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In view of Sarnak’s conjecture in number theory, we investigate orders of oscillating sequences. For oscillating sequences (of order 1), we have proved that they are linearly disjoint from all MMA and MMLA flows. We define oscillating sequences of order d and oscillating sequences of order d in the arithmetic sense for d ≥ 2 in this paper. Moreover, we prove that oscillating sequences of order d are linearly disjoint from all affine distal flows as well as all nonlinear affine distal flows with Diophantine translations on the d-torus. We prove that oscillating sequences of order d in the arithmetic sense are linearly disjoint from all nonlinear distal flows with rational translations on the d-torus, too. Furthermore, the linear disjointness of oscillating sequences of order d in the arithmetic sense from other affine flows with zero topological entropy as well as associated nonlinear flows with Diophantine translations on the d-torus can be treated as a consequence of the main result in this paper. One of the consequences is that Sanark’s conjecture holds for all the flows discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper studies vanishing of Ext modules over Cohen--Macaulay local rings. The main result of this paper implies that the Auslander--Reiten conjecture holds for maximal Cohen--Macaulay modules of rank one over Cohen--Macaulay normal local rings. It also recovers a theorem of Avramov--Buchweitz--Şega and Hanes--Huneke, which shows that the Tachikawa conjecture holds for Cohen--Macaulay generically Gorenstein local rings. [ABSTRACT FROM AUTHOR]

The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan- Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon's conjecture by the Pieri rule. [ABSTRACT FROM AUTHOR]

In this paper we prove the containment I(nm) ⊂ M(n-1)mIm, for a radical ideal I of s general points in Pn, where s ≥ 2n. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least 2n points in Pn. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following “weak bounded negativity conjecture", which says that given a complex smooth projective surface X, for any reduced curve C in X and integer g, assume that the geometric genus of each component of C is bounded from above by g; then the self-intersection number C2 is bounded from below. [ABSTRACT FROM AUTHOR]

Let G be a finite solvable group, and let Δ(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P. P. Pálfy asserts that the complement Δ(G)... of the graph Δ(G) does not contain any cycle of length 3. In this paper we generalize Pálfy’s result, showing that Δ(G)... does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of Δ(G) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if n is the clique number of Δ(G), then Δ(G) has at most 2n vertices. This confirms a conjecture by Z. Akhlaghi and H. P. Tong-Viet, and provides some evidence for the famous ρ-σ conjecture by B. Huppert. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the restriction problem in harmonic analysis. Based on the spherical harmonics expansion and analyzing the asymptotic behavior of the Bessel function, we show that a modified linear adjoint restriction estimate holds for all Schwartz functions compactly supported on the cone, which generalizes Shao's result. [ABSTRACT FROM AUTHOR]

We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in 2006. Another one of our main results, in the special setting of hypersurfaces, was also explicitly stated as a conjecture by B. Lamel and N. Mir in a paper from 2006. [ABSTRACT FROM AUTHOR]

The authors have conjectured that if a normalized generalized modular function (GMF) f, defined on a congruence subgroup Γ, has integral Fourier coefficients, then f is classical in the sense that some power fm is a modular function on Γ. A strengthened form of this conjecture was proved in case the divisor of f is empty. In the present paper we study the canonical decomposition of a normalized parabolic GMF f = f1f0 into a product of normalized parabolic GMFs f1, f0 such that f1 has unitary character and f0 has empty divisor. We show that the strengthened form of the conjecture holds if the first "few" Fourier coefficients of f1 are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either f0 = 1 or the divisor of f is concentrated at the cusps of Γ. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new expected dimension formula for certain rank two Brill-Noether loci with fixed special determinant. This answers a question asked by Osserman and also leads to a new and much simpler proof of a theorem in his 2015 work. Our result generalizes the well-known result by Bertram, Feinberg and independently Mukai on expected dimension of rank two Brill-Noether loci with canonical determinant and partially verifies a conjecture (in rank two) of Grzegorczyk and Newstead on coherent systems. [ABSTRACT FROM AUTHOR]

