Lee, I will construct interval Garside structures related to this group. This construction is the first successful attempt to establish interval Garside structures not related to the dual presentations in the case of affine Artin groups.

By work of Belyĭ [2] , the absolute Galois group G Q = Gal ( Q ‾ / Q ) of the field Q of rational numbers can be embedded into A = Aut ( F ˆ 2 ) , the automorphism group of the free profinite group F ˆ 2 on two generators. The image of G Q lies inside G T ˆ , the Grothendieck-Teichmuller group. While it is known that every abelian representation of G Q can be extended to G T ˆ , Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of G T ˆ . We do this virtually, namely by showing that a rich class of arithmetically defined representations of G Q can be extended to finite index subgroups of G T ˆ . This is achieved, in fact, by extending these representations all the way to finite index subgroups of A = Aut ( F ˆ 2 ) . We do this by developing a profinite version of the work of Grunewald and Lubotzky [7] , which provided a rich collection of representations for the discrete group Aut ( F d ) .

An even Artin group is a group which has a presentation with relations of the form ( s t ) n = ( t s ) n with n ≥ 1 . With a group G we associate a Lie Z -algebra TG r ( G ) . This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group G we determine a presentation for TG r ( G ) . By means of this presentation we obtain information about the diagram of G. We then prove an isomorphism criterion for Coxeter matrices that ensures that the diagram of G is uniquely determined by this information. Let d ≥ 2 . We show that, if two even Artin groups G and H having presentations with relations of the form ( s t ) d k = ( t s ) d k with k ≥ 0 are such that TG r ( G ) ≃ TG r ( H ) , then G and H have the same presentation up to permutation of the generators. On the other hand, we show an example of two non-isomorphic even Artin groups G and H such that TG r ( G ) ≃ TG r ( H ) .

Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $\mu = (\mu_1,\dots,\mu_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik-Dunkl operators $Y_i$ for all $1 \leq i \leq n$, and are thus equal to nonsymmetric Macdonald polynomials $E_{\mu}$. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such path ensembles, we show how to recover the well-known combinatorial formula for $E_{\mu}$ due to Haglund-Haiman-Loehr., Comment: 36 pages

We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$., Comment: 23 pages

A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $\mathcal{E}_d$ of elliptic curves defined over number fields and with degree $d$ $j$-invariant., 17 pages, to appear in Journal of Number Theory

We prove that pseudorandom sets in the Grassmann graph have near-perfect expansion. This completes the last missing piece of the proof of the 2-to-2-Games Conjecture (albeit with imperfect completeness). The Grassmann graph has induced subgraphs that are themselves isomorphic to Grassmann graphs of lower orders. A set of vertices is called pseudorandom if its density within all such subgraphs (of constant order) is at most slightly higher than its density in the entire graph. We prove that pseudorandom sets have almost no edges within them. Namely, their edge-expansion is very close to 1.

We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$\tau(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $\tau(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations., Comment: To appear in JNT Prime. For more general results, see arXiv:2005.10354

For q an odd prime power, A = F q [ T ] , and F = F q ( T ) , let ψ : A → F { τ } be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p = p A be a prime of A of good reduction for ψ, with residue field F p . We study the growth of the absolute value | Δ p | of the discriminant of the F p -endomorphism ring of the reduction of ψ modulo p and prove that, for all p , | Δ p | grows with | p | . Moreover, we prove that, for a density 1 of primes p , | Δ p | is as close as possible to its upper bound | a p 2 − 4 μ p p | , where X 2 + a p X + μ p p ∈ A [ X ] is the characteristic polynomial of τ deg p .

A graph database is a digraph whose arcs are labeled with symbols from a fixed alphabet. A regular graph pattern (RGP) is a digraph whose edges are labeled with regular expressions over the alphabet. RGPs model navigational queries for graph databases called conjunctive regular path queries (CRPQs). A match of a CRPQ in the database is witnessed by a special navigational homomorphism of the corresponding RGP to the database. We study the complexity of deciding the existence of a homomorphism between two RGPs. Such homomorphisms model a strong type of containment between the two corresponding CRPQs. We show that this problem can be solved by an EXPTIME algorithm (while general query containmement in this context is EXPSPACE-complete). We also study the problem for restricted RGPs over a unary alphabet, that arise from some applications like XPath or SPARQL. For this case, homomorphism-based CRPQ containment is in NP. We prove that certain interesting cases are in fact polynomial-time solvable., 15 pages. Short version appeared in the proceedings of the 15th Conference on Computability in Europe (CIE 2019)

