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In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

The hyperbolic structure on a 3–dimensional cone–manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number, which is reminiscent of Sabitov's theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated. [ABSTRACT FROM AUTHOR]

Hovey and Palmieri [ The structure of the Bousfield lattice , Amer. Math. Soc., Providence, RI, 1999] proposed the retract conjecture on the Bousfield lattice of the stable homotopy category. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), pp. 497–513] defined the notion of monoidal posets as a generalization of the Bousfield lattice. In this paper, we prove that an analogue of the retract conjecture holds on a sublattice of monoidal posets. [ABSTRACT FROM AUTHOR]

Let G be a group and let k be a field. Kaplansky's direct finiteness conjecture states that every one-sided unit of the group ring k[G] must be a two-sided unit. In this paper, we establish a geometric direct finiteness theorem for endomorphisms of symbolic algebraic varieties. Whenever G is a sofic group or more generally a surjunctive group, our result implies a generalization of Kaplansky's direct finiteness conjecture for the near ring R(k,G) which is k[X_g\colon g \in G] as a group and which contains naturally k[G] as the subring of homogeneous polynomials of degree one. We also prove that Kaplansky's stable finiteness conjecture is a consequence of Gottschalk's Surjunctivity Conjecture. [ABSTRACT FROM AUTHOR]

Given a fixed hypergraph H, let \operatorname {wsat}(n,H) denote the smallest number of edges in an n-vertex hypergraph G, with the property that one can sequentially add the edges missing from G, so that whenever an edge is added, a new copy of H is created. The study of \operatorname {wsat}(n,H) was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H very little is known regarding \operatorname {wsat}(n,H), Alon proved in 1985 that for every graph H there is a limiting constant C_H so that \operatorname {wsat}(n,H)=(C_H+o(1))n. Tuza conjectured in 1992 that Alon's theorem can be (appropriately) extended to arbitrary r-uniform hypergraphs. In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage's Liouville-type theorem. [ABSTRACT FROM AUTHOR]

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded. [ABSTRACT FROM AUTHOR]

Let \mathcal {A} denote a central hyperplane arrangement of rank n in affine space \mathbb {K}^n over a field \mathbb {K} of characteristic zero and let l_1,\ldots, l_m\in R≔\mathbb {K}[x_1,\ldots,x_n] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f≔l_1\cdots l_m\in R. The focus of the paper is on the ideal J_f\subset R generated by the partial derivatives of f. We conjecture that J_f is a minimal reduction of the ideal \mathbb {I}\subset R generated by the (m-1)-fold products of distinct forms among l_1,\ldots, l_m. We prove this conjecture for an almost generic \mathcal {A} (i.e., any n-1 among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for n=3. We also conjecture that J_f is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for n=3. In the sequel we explain the tight relationship between the two ideals J_f, \mathbb {I}\subset R; in particular, we show that in the generic case (J_f)^{\text {sat}}=\mathbb I. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of R/J_f. [ABSTRACT FROM AUTHOR]

In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

Let K_{\nu } be the modified Bessel functions of the second kind of order \nu. The ratio Q_{\nu }\left (x\right) =xK_{\nu -1}\left (x\right) /K_{\nu }\left (x\right) appeared in physics and probability. In this paper, we collate properties of this ratio, and prove the conjecture that \left (-1\right) ^{n}Q_{\nu }^{\left (n\right) }\left (x\right) >\left (<\right) 0 for x>0 and n=2,3 if \left \vert \nu \right \vert >\left (<\right) 1/2 holds for n=2. This yields several new consequences and improves some known results. Finally, two conjectures are proposed. [ABSTRACT FROM AUTHOR]

Let B be an integral domain over a field K of characteristic 0. The derivation δ of B[Yd] = B[y1, . . . ,yd] is elementary if δ (B) = 0 and δ (yi) ∈ B, i = 1, . . . ,d. Then for d ≥ 2 the elements uij = δ (yi)yj-δ (yj)yi, 1 ≤ i < j ≤ d, belong to the algebra B[Yd]δ of constants of δ, and it is a natural question whether the B-algebra B[Yd]δ is generated by all uij. In this paper we consider the special case of B = K[Xd] = K[x1, . . . ,xd]. If δ (yi) = xi, i = 1, . . . ,d, this is the Nowicki conjecture from 1994, which was confirmed in several papers based on different methods. The case δ (yi) = xini, ni > 0, i = 1, . . . ,d, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009, if δ (yi) = ƒi(xi), for any nonconstant polynomials ƒi(xi), i = 1, . . . ,d, then B[Yd]δ = K[Xd,Yd]δ is generated by Xd and Ud = uij = ƒi(xi)yj-yiƒj(xj) mid 1 ≤ i < j ≤ d. In the present paper we have found a presentation of the algebra K[Xd,Yd]δ = K[Xd,Ud mid R = S = 0],d ≥ 4, R = r(i,j,k,l)mid 1 ≥ i < j < k < l \≤ d, S = s(i,j,k)mid 1 ≤ i < j < k ≤ d, and a basis of K[Xd,Yd]δ as a vector space. As a corollary we have shown that the defining relations R ∩ S form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009. The algebras K[Xd,Yd]δ, d < 4, are also described. [ABSTRACT FROM AUTHOR]