Bringmann and Dousse recently established a conjecture of Dyson dealing with the limiting asymptotics of the Andrews-Garvan crank statistic for integer partitions. This note presents a direct "sieving" technique to establish this conjecture. The technique readily yields the analogous result for Dyson's partition rank and all of Garvan's k-rank statistics. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove Bloch's conjecture for Inoue surfaces with pg = 0 and K2 = 7. These surfaces can also be described as bidouble covers of the four nodal cubic, which allows one to use the method of "enough automorphisms" due to Inose and Mizukami. [ABSTRACT FROM AUTHOR]

Sendov's conjecture says that if all zeros of a complex polynomial P lie in the closed unit disk and a denotes one of them, then the closed disk of center a and radius 1 contains a critical point of P (i.e. a zero of its derivative P'). The main result of this paper is to prove that, for each a, there exists an integer N such that the disk |ζ -a| ≤ 1 contains a critical point of P when the degree of P is larger than N. We obtain this by studying the geometry of the zeros and critical points of a polynomial which would eventually contradict Sendov's conjecture. [ABSTRACT FROM AUTHOR]

We determine the numerical invariants of blocks with defect group D2n "C2m ≃ Q2n "C2m (central product), where n ⩾ 3 and m ⩾ 2. As a consequence, we prove Brauer's k(B)-conjecture, Olsson's conjecture (and more generally Eaton's conjecture), Brauer's height zero conjecture, the Alperin- McKay conjecture, Alperin's weight conjecture and Robinson's ordinary weight conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper continues B. Sambale, Blocks with defect group D2n × C2m, J. Pure Appl. Algebra 216 (2012), 119-125. [ABSTRACT FROM AUTHOR]

In the present paper, we show that the Morimoto Conjecture on the super additivity of the tunnel numbers of knots in S3 is true for knots K1,K2 in S3 in which each Ki is either a tunnel number one or m-small, i = 1, 2. This extends two known results by Morimoto. [ABSTRACT FROM AUTHOR]

Let K be a field and S = K[x1, . . .,xn] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G1, . . .,Gp and let I = I(G) be its edge ideal in S. Suppose that di is the diameter of Gi, 1 ≤ i ≤ p, and consider d = max{di | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max{...} is a lower bound for depth(S/It). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/It). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem. [ABSTRACT FROM AUTHOR]

Let Gn denote either the group Sp(n, F) or SO(2n + 1, F) over a non-archimedean local field F. We explicitly determine the Aubert duals of strongly positive discrete series representations of the group Gn. This enables us to construct a large class of unitarizable representations of this group. [ABSTRACT FROM AUTHOR]

A recent nice result due to I. Pak and G. Panova is the strict unimodality of the q-binomial coefficients ... . Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of ... can get arbitrarily large, when we assume that b is fixed and a is large enough. [ABSTRACT FROM AUTHOR]

We determine the numerical invariants of blocks with defect group D2n "C2m ≃ Q2n "C2m (central product), where n ⩾ 3 and m ⩾ 2. As a consequence, we prove Brauer's k(B)-conjecture, Olsson's conjecture (and more generally Eaton's conjecture), Brauer's height zero conjecture, the Alperin- McKay conjecture, Alperin's weight conjecture and Robinson's ordinary weight conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper continues B. Sambale, Blocks with defect group D2n × C2m, J. Pure Appl. Algebra 216 (2012), 119-125. [ABSTRACT FROM AUTHOR]

We consider a conjecture attributed to Muckenhoupt and Wheeden which suggests a positive relationship between the continuity of the Hardy-Littlewood maximal operator and the Hilbert transform in the weighted setting. Although continuity of the two operators is equivalent for Ap weights with 1 < p < 8, through examples we illustrate this is not the case in more general contexts. In particular, we study weights for which the maximal operator is bounded on the corresponding Lp spaces while the Hilbert transform is not. We focus on weights which take the value zero on sets of nonzero measure and exploit this lack of strict positivity in our constructions. These types of weights and techniques have been explored previously by the first author and independently with C. Thiele. [ABSTRACT FROM AUTHOR]

The holomorphy conjecture predicts that the topological zeta function associated to a polynomial $ f \in \mathbb{C}[x_1,\ldots,x_n]$ is holomorphic unless $ d$. In this paper, we generalise the holomorphy conjecture to the setting of arbitrary ideals in $ \mathbb{C}[x_1,\ldots,x_n]$. [ABSTRACT FROM AUTHOR]

