Math Soc Research Articles

Mixing time of the Chung–Diaconis–Graham random process

Define $$(X_n)$$ on $${\mathbf {Z}}/q{\mathbf {Z}}$$ by $$X_{n+1} = 2X_n + b_n$$ , where the steps $$b_n$$ are chosen independently at random from $$-1, 0, +1$$ . The mixing time of this random walk is known to be at most $$1.02 \log _2 q$$ for almost all odd q (Chung, Diaconis, Graham in Ann Probab 15(3):1148–1165, 1987), and at least $$1.004 \log _2 q$$ (Hildebrand in Proc Am Math Soc 137(4):1479–1487, 2009). We identify a constant $$c = 1.01136\dots $$ such that the mixing time is $$(c+o(1))\log _2 q$$ for almost all odd q. In general, the mixing time of the Markov chain $$X_{n+1} = a X_n + b_n$$ modulo q, where a is a fixed positive integer and the steps $$b_n$$ are i.i.d. with some given distribution in $${\mathbf {Z}}$$ , is related to the entropy of a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution). We estimate the mixing time up to a $$1+o(1)$$ factor whenever the entropy exceeds $$(\log a)/2$$ .

The fourth moment of quadratic Dirichlet L-functions

We study the fourth moment of quadratic Dirichlet L-functions at $$s= \frac{1}{2}$$ . We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally. The proofs of these results follow closely arguments of Soundararajan and Young (J Eur Math Soc 12(5):1097–1116, 2010) and Soundararajan (Ann Math (2) 152(2):447–488, 2000).

Periodic Solutions of Discontinuous Duffing Equations

In this paper, we investigate the multiplicity problem of periodic solutions for a class of periodically forced Duffing equations allowing for discontinuities. By using a generalized form of the Poincare–Birkhoff theorem due to Ding (Proc Am Math Soc 88:341–346, 1983), we demonstrate that the discontinuous equation has an infinite number of periodic solutions with large amplitude.

Lacunarity of Han\u2013Nekrasov\u2013Okounkov q-Series

A power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms Fa,b,c(z):=η(24az)aη(24acz)b-aη(24z),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}F_{a,b,c}(z) :=\\frac{\\eta (24az)^a \\eta (24acz)^{b-a}}{\\eta (24z)},\\end{aligned}$$\\end{document}defined in terms of the Dedekind eta -function, for integers a,c ge 1, where b ge 1 is odd throughout. Serre (Publications Mathématiques de l’IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when a = c = 1. Later, Clader et al. (Am Math Soc 137(9):2959–2968, 2009) extended this result by allowing a to be general and completely classified the F_{a,b,1}(z) which are lacunary. Here, we consider all c and show that for {a in {1,2,3}}, there are infinite families of lacunary series. However, for a ge 4, we show that there are finitely many triples (a, b, c) such that F_{a,b,c}(z) is lacunary. In particular, if a ge 4, b ge 7, and c ge 2, then F_{a,b,c}(z) is not lacunary. Underlying this result is the proof the t-core partition conjecture proved by Granville and Ono (Trans Am Math Soc 348(1):331–347, 1996).

On Finch’s Conditions for the Completion of Orthomodular Posets

In this paper, we aim at highlighting the significance of the A- and B-properties introduced by Finch (Bull Aust Math Soc 2:57–62, 1970b). These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logico-algebraic approach to quantum mechanics.

Stochastic expected utility for binary choice: a ‘modular’ axiomatic foundation

We present new axiomatisations for various models of binary stochastic choice that may be characterised as “expected utility maximisation with noise”. These include axiomatisations of simple scalability (Tversky and Russo in J Math Psychol 6:1–12, 1969) with respect to a scale having the expected utility (EU) form, and strong utility (Debreu in Econometrica 26(3):440–444, 1958) of the EU form. The latter model features Fechnerian “noise”: choice probabilities depend on EU differences. Our axiomatisations complement the important contributions of Blavatskyy (J Math Econ 44:1049–1056, 2008) and Dagsvik (Math Soc Sci 55:341–370, 2008). Our representation theorems set all models on a common axiomatic foundation, with additional axioms added in modular fashion to characterise successively more restrictive models. The key is a decomposition of Blavatskyy’s (2008) common consequence independence axiom into two parts: one (which we call weak independence) that underwrites the EU form of utility and another (stochastic symmetry) than underwrites the Fechnerian structure of noise. We also show that in many cases of interest (which we call preference-bounded domains) stochastic symmetry can be replaced with weak transparent dominance (WTD). For choice between lotteries, WTD only restricts behaviour when choosing between probability mixtures of a “best” and a “worst” possible outcome.

