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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]

Let \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of \Gamma which depends only on the width of \Gamma, that is, the difference between the largest and the smallest generator of \Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width. [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]

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 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G is a finite group, p is a prime dividing the group order, and k(G) is the number of conjugacy classes of G, then k(G)\geq 2\sqrt {p-1}, and they proved this conjecture for solvable G and showed that it is sharp for those primes p for which \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p, and we prove it for solvable groups, and when p is large, also for arbitrary groups. [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]

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]

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]

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]

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]

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]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

Over a field of characteristic 0, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of n \times n matrices. This modifies a conjecture of Weyman and provides a complete answer to it. [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]

We prove a strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far. [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]

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 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 show that Woodin's \mathsf {AD}^+ Conjecture follows from various hypotheses extending the Continuum Hypothesis (\mathsf {CH}). These results complement Woodin's original result that the \mathsf {AD}^+ Conjecture follows from \mathsf {MM}(\mathfrak {c}). [ABSTRACT FROM AUTHOR]

Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a "Moonshine Conjecture at Landau-Ginzburg points" [arXiv:2107.12405, 2021] for Klein's modular j-function at j=0 and j=1728. The conjecture asserts that the j-function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical _2F_1-hypergeometric inversion formulae for the j-function. [ABSTRACT FROM AUTHOR]

A recent conjecture of Di Francesco states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted \mathcal {T}_n, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions \mathcal {T}_n, we construct a family C_{m,n}^{a,b,c,d} of cruciform regions generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of \mathcal {T}_n is a divisor of the number of tilings of the cruciform region C_{2n-1,2n-1}^{n-1,n,n,n-2}, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture. [ABSTRACT FROM AUTHOR]

In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin resolution of a conjecture of Pólya-Szegö: when the Gaussian torsional rigidity of a domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a "modified" torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle. [ABSTRACT FROM AUTHOR]

We describe an explicit formula of the canonical pairing on the twisted de Rham cohomology associated with the category of local matrix factorizations and by characterizing its relation to Saito's higher residue pairings we reprove the conjecture of Shklyarov [Adv. Math. 292 (2016), pp. 181–209]. [ABSTRACT FROM AUTHOR]

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomologies of differential graded Leavitt algebras. [ABSTRACT FROM AUTHOR]

We establish a Liouville type result for stable solutions for a wide class of second order semilinear elliptic equations in \mathbb {R}^{n} with sign-changing nonlinearity f. Under the hypothesis that the equation does not have any nonconstant one dimensional stable solution, and a further nondegeneracy condition of f at its zero points, we show that in any dimension, stable solutions of the equation must be constant. This partially answers a question raised by Dancer. [ABSTRACT FROM AUTHOR]

In 1996 Stolz [Math. Ann. 304 (1996), pp. 785–800] conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature. [ABSTRACT FROM AUTHOR]

Let (K,d) be a connected compact metric space and p\in (1, \infty). Under the assumption of Kigami [ Conductive homogeneity of compact metric spaces and construction of p-energy , Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that \mathcal {W}^p(K)\subset C(K) holds if and only if p>\operatorname {dim}_{AR}(K,d), where \mathcal {W}^p(K) is Kigami's (1,p)-Sobolev space and \operatorname {dim}_{AR}(K,d) is the Ahlfors regular dimension. [ABSTRACT FROM AUTHOR]

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules. [ABSTRACT FROM AUTHOR]

Our main goal in this note is to use a version of a lemma by Hanes and Huneke to provide characterizations of when certain one-dimensional reduced local rings are regular. This is of interest in view of the long-standing Berger's Conjecture (the ring is predicted to be regular if its universally finite differential module is torsion-free), which in fact we show to hold under suitable additional conditions, mostly toward the G-regular case of the conjecture. Furthermore, applying the same lemma to a Cohen-Macaulay local ring which is locally Gorenstein on the punctured spectrum but of arbitrary dimension, we notice a numerical characterization of when an ideal is strongly non-obstructed and of when a given semidualizing module is free. [ABSTRACT FROM AUTHOR]