For non-zero integers n and k\geq 2, a generalized Diophantine m-tuple with property D_k(n) is a set of m positive integers \{a_1,a_2,\ldots, a_m\} such that a_ia_j + n is a k-th power for 1\leq i< j\leq m. Define M_k(n)≔\sup \{|S| : S has property D_k(n)\}. In this paper, we study upper bounds on M_k(n), as we vary n over positive integers. In particular, we show that for k\geq 3, M_k(n) is O(\log n) and further assuming the Paley graph conjecture, M_k(n) is O((\log n)^{\epsilon }). The problem for k=2 was studied by a long list of authors that goes back to Diophantus. [ABSTRACT FROM AUTHOR]

In this paper we classify varieties of Picard number two having two projective bundle structures of any relative dimension, under the assumption that these structures are mutually uniform. As an application we prove the Campana–Peternell conjecture for varieties of Picard number one admitting \mathbb {C}^*-actions of a certain kind. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer. [ABSTRACT FROM AUTHOR]

In this paper, we provide a sufficient condition for a curve on a surface in R3 to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball B3. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture. [ABSTRACT FROM AUTHOR]

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group Γ implies that Γ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition. [ABSTRACT FROM AUTHOR]

A well known conjecture due to Hassett asserts that a cubic fourfold X whose transcendental cohomology TX cannot be realized as the transcendental cohomology of a K3 surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic 2-cycles on them, it is natural to ask for the most algebraic cubic fourfolds X to which this conjecture is still applicable. In this paper, we show that for an appropriate "algebraicity index" κX ∈ Q+, there exists a unique class of cubics maximizing κX, not having an associated K3 surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett's irrationality conjecture for cubic fourfolds. [ABSTRACT FROM AUTHOR]

Given a poset (P,≤), an antichain is a subset of pairwise incomparable elements of P. Let (P,w) be a graded, weighted poset. If the maximum weight of an antichain of P is equal to the weight of the largest rank of P, then P is said to be Sperner. In 1967, Rota conjectured that the poset of partitions, ordered by refinement of blocks, is Sperner; this conjecture was later disproved by Canfield. In this paper, we consider a generalization of Rota's conjecture and show that Sn, partially ordered by refinement of cycles, is strongly Sperner. [ABSTRACT FROM AUTHOR]

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger-Bressoud q-Dyson theorem or the q-Dyson constant term identity. This conjecture was proved by Károlyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the q-Dyson constant term identity. This new generalized constant term admits a simple product-form expression. [ABSTRACT FROM AUTHOR]

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]

The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic (3-valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non- 3-edge-colorable cubic graph. In contrast to Tutte's 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5-edge-connected. [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]

In this paper we construct an infinite family of paramodular forms of weight 2 which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta block conjecture of Gritsenko-Poor-Yuen (2013) related to the most important infinite series of theta blocks of weight 2 and q-order 1. We also consider some applications of this result. [ABSTRACT FROM AUTHOR]

As a natural generalization of the Euler-Mascheroni constant γ, Ihara introduced the Euler-Kronecker constant γKattached to any number field K. In this paper, we prove that a certain bound on γK in a tower of number fields K implies the generalized Brauer-Siegel conjecture for K as formulated by Tsfasman and Vladuţ. Moreover, we use known bounds on γK for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function ζK(s) in the critical strip. [ABSTRACT FROM AUTHOR]

In this paper, we compute categorical entropy of spherical twists. In particular, we prove that the Gromov-Yomdin-type conjecture holds for spherical twists. Moreover, we construct counterexamples of Gromov-Yomdin type conjecture for K3 surfaces modifying Fan's construction for even higher-dimensional Calabi-Yau manifolds. The appendix, by Arend Bayer, shows the nonemptiness of complements of a number of spherical objects in the derived categories of K3 surfaces. [ABSTRACT FROM AUTHOR]

In 1976 during a conference in Halifax, A. Lau raised the question on whether left amenability property for a semitopological semigroup is sufficient to ensure the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex set in a dual Banach space. In this paper we establish a positive answer for discrete semigroups and for strongly left amenable semitopological semigroups. [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 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]

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 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]

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]