On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity

We address the questions (P1), (P2) asked in Kirchheim et al. (Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 1986) concerning the structure of the Rank-1 convex hull of a submanifold $$\mathcal {K}_1\subset M^{3\times 2}$$ that is related to weak solutions of the two by two system of Lagrangian equations of elasticity studied by DiPerna (Trans Am Math Soc 292(2):383–420, 1985) with one entropy augmented. This system serves as a model problem for higher order systems for which there are only finitely many entropies. The Rank-1 convex hull is of interest in the study of solutions via convex integration: the Rank-1 convex hull needs to be sufficiently non-trivial for convex integration to be possible. Such non-triviality is typically shown by embedding a $$\mathbb {T}_4$$ (Tartar square) into the set; see for example Muller et al. (Attainment results for the two-well problem by convex integration. Geometric analysis and the calculus of variations, Int. Press, Cambridge, 1996) and Muller and Sverak (Ann Math (2) 157(3):715–742, 2003). We show that in the strictly hyperbolic, genuinely nonlinear case considered by DiPerna (1985), no $$\mathbb {T}_4$$ configuration can be embedded into $$\mathcal {K}_1$$ .

Equivalence After Extension and Schur Coupling for Relatively Regular Operators

It was recently shown in Ter Horst et al. (Bull Lond Math Soc 51:1005–1014, 2019) that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular operators, the latter being operators with complementable kernels and ranges. In this paper we analyse the relations EAE and SC for the class of relatively regular operators, leading to an equivalent Banach space operator problem from which we derive new cases where EAE and SC coincide and provide a new example for which EAE and SC do not coincide and where the Banach spaces are not essentially incomparable.

The infinitesimal generator of the stochastic Burgers equation

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially L^2-ergodic, and that the stochastic Burgers equation on the real line is ergodic.

Polarization Constant for the Numerical Radius

We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which does not have an almost isometric copy of $$\ell _1^2$$ , such that the second polarization constant for the numerical radius is maximal. We also give a negative answer for complex Banach spaces to the question of Choi and Kim (J Lond Math Soc (2) 54(1):135–147, 1996) whether $$\frac{\sum _{k=1}^m k^m ~\left( {\begin{array}{c}m\\ k\end{array}}\right) }{m!}$$ is the optimal upper bound for the mth polarization constant for arbitrary $$m\in \mathbb {N}$$ . Finally, we generalize the result of Garcia et al. (Proc Am Math Soc 142(4):1229–1235, 2014) that, for 2-homogeneous polynomials, the numerical index of a complex lush space is greater or equal than 1/3.

Asymptotic Behavior of n-Jordan Homomorphisms

The purpose of the present work is to obtain some results concerning approximate n-Jordan homomorphisms on Banach algebras. The results of this paper correct and improve the main results from Ghaleh and Ghasemi (Bull Iran Math Soc 39:347–353, 2013) and improve some results obtained in Gordji et al. (J Ineq Appl 870843, 2009), Miura et al. (J Ineq Appl 2005(4):435–441, 2005). Applications in connection with classical results are also provided.

Deletion correcting codes meet the Littlewood–Offord problem

In this paper, we make a novel connection between information theory and additive combinatorics; more specifically, between deletion/insertion correcting codes and the celebrated Littlewood–Offord problem. We see how results from one area can have an impact on the other area and vice versa. In particular, a result on the Littlewood–Offord problem gives a nice upper bound for the size of the Levenshtein code and the Helberg code (and possibly other variants of these codes). Also, a recent result on the deletion correcting codes gives a modular analogue of the Littlewood–Offord problem which generalizes the results of Vaughan and Wooley (Q J Math Oxf Ser (2) 42(2):379–386, 1991) (obtained using tools from analytic number theory and properties of exponential sums) and of Griggs (Bull Am Math Soc (N.S.) 28:329–333, 1993) (obtained using a combinatorial argument). This novel connection might opens up new doors to research in these or other related areas.

Diagonalizing the Ricci Tensor

We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being “nice”. Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Lauret and Will (Proc Am Math Soc 141(10):3651–3663, 2013) on nilpotent Lie algebras. The result follows from a more general characterization for diagonalizing the Ricci tensor for homogeneous spaces. Finally, we also study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds.

A classification of SNC log symplectic structures on blow-up of projective spaces

It is commonly recognized that the classfication of Poisson manifold is a major problem. From the viewpoint of algebraic geometry, holomorphic projective Poisson manifold is the most important target. Poisson structures on the higher dimensional projective varieties was first studied by Lima and Pereira (Lond Math Soc 46(6):1203–1217, 2014). They proved that any Poisson structures with the reduced and simple normal crossing degeneracy divisor, we call SNC log symplectic structure, on the $$2n\ge 4$$ dimensional Fano variety with the cyclic Picard group must be a diagonal Poisson structure on the projective space. However, it remains to be elucidated when the Picard rank of the variety is greater or equals to 2. Here, we studied SNC log symplectic structures on blow-up of a projective space along a linear subspace, whose Picard rank equals to 2. Using Pym’s method, we have found that there are conditions on the irreducible decomposition of the degeneracy divisor and applying Polishchuk’s study Polishchuk (J Math Sci 84(5):1413–1444, 1997), we concretely described the Poisson structures corresponding to each classification result.