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k sufficiently small with respect to the Euler characteristic of S. We prove the same result for the so-called systolic complex, a variant of the curve graph whose complete subgraphs encode the intersection patterns for any collection of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller., Comment: 33 pages, 23 figures, 1 table. Incorporated referee comments. To appear in the Michigan Mathematical Journal

Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation number for the butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $\mathcal N$., Comment: 13 pages

We show that there is a measure-preserving system $(X,\mathscr{B}, \mu, T)$ together with functions $F_0, F_1, F_2 \in L^{\infty}(\mu)$ such that the correlation sequence $C_{F_0, F_1, F_2}(n) = \int_X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 d\mu$ is not an approximate integral combination of $2$-step nilsequences., Comment: 14 pages

In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $\alpha n^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations., Comment: 132 pages

Let F be a local field of characteristic zero. Let μ be a good character of GL p ( F ) × GL p + 1 ( F ) . We study the generalized linear period problem for the pair ( G , H p , p + 1 ) = ( GL 2 p + 1 ( F ) , GL p ( F ) × GL p + 1 ( F ) ) and we prove that any bi- ( H p , p + 1 , μ ) -equivariant tempered generalized function on G is invariant under the matrix transpose. We also show that any P ∩ H p , p + 1 -invariant linear functional on an H p , p + 1 -distinguished irreducible smooth representation of G is also H p , p + 1 -invariant if F is nonarchimedean, where P is the standard mirabolic subgroup of G consisting of matrices with last row vector ( 0 , ⋯ , 0 , 1 ) .

Let q be a power of the prime number p, let K = F q ( t ) , and let r ⩾ 2 be an integer. For points a , b ∈ K which are F q -linearly independent, we show that there exist positive constants N 0 and c 0 such that for each integer l ⩾ N 0 and for each generator τ of F q l / F q , we have that for all except N 0 values λ ∈ F q ‾ , the corresponding specializations a ( τ ) and b ( τ ) cannot have orders of degrees less than c 0 log ⁡ log ⁡ l as torsion points for the Drinfeld module Φ ( τ , λ ) : F q [ T ] ⟶ End F q ‾ ( G a ) (where G a is the additive group scheme), given by Φ T ( τ , λ ) ( x ) = τ x + λ x q + x q r .

Maximal mediated sets (MMS), introduced by Reznick, are distinguished subsets of lattice points in integral polytopes with even vertices. MMS of Newton polytopes of AGI-forms and nonnegative circuit polynomials determine whether these polynomials are sums of squares. In this article, we take initial steps in classifying MMS both theoretically and practically. Theoretically, we show that MMS of simplices are isomorphic if and only if the simplices generate the same lattice up to permutations. Furthermore, we generalize a result of Iliman and the third author. Practically, we fully characterize the MMS for all simplices of sufficiently small dimensions and maximal 1-norms. In particular, we experimentally prove a conjecture by Reznick for 2 dimensional simplices up to maximal 1-norm 150 and provide indications on the distribution of the density of MMS., Minor revision; final version; 26 pages, 7 figures, 6 tables

Let p be a rational prime and q a power of p. Let n be a non-constant monic polynomial in F q [ t ] which has a prime factor of degree prime to q − 1 . In this paper, we define a Drinfeld modular curve Y 1 Δ ( n ) over A [ 1 / n ] and study the structure around cusps of its compactification X 1 Δ ( n ) , in a parallel way to Katz-Mazur's work on classical modular curves. Using them, we also define a Hodge bundle over X 1 Δ ( n ) such that Drinfeld modular forms of level Γ 1 ( n ) , weight k and some type are identified with global sections of its k-th tensor power.