Let (R,\mathfrak {m}_R,\mathbb {k}) be a one-dimensional complete local reduced \mathbb {k}-algebra over a field of characteristic zero. The ring R is said to be quasihomogeneous if there exists a surjection \Omega _R\twoheadrightarrow \mathfrak {m} where \Omega _R denotes the module of differentials. We present two characterizations of quasihomogeniety of R in the case when R is a domain. The first one on the valuation semigroup of R and the other on the trace ideal of the module \Omega _R. [ABSTRACT FROM AUTHOR]

We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave. [ABSTRACT FROM AUTHOR]

In this short note we prove a lower bound for the MaxCut of a graph in terms of the Lovász theta function of its complement. We combine this with known bounds on the Lovász theta function of complements of H-free graphs to recover many known results on the MaxCut of H-free graphs. In particular, we give a new, very short proof of a conjecture of Alon, Krivelevich and Sudakov about the MaxCut of graphs with no cycles of length r. [ABSTRACT FROM AUTHOR]

Let f^{(r)}(n;s,k) be the maximum number of edges of an r-uniform hypergraph on n vertices not containing a subgraph with k edges and at most s vertices. In 1973, Brown, Erdős and Sós conjectured that the limit \begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} exists for all positive integers k\ge 2. They proved this for k=2. In 2019, Glock proved this for k=3 and determined the limit. Quite recently, Glock, Joos, Kim, Kühn, Lichev and Pikhurko proved this for k=4 and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers k\ge 2. [ABSTRACT FROM AUTHOR]

The notion of strict closedness of rings was given by J. Lipman [Amer. J. Math. 93 (1971), pp. 649–685] in connection with a conjecture of O. Zariski. The present purpose is to give a practical method of construction of strictly closed rings. It is also shown that the Stanley-Reisner rings of simplicial complexes (resp. F-pure rings satisfying the condition (S2) of Serre) are strictly closed (resp. weakly Arf) rings. [ABSTRACT FROM AUTHOR]

From Carlitz's identity, we deduce two new q-supercongruences modulo the square of a cyclotomic polynomial, which were originally conjectured by Guo. These results establish new q-analogues of a supercongruence of Sun. [ABSTRACT FROM AUTHOR]

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui–Hà–Jayanthan–Thomas [Collect. Math. 72 (2021), pp. 605–614]. Complete solutions are delivered for two conjectures proposed by these authors. For given real numbers a and b, we consider the set Res(a,b) of possible values of the resurgence of I+J where I and J are ideals in disjoint sets of variables having resurgence a and b, respectively. Some questions and partial results about Res(a,b) are discussed. [ABSTRACT FROM AUTHOR]

We present a very simple proof of some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear Neumann boundary condition, which confirms partially the Wang's conjecture in case \frac {n-2}{2}<\lambda. [ABSTRACT FROM AUTHOR]

The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area \pi. In this note we confirm two cases of the conjecture: perturbations of the unit disk and domains bounded by curves symmetric to both coordinate axes and having exactly four vertices. [ABSTRACT FROM AUTHOR]

The Farrell-Jones conjecture for lax monoidal finitary localising invariants was recently proved by Bunke-Kasprowski-Winges [ On the Farrell-Jones conjecture for localising invariants , 2021, arXiv:2111.02490]. In this short note, making use of the theory of noncommutative motives, we prove that the lax monoidal assumption is not necessary. [ABSTRACT FROM AUTHOR]

For each positive integer n, function f, and point c, the GGR Theorem states that f is n times Peano differentiable at c if and only if f is n-1 times Peano differentiable at c and the following n-th generalized Riemann derivatives of f at c exist: \[ \lim _{h\rightarrow 0}\frac 1{h^{n}}\sum _{i=0}^n(-1)^i\binom {n}{i}f(c+(n-i-k)h), \] for k=0,\ldots,n-1. The theorem has been recently proved by Ash and Catoiu [Int. Math. Res. Not. IMRN 2022, no. 10, pp. 7893–7921] and has been a conjecture by Ginchev, Guerraggio, and Rocca [ Equivalence of Peano and Riemann derivatives , Generalized convexity and optimization for economic and financial decisions (Verona, 1998), Pitagora, Bologna, 1999] since 1998. We provide a new proof of this theorem, based on a generalization of it that produces numerous new sets of n-th Riemann smoothness conditions that can play the role of the above set in the GGR Theorem. [ABSTRACT FROM AUTHOR]