A formula on the scattering length for 3-dimensional Brownian motion was conjectured by M. Kac and proved by others later. It was recently proved under the framework of symmetric Markov processes by Takeda. In this paper, we shall prove that this formula holds for Markov processes under weak duality by the machinery developed mainly by Fitzsimmons and Getoor. [ABSTRACT FROM AUTHOR]

In a recent paper, Evans relates twisted Kloosterman sheaf sums to Gaussian hypergeometric functions, and he formulates a number of conjectures relating certain twisted Kloosterman sheaf sums to the coefficients of modular forms. Here we prove one of his conjectures for a fourth order twisted Kloosterman sheaf sum $T_n$ of the quadratic character on $mathbf {F}_p^times $. In the course of the proof we develop reductions for twisted moments of Kloosterman sums and apply these in the end to derive a congruence relation for $T_n$ with generalized Apéry numbers. [ABSTRACT FROM AUTHOR]

In this paper we address the issue of positive scalar curvature on oriented nonspin compact manifolds whose universal cover is also nonspin. We provide a conjecture for an obstruction to such curvature in this venue that takes into account all the data known to date. The conjecture is proved for a wide class of closed manifolds based on their fundamental group structure. [ABSTRACT FROM AUTHOR]

Recently Alonso, Martín and Papini conjectured that the value of the von Neumann-Jordan constant is less than or equal to that of the James constant. This paper presents an affirmative answer to such a conjecture. Moreover, we obtain a sharp estimate for the von Neumann-Jordan constant. [ABSTRACT FROM AUTHOR]

The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this paper is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over $p$-adic number fields and the estimates of character sums over Galois rings. Given $n$ we prove, for large enough $q$, the Hansen-Mullen conjecture that there exists a primitive polynomial $f(x)=x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ over $ F_{q}$ of degree $n$ with the $m$-th ($0

In this paper, we prove Vojta's Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce from the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces. [ABSTRACT FROM AUTHOR]

In this paper we prove Pardue's conjecture on the regularity of principal $p$-Borel ideals. As a consequence we obtain an upper bound for the regularity of general $p$-Borel ideals. [ABSTRACT FROM AUTHOR]

In this paper, we borrowed some ideas from general relativity and find a Robinson-type identity for the overdetermined system of partial differential equations in the Fischer-Marsden conjecture. We proved that if there is a nontrivial solution for such an overdetermined system on a 3-dimensional, closed manifold with positive scalar curvature, then the manifold contains a totally geodesic 2-sphere. [ABSTRACT FROM AUTHOR]

It is proved that for any domain in Cn the Carath'eodory-Eisenman volume is comparable with the volume of the indicatrix of the Carath'eodory metric up to small/large constants depending only on n. Then the "multidimensional Suita conjecture" theorem of Blocki and Zwonek implies a comparable relationship between these volumes and the Bergman kernel. [ABSTRACT FROM AUTHOR]

To study the analog of Suita's conjecture for domains D ⊂ ℂn,n ≥ 2, Bɫocki introduced the invariant FkD(z) = KD(z)λ(IkD(z)), where KD(z) is the Bergman kernel of D along the diagonal and λ(IkD(z)) is the Lebesgue measure of the Kobayashi indicatrix at the point z. In this note, we study the behaviour of FkD(z) (and other similar invariants using different metrics) on strongly pseudconvex domains and also compute its limiting behaviour explicitly at certain points of decoupled egg domains in ℂ2 . [ABSTRACT FROM AUTHOR]

A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with some non-anticanonical height functions. As a special case, we verify these conjectures for the first time for some smooth cubic surfaces for height functions associated to certain ample line bundles. [ABSTRACT FROM AUTHOR]

We provide a counterexample to the 1980 conjecture that the CI (Complete Intersection) operators can be chosen to commute on a sufficiently high truncation of the minimal free resolution of a module over a complete intersection. [ABSTRACT FROM AUTHOR]

In this article we study the stochastic six vertex model under the scaling proposed by Borodin and Gorin, where the weights of corner-shape vertices are tuned to zero, and prove their conjecture that the height fluctuation converges in finite dimensional distributions to the solution of the stochastic telegraph equation. [ABSTRACT FROM AUTHOR]