For a graph G and c R : E ( G ) → [ k ] , a path P between u , v ∈ V ( G ) is a rainbow path if for distinct e , e ′ ∈ E ( P ) , we have c R ( e ) ≠ c R ( e ′ ) . Rainbow k -Coloring takes a graph G and the objective is to check if there is c R : E ( G ) → [ k ] such that for all u , v ∈ V ( G ) there is a rainbow path between u and v. Two variants of the above problem are Subset Rainbow k -Coloring and Steiner Rainbow k -Coloring , where we are additionally given a subset S ⊆ V ( G ) × V ( G ) and S ′ ⊆ V ( G ) , respectively. Moreover, the objective is to check if there is c R : E ( G ) → [ k ] , such that there is a rainbow path for each ( u , v ) ∈ S and u , v ∈ S ′ , respectively. Under ETH, we obtain that for each k ≥ 3 : 1. Rainbow k -Coloring has no 2 o ( | E ( G ) | ) n O ( 1 ) -time algorithm. 2. Steiner Rainbow k -Coloring has no 2 o ( | S | 2 ) n O ( 1 ) -time algorithm. We also obtain that Subset Rainbow k -Coloring and Steiner Rainbow k -Coloring admit 2 O ( | S | ) n O ( 1 ) - and 2 O ( | S | 2 ) n O ( 1 ) -time algorithms, respectively.

There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n) \hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) = \text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ is a natural generalization of the parafermion algebra of $\mathfrak{sl}_2$. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,3,\dots)$. In this paper, we consider an analogous structure of orthogonal type, namely $\mathcal{D}^k(n) = \text{Com}(V^k(\mathfrak{so}_{2n}), V^k(\mathfrak{so}_{2n+1}))^{\mathbb{Z}_2}$. We realize this algebra as a one-parameter quotient of the two-parameter even spin $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,4,\dots)$, and we classify all coincidences between its simple quotient $\mathcal{D}_k(n)$ and the algebras $\mathcal{W}_{\ell}(\mathfrak{so}_{2m+1})$ and $\mathcal{W}_{\ell}(\mathfrak{so}_{2m})^{\mathbb{Z}_2}$. As a corollary, we show that for the admissible levels $k = -(2n-2) + \frac{1}{2} (2 n + 2 m -1)$ for $\widehat{\mathfrak{so}}_{2n}$ the simple affine algebra $L_k(\mathfrak{so}_{2n})$ embeds in $L_k(\mathfrak{so}_{2n+1})$, and the coset is strongly rational. As a consequence, the category of ordinary modules of $L_k(\mathfrak{so}_{2n+1})$ at such a level is a braided fusion category., Comment: Minor corrections, final version to appear in J. Algebra

We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference of the two preceding terms where the pluses and minuses follow a certain pattern. In 2012, McLellan proved that if the pluses and minuses follow a periodic pattern and G n is the nth term of the resulting generalised Fibonacci sequence, then lim n → ∞ ⁡ | G n | 1 / n exists. We extend her results to recurrences of the form G m + 2 = α m G m + 1 ± G m if the choices of pluses and minuses, and of the α m follow a balancing word type pattern.

For any integer K ≥ 1 , let s ( K ) be the smallest integer such that when the set of squares of the prime numbers is coloured in K colours, each sufficiently large integer can be written as a sum of no more than s ( K ) squares of primes, all of the same colour. We show that s ( K ) ≪ K exp ⁡ ( ( 3 log ⁡ 2 + o ( 1 ) ) log ⁡ K log ⁡ log ⁡ K ) for K ≥ 2 . This upper bound for s ( K ) is close to optimal and improves on s ( K ) ≪ ϵ K 2 + ϵ , which is the best available upper bound for s ( K ) .

Let l be an odd prime number and k = Q ( ζ + ζ − 1 ) where ζ is a primitive l-th root of unity. We provide a sufficient condition for the monogenity of a cyclic extension K s / k of degree l, where s is an integer of k and K s is a field defined by Rikuna's generic cyclic polynomial. As an application, we prove that there exist infinitely many monogenic extensions K s / k for l ≥ 5 .

We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup Γ H ( N , t ) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about Γ 1 ( m ) ⋂ Γ 0 ( m N ) . Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with [ Γ : Γ H ( N , t ) ] = 2 . Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.

The Hilbert L-matrix A s = [ a i j ( s ) ] , where a i j ( s ) = 1 / ( max ⁡ { i , j } + s ) with i , j ≥ 0 , was introduced in [3] . As a surprising property, we showed that its 2-norm is constant for s ≥ s 0 , where the critical point s 0 is unknown but relies in the interval ( 1 / 4 , 1 / 2 ) . In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding s 0 . Moreover, we establish that the same property persists for the p-norm of A s matrices.