Asymptotic trace formula for the Hecke operators

Given integers m, n and k, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the mth and nth Fourier coefficients of an orthonormal basis of $$S_k\left( N\right) ^*$$ (the weight k newforms with fixed square-free level N) provided that $$|4 \pi \sqrt{mn}- k|=o\left( k^{\frac{1}{3}}\right) $$ . Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator $$\mathcal {T}_n^*$$ on $$S_k\left( N\right) ^*$$ averaged over k in a short interval. By bounding the second moment of the trace of $$\mathcal {T}_{n}$$ over a larger interval, we show that the trace of $$\mathcal {T}_n$$ is unusually large in the range $$|4 \pi \sqrt{n}- k| = o\left( n^{\frac{1}{6}}\right) $$ . As an application, for any fixed prime p coprime to N, we show that there exists a sequence $$\{k_n\}$$ of weights such that the error term of Weyl’s law for $$\mathcal {T}_p$$ is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd et al. (J Eur Math Soc 1(1):51–85, 1999) [Theorem 1.4] with an improved exponent.

Meromorphic Functions Sharing four Values with their Difference Operators

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and $$f(z+\eta )$$ share a, b, c IM, where $$\eta $$ is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then $$f(z)=f(z+\eta )$$ for all $$z\in \mathbb {C}$$ . We prove that if a non-constant finite order entire function f and its n-th difference operator $$\Delta ^n_{\eta }f$$ share $$a_1$$ , $$a_2$$ , $$a_3$$ IM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, while $$a_1$$ , $$a_2$$ , $$a_3$$ are three distinct finite complex values, then $$f=\Delta ^n_{\eta }f$$ . The main results in this paper also improve Theorems 1.1 and 1.2 from Li and Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).

Some remarks on Hermitian manifolds satisfying Kähler-like conditions

We study Hermitian metrics whose Bismut connection $$\nabla ^B$$ satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition $$R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$$ , for every tangent vectors x, y, z, w, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in Angella et al. (Commun Anal Geom, to appear, 2018), Yau et al. (2019) and Zhao and Zheng (2019). Using the characterization of SKT almost abelian Lie groups in Arroyo and Lafuente (Proc Lond Math Soc (3) 119:266–289, 2019), we construct new examples of Hermitian manifolds satisfying the Bismut Kähler-like condition. Moreover, we prove some results in relation to the pluriclosed flow on complex surfaces and on almost abelian Lie groups. In particular, we show that, if the initial metric has constant scalar curvature, then the pluriclosed flow preserves the Vaisman condition on complex surfaces.

Prophet secretary through blind strategies

In the classic prophet inequality, a well-known problem in optimal stopping theory, samples from independent random variables (possibly differently distributed) arrive online. A gambler who knows the distributions, but cannot see the future, must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet that sees all the realizations in advance gets. In the late seventies, Krengel and Sucheston (Bull Am Math Soc 83(4):745–747, 1977), established that this worst case ratio is 0.5. A particularly interesting variant is the so-called prophet secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms are known to achieve a constant of $$1-1/e \approx 0.632$$ and very recently this barrier was slightly improved by Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). In this paper we introduce a new type of multi-threshold strategy, called blind strategy. Such a strategy sets a nonincreasing sequence of thresholds that depends only on the distribution of the maximum of the random variables, and the gambler stops the first time a sample surpasses the threshold of the stage. Our main result shows that these strategies can achieve a constant of 0.669 for the prophet secretary problem, improving upon the best known result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018), and even that of Beyhaghi et al. (Improved approximations for posted price and second price mechanisms. CoRR arXiv:1807.03435 , 2018) that works in the case in which the gambler can select the order of the samples. The crux of the result is a very precise analysis of the underlying stopping time distribution for the gambler’s strategy that is inspired by the theory of Schur-convex functions. We further prove that our family of blind strategies cannot lead to a constant better than 0.675. Finally we prove that no algorithm for the gambler can achieve a constant better than $$\sqrt{3}-1 \approx 0.732$$ , which also improves upon a recent result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). This implies that the upper bound on what the gambler can get in the prophet secretary problem is strictly lower than what she can get in the i.i.d. case. This constitutes the first separation between the prophet secretary problem and the i.i.d. prophet inequality.

Rings in which every 2-absorbing primary ideal is primary

Let R be a commutative ring with a nonzero identity. Badawi et al. (Bull Korean Math Soc 51(4):1163–1173, 2014) defined the concept of 2-absorbing primary ideal as follows: a proper ideal P of R is said to be a 2-absorbing primary ideal if whenever $$xyz\in P$$ for some $$x,y,z\in R,$$ then either $$xy\in P$$ or $$xz\in \sqrt{P}$$ or $$yz\in \sqrt{P}.$$ It is clear that every primary ideal is also a 2-absorbing primary ideal. The author in this paper is to study rings in which every 2-absorbing primary ideal is primary. A ring R is said to be $$2-ABP$$ -ring if every 2-absorbing primary ideal is primary.

Differentiability of distance function and the proximinal condition implying convexity

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles (Proc Am Math Soc 104: 458–464, 1988). Hence, our result advances that of Giles (Proc Am Math Soc 104: 458–464, 1988). We prove that the proximinal condition of Giles (Proc Am Math Soc 104: 458–464, 1988) is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A necessary and sufficient condition is obtained for the convexity of Chebyshev sets in Banach spaces with rotund dual.