In 1975 Stein conjectured that in every n × n array filled with the numbers 1, . . ., n with every number occuring exactly n times, there is a partial transversal of size n - 1. In this note we show that this conjecture is false by constructing such arrays without partial transverals of size n- 1/42 ln n. [ABSTRACT FROM AUTHOR]

We show that the real parts of the roots of the Alexander polynomial for any 2-bridge knot are greater than -1. This is a specific case of Hoste's conjecture. [ABSTRACT FROM AUTHOR]

We prove the weight-monodromy conjecture for varieties which are p-adically uniformized by a product of the Drinfeld upper half-spaces. It is an easy consequence of Dat's work on the cohomology complex of the Drinfeld upper half-space. [ABSTRACT FROM AUTHOR]

What is the minimal closed cone containing all ƒ-vectors of cubical d-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical g-vector coordinates, contains the nonnegative g-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera, and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold. [ABSTRACT FROM AUTHOR]

Using some formulas of hypergeometric series and properties of the gamma function, we deduce certain p-adic supercongruence relations concerning truncated hypergeometric series. As consequences, we confirm the conjectural supercongruences (see Conjecture 1.2 [Proc. Amer. Math. Soc. 145 (2017), no. 2, 501-508]) posed by Bing He and extend them to all primes. [ABSTRACT FROM AUTHOR]

The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form PGL2(R), where R is some finite commutative ring with identity. We give an equivalent formulation of Luo's conjecture via faithful representations and provide various examples and a counterexample. [ABSTRACT FROM AUTHOR]

It is shown that the original Andrews–Curtis conjecture on balanced presentations of the trivial group is equivalent to its “cyclic” version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis [Amer. Math. Monthly 73 (1966), 21–28]. We also consider a more restrictive “cancellative” version of the cyclic Andrews–Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews– Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed. [ABSTRACT FROM AUTHOR]

We shall confirm a conjecture of Gosper on the q-analogue of the function cos(2z) and we shall give a short proof for his other related identity on the q-analogue of sin(2z) which was recently proved by Mező. [ABSTRACT FROM AUTHOR]

We study the Lq-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq-dimensions are equal to the value predicted by Falconer for a range of q. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets. [ABSTRACT FROM AUTHOR]

We prove that the multiplicities of certain maximal weights of g(An(1) )-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Jayne-Misra. We also prove similar phenomena in types A2n(2) and Dn+1(2). Both proofs are applications of Kashiwara’s crystal theory. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over a number field K, cv the Tamagawa number of E at v, and let cE = ∏v cv. Lorenzini proved that v13(cE) is positive for all elliptic curves over quadratic fields with a point of order 13. Krumm conjectured, based on extensive computation, that the 13-adic valuation of cE is even for all such elliptic curves. In this note we prove this conjecture and furthermore prove that there is a unique such curve satisfying v13(cE)=2. [ABSTRACT FROM AUTHOR]

Let G be a Métivier group and let N be any homogeneous norm on G. For α > 0 denote by w α the function ... and consider the weighted sub-Laplacian ... associated with the Dirichlet form φ → ∫G ||∇ H φ(y)||²wα (y) dy, where ∇H is the horizontal gradient on G. Consider ... with domain Cc∞. We prove that ... is essentially self-adjoint when α ≥ 1. For a particular N, which is the norm appearing in L's fundamental solution when G is an H-type group, we prove that L ... has purely discrete spectrum if and only if α > 2, thus proving a conjecture of J. Inglis. [ABSTRACT FROM AUTHOR]

Let A be a graded complete intersection over a field and B the monomial complete intersection with the generators of the same degrees as A. The EGH conjecture says that if I is a graded ideal in A, then there should be an ideal J in B such that B/J and A/I have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property. [ABSTRACT FROM AUTHOR]

We show that there is only one natural Turing-degree invariant, analytic equivalence relation with ℵ1 many equivalence classes: the equivalence X ≡ω1 Y ⇐⇒ ωX 1 = ωY 1 . More precisely, under PD + ¬CH, we show that every Turing-degree invariant, analytic equivalence relation with ℵ1 many equivalence classes is equal to ≡ω1 on a Turing cone. [ABSTRACT FROM AUTHOR]