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of $n$ disjoint growing squares. Our data structure uses $O(n (\log n \log\log n)^2)$ space, supports queries in worst case $O(\log^3 n)$ time, and updates in $O(\log^7 n)$ amortized time. This leads to an $O(n\alpha(n)\log^7 n)$ time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $O(n^2)$ time algorithms., Comment: 14 pages, 7 figures

We show that any $m\times m$ matrix $M$ with integer entries and $\det M =\Delta \neq 0$ can be equipped by a finite digit set $\mathcal{D}\subset\mathbb{Z}^m$ such that any integer $m$-dimensional vector belongs to the set $$ {\rm Fin}_{\mathcal{D}}(M)= \Bigl\{\sum_{k\in I}M^k {d}_k : \emptyset\neq I \text{ finite subset of } \mathbb{Z} \text{ and } {d}_k \in \mathcal{D} \text{ for each } k \in I\Bigr\} \subset \bigcup\limits_{k\in \mathbb{N}} \frac{1}{\Delta^k}\mathbb{Z}^{m} \,. $$ We also characterize the matrices $M$ for which the sets $ {\rm Fin}_{\mathcal{D}}(M)$ and $ \bigcup\limits_{k\in \mathbb{N}} \frac{1}{\Delta^k}\mathbb{Z}^{m}$ coincide.

We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a bounded domain $\Omega\subset X$ with $\mu_X(X\setminus\Omega)>0$, and a function $f$ in the Besov class $B^\theta_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^\theta_{2,2}(X)$ such that $u=f$ in $X\setminus\Omega$ and $\mathcal{E}_\theta(u,u)\le \mathcal{E}_\theta(h,h)$ whenever $h\in B^\theta_{2,2}(X)$ with $h=f$ in $X\setminus\Omega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally H\"older continuous on $\Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups., Comment: 42 pages, comments welcome, submitted. Revision to add crucial references and attributions to the introduction

The directed power graph $$\vec {\mathcal {G}}( G)$$ of a group G is the simple digraph with vertex set G in which $$x\rightarrow y$$ if y is a power of x, and the power graph is the underlying simple graph $$\mathcal {G}( G)$$ . In this paper, three versions of the definition of the power graph are discussed, and it is proved that the power graph by any of the three versions of the definition determines the other two up to isomorphism. It is also proved that if G is a torsion-free group of nilpotency class 2 and if H is a group such that $$\mathcal {G}( H)\cong \mathcal {G}( G)$$ , then G and H have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina [9].

A graph where each vertex $v$ has a list $L(v)$ of available colors is $L$-colorable if there is a proper coloring such that the color of $v$ is in $L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at least $k$ colors to each vertex guarantees an $L$-coloring. Given a list assignment $L$, an $L$-request for a vertex $v$ is a color $c\in L(v)$. In this paper, we look at a variant of the widely studied class of precoloring extension problems from [Z. Dvo\v{r}\'ak, S. Norin, and L. Postle: List coloring with requests. J. Graph Theory 2019], wherein one must satisfy "enough", as opposed to all, of the requested set of precolors. A graph $G$ is $\varepsilon$-flexible for list size $k$ if for any $k$-list assignment $L$, and any set $S$ of $L$-requests, there is an $L$-coloring of $G$ satisfying an $\varepsilon$-fraction of the requests in $S$. It is conjectured that planar graphs are $\varepsilon$-flexible for list size $5$, yet it is proved only for list size $6$ and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masa\v{r}\'ik and show that planar graphs without $K_4^-$ are $\varepsilon$-flexible for list size $5$. We also prove that planar graphs without $4$-cycles and $3$-cycle distance at least 2 are $\varepsilon$-flexible for list size $4$. Finally, we introduce a new (slightly weaker) form of $\varepsilon$-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are $\varepsilon$-flexible for list size $5$., Comment: 18 pages, 4 figures

Funding: The first author is supported by an Engineering and Physical Sciences Research Council grant (EP/P015638/1). The second author is supported by a Royal Society University Research Fellowship (URF\R\180015). Let H ≤ Sn be an intransitive group with orbits Ω1, Ω2, ... , Ωk. Then certainly H is a subdirect product of the direct product of its projections on each orbit, H|Ω1 x H|Ω2 x ... x H|Ωk. Here we provide a polynomial time algorithm for computing the finest partition P of the H-orbits such that H = Πc∈P H|c and we demonstrate its usefulness in some applications. Postprint