In this note, we prove some new stability results for plethysm coefficients. As special cases, we verify a conjecture of Wildon, and show the stability of sequences recently predicted by Bessenrodt, Bowman and Paget [Trans. Amer. Math. Soc. 375 (2022), pp. 5151–5194] to be weakly increasing. [ABSTRACT FROM AUTHOR]

We show how to obtain minimal projective resolutions of finitely generated modules over an idempotent subring \Gamma _e ≔(1-e)R(1-e) of a semiperfect noetherian basic ring R by a construction inside \mathsf {mod}\,R. This is then applied to investigate homological properties of idempotent subrings \Gamma _e under the assumption of R/\langle 1-e\rangle being a right artinian ring. In particular, we prove the conjecture by Ingalls and Paquette that a simple module S_e ≔eR /\operatorname {rad}eR with \operatorname {Ext}_R^1(S_e,S_e) = 0 is self-orthogonal, that is \operatorname {Ext}^k_R(S_e,S_e) vanishes for all k \geq 1, whenever \operatorname {gldim}R and \operatorname {pdim}eR(1-e)_{\Gamma _e} are finite. Indeed, a slightly more general result is established, which applies to sandwiched idempotent subrings: Suppose e \in R is an idempotent such that all idempotent subrings \Gamma sandwiched between \Gamma _e and R, that is \Gamma _e \subseteq \Gamma \subseteq R, have finite global dimension. Then the simple summands of S_e can be numbered S_1, \dots, S_n such that \operatorname {Ext}_R^k(S_i, S_j) = 0 for 1 \leq j \leq i \leq n and all k > 0. [ABSTRACT FROM AUTHOR]

In this work, we start an investigation of asymmetric Rogers–Ramanujan type identities. The first object is the following unexpected relation \begin{equation*} \sum _{n\ge 0} \frac {(-1)^n q^{3\binom {n}{2}+4n}(q;q^3)_n}{(q^9;q^9)_n} = \frac {(q^{4};q^{6})_\infty (q^{12};q^{18})_\infty }{(q^{5};q^{6})_\infty (q^{9};q^{18})_\infty } \end{equation*} and its a-generalization. We then use this identity as a key ingredient to confirm a recent conjecture of G. E. Andrews and A. K. Uncu [Ramanujan J. (2023), DOI 10.1007/s11139-022-00685-y]. [ABSTRACT FROM AUTHOR]

In 2008, Schmidt and Tuller stated a conjecture concerning optimal packing and covering of integers by translates of a given three-point set. In this note, we confirm their conjecture and relate it to several other problems in combinatorics. [ABSTRACT FROM AUTHOR]

For a definable continuous mapping f from a definable connected open subset \Omega of \mathbb R^n into \mathbb R^n, we show that the following statements are equivalent: The mapping f is open. The fibers of f are finite and the Jacobian of f does not change sign on the set of points at which f is differentiable. The fibers of {f} are finite and the set of points at which f is not a local homeomorphism has dimension at most n - 2. As an application, we prove that Whyburn's conjecture is true for definable mappings: A definable surjective open continuous mapping of one closed ball into another which maps boundary homeomorphically onto boundary is necessarily a homeomorphism. [ABSTRACT FROM AUTHOR]

We study the Picard rank of smooth toric Fano varieties constrained to possess families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen–Fu–Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes. [ABSTRACT FROM AUTHOR]

For the ideal \mathfrak {p} in k[x, y, z] defining a space monomial curve, we show that \mathfrak {p}^{(2 n - 1)} \subseteq \mathfrak {m} \mathfrak {p}^{n} for some positive integer n, where \mathfrak {m} is the maximal ideal (x, y, z). Moreover, the smallest such n is determined. It turns out that there is a counterexample to a claim due to Grifo, Huneke, and Mukundan, which states that \mathfrak {p}^{(3)} \subseteq \mathfrak {m} \mathfrak {p}^2 if k is a field of characteristic not 3; however, the stable Harbourne conjecture holds for space monomial curves as they claimed. [ABSTRACT FROM AUTHOR]

We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [ Classical and quantum orthogonal polynomials in one variable , Cambridge University Press, Cambridge, 2005]. These results generalize the classical problems of Sonine and Hahn. [ABSTRACT FROM AUTHOR]

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the L-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients. [ABSTRACT FROM AUTHOR]