We give a simple combinatorial description of an (n-2k+2)-chromatic edge-critical subgraph of the Schrijver graph SG(n,k), itself an induced vertex-critical subgraph of the Kneser graph KG(n,k). This extends the main result of Kaiser and Stehlík (2020) to all values of k, and sharpens the classical results of Lovász and Schrijver from the 1970s. We give a simple combinatorial description of an (n-2k+2)-chromatic edge-critical subgraph of the Schrijver graph SG(n,k), itself an induced vertex-critical subgraph of the Kneser graph KG(n,k). This extends the main result of Kaiser and Stehlík (2020) to all values of k, and sharpens the classical results of Lovász and Schrijver from the 1970s.

In a Note in this Monthly, Klazar raised the question of whether the alternating sum of the Stirling numbers of the second kind $B^\pm(n)=\sum_{k=0}^n(-1)^kS(n,k)$ is ever zero for $n\neq 2$. In this article, we present an exposition of the history of this problem, and an economical account of a recent proof that there is at most one $n\neq 2$ for which $B^\pm(n)=0$., Comment: 19 page, 1 figure

We assume that the diffusion $X$ satisfies a stochastic differential equation of the form: $dX_t=\mu(X_t, \theta)dt+\sigma_0\nu(X_t)dW_t$, with unknown drift parameter $\theta$ and known diffusion coefficient parameter $\sigma_0$. We prove that approximate maximum likelihood estimator of drift parameter $\bar{; ; ; ; ; ; \theta}; ; ; ; ; ; _n$ obtained from discrete observations $(X_{; ; ; ; ; ; i\Delta_n}; ; ; ; ; ; , 0\leq i\leq n)$ along fixed time interval $[0, T]$, and when $\Delta_n =\frac{; ; ; ; ; ; T}; ; ; ; ; ; {; ; ; ; ; ; n}; ; ; ; ; ; $ tends to zero, is locally asymptotic mixed normal, with covariance matrix which depends on MLE $\hat{; ; ; ; ; ; \theta}; ; ; ; ; ; $ obtained from continuous observations $(X_t, 0\leq t\leq T)$ along fixed time interval $[0, T]$, and on path $(X_t, 0\leq t\leq T)$.

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M\_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M\_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E\_i\times E\_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M\_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part., 15 pages

Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol--Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups., 22 pages, 2 figures; V3: Final version, to appear in Izvestiya: Mathematics

The Chevalley–Warning Theorem is a result on the solution set of a system of polynomial equations f 1 , … , f r in n variables over a finite field F q in the low degree case d ≔ ∑ j = 1 r deg ( f j ) n . In this note we reformulate that result in terms of fibers of the associated polynomial map and, following Heath-Brown, show that something weaker continues to hold when d = n . This result invites a search for homogeneous degree n polynomials in n variables over F q for which the associated polynomial function F q n → F q is not surjective, and we exhibit several families of such polynomials.

In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular overpartition function C ‾ k , i ( n ) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ± i ( mod k ) may be overlined. Andrews also proved two beautiful Ramanujan type congruences modulo 3 satisfied by C ‾ 3 , 1 ( n ) . Later on, Aricheta proved that for an infinite family of l, C ‾ 3 l , l ( n ) is almost always divisible by 2. In this article, for an infinite subfamily of l considered by Aricheta, we prove that C ‾ 3 l , l ( n ) is almost always divisible by arbitrary powers of 2. We also prove that C ‾ 3 l , l ( n ) is almost always divisible by arbitrary powers of 3 when l = 3 , 6 , 12 , 24 . Proofs of our density results rely on the modularity of certain eta-quotients which arise naturally as generating functions for the Andrews' singular overpartition functions.

We complete the discrete cluster categories of type A as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom‐finite Krull–Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom‐spaces in this new category can be described geometrically, even though the category is not 2‐Calabi–Yau and Ext‐spaces are not always symmetric. We describe all cluster‐tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite‐dimensional subrepresentations of infinite‐dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi–Yau conditions.