On the evolution of continued fractions in a fixed quadratic field

We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the ‘normal’ statistics given by the Gauss-Kuzmin measure. As a byproduct, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer k and a quadratic irrational α, the length of the period of the continued fraction expansion of k ⁿ α equals ck ⁿ + o(k^15n/16) for some positive constant c. This improves results of Cohn, Lagarias, and Grisel, and settles a conjecture of Hickerson. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on S-arithmetic homogeneous spaces.     

On vanishing coefficients of algebraic power series over fields of positive characteristic

Let K be a field of characteristic p>0 and let f(t ₁,…,t _d ) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t ₁,…,t _d ). We prove a generalization of both Derksen’s recent analogue of the Skolem–Mahler–Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices (n ₁,…,n _d )∈? ^d for which the coefficient of \(t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\) in f(t ₁,…,t _d ) is zero is a p-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to S-unit equations and more generally to the Mordell–Lang Theorem over fields of positive characteristic.     

Planar Posets,Dimension, Breadth and the Number of Minimal Elements

In recent years, researchers have shown renewed interest in combinatorial properties of posets determined by geometric properties of its order diagram and topological properties of its cover graph. In most cases, the roots for the problems being studied today can be traced back to the 1970’s, and sometimes even earlier. In this paper, we study the problem of bounding the dimension of a planar poset in terms of the number of minimal elements, where the starting point is the 1977 theorem of Trotter and Moore asserting that the dimension of a planar poset with a single minimal element is at most 3. By carefully analyzing and then refining the details of this argument, we are able to show that the dimension of a planar poset with t minimal elements is at most 2t + 1. This bound is tight for t = 1 and t = 2. But for t ≥ 3, we are only able to show that there exist planar posets with t minimal elements having dimension t + 3. Our lower bound construction can be modified in ways that have immediate connections to the following challenging conjecture: For every d ≥ 2, there is an integer f(d) so that if P is a planar poset with dim(P) ≥ f(d), then P contains a standard example of dimension d. To date, the best known examples only showed that the function f, if it exists, satisfies f(d) ≥ d + 2. Here, we show that lim _d→∞ f(d)/d ≥ 2.     

On the divergence of Fourier series in the spaces ϕ(<Emphasis Type="Italic">L</Emphasis>) containing <Emphasis Type="Italic">L</Emphasis>

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[?π, π], but narrower than L^p = L^p[?π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ?(L) of Kolmogorov’s theorem on the convergence of Fourier series in L^p, p ∈ (0, 1), cannot be improved.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Complexity of the satisfiability problem for multilinear forms over a finite field

Multilinear forms over finite fields are considered. Multilinear forms over a field are products in which each factor is the sum of variables or elements of this field. Each multilinear form defines a function over this field. A multilinear form is called satisfiable if it represents a nonzero function. We show the N P-completeness of the satisfiability recognition problem for multilinear forms over each finite field of q elements for q ≥ 3. A theorem is proved that distinguishes cases of polynomiality and NP-completeness of the satisfiability recognition problem for multilinear fields for each possible q ≥ 3.     

Positive existential definability of multiplication from addition and the range of a polynomial

We consider the problem of recovering multiplication in the integers from enrichments of its additive structure, in the positive existential context. We prove that if a conjecture by Caporaso–Harris–Mazur holds, then for all integer-valued polynomials F of degree at least 2, multiplication is positive-existentially definable in (Z; 0, 1,+, R_F, =) where R_F is the unary relation F(Z). Similar results were only known for the polynomials F(t) = t² (under the Bombieri–Lang conjecture) and F(t) = tⁿ (under a generalization of the abc conjecture).     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Maximal Quadratic Modules on ∗-rings

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to ?-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime ?-ideal, that every maximal proper quadratic module in a Noetherian ?-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schmüdgen’s Strict Positivstellensatz for the Weyl algebra: Let c be an element of the Weyl algebra \(\mathcal{W}(d)\) which is not negative semidefinite in the Schrödinger representation. It is shown that under some conditions there exists an integer k and elements \(r_1,\ldots,r_k \in \mathcal{W}(d)\) such that ∑ _j=1 ^k r _j c r _j ^? is a finite sum of hermitian squares. This result is not a proper generalization however because we don’t have the bound k ≤d.     

On the periodicity of continued fractions in elliptic fields

Article [1] raised the question of the finiteness of the number of square-free polynomials f ∈ ?[h] of fixed degree for which \(\sqrt f \) has periodic continued fraction expansion in the field ?((h)) and the fields ?(h)(\(\sqrt f \)) are not isomorphic to one another and to fields of the form ?(h)\(\left( {\sqrt {c{h^n} + 1} } \right)\), where c ∈ ?* and n ∈ ?. In this paper, we give a positive answer to this question for an elliptic field ?(h)(\(\sqrt f \)) in the case deg f = 3.     

Remarks on the stochastic integral

In Karandikar-Rao [11], the quadratic variation [M, M] of a (local) martingale was obtained directly using only Doob’s maximal inequality and it was remarked that the stochastic integral can be defined using [M, M], avoiding using the predictable quadratic variation 〈M, M〉 (of a locally square integrable martingale) as is usually done. This is accomplished here- starting with the result proved in [11], we construct ∫ f dX where X is a semimartingale and f is predictable and prove dominated convergence theorem (DCT) for the stochastic integral. Indeed, we characterize the class of integrands f for this integral as the class L(X) of predictable processes f such that |f| serves as the dominating function in the DCT for the stochastic integral. This observation seems to be new.We then discuss the vector stochastic integral ∫ 〈f, dY〉 where f is ? ^d valued predictable process, Y is ? ^d valued semimartingale. This was defined by Jacod [6] starting from vector valued simple functions. Memin [13] proved that for (local) martingales M¹, … M ^d : If N ⁿ are martingales such that N _t ⁿ → N _t for every t and if ?f ⁿ such that N _t ⁿ = ∫ 〈f ⁿ , dM〉, then ?f such that N = ∫ 〈f, dM〉.Taking a cue from our characterization of L(X), we define the vector integral in terms of the scalar integral and then give a direct proof of the result due to Memin stated above.This completeness result is an important step in the proof of the Jacod-Yor [4] result on martingale representation property and uniqueness of equivalent martingale measure. This result is also known as the second fundamental theorem of asset pricing.     

Strong factorization property of Macdonald polynomials and higher-order Macdonald’s positivity conjecture

We prove a strong factorization property of interpolation Macdonald polynomials when q tends to 1. As a consequence, we show that Macdonald polynomials have a strong factorization property when q tends to 1, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate q, t-Kostka numbers and we show that they are polynomials in q, t with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate q, t-Kostka numbers are in fact polynomials in q, t with nonnegative integer coefficients, which generalizes the celebrated Macdonald’s positivity conjecture.     

An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions

We obtain an integro-local limit theorem for the sum S(n) = ξ(1)+?+ξ(n) of independent identically distributed random variables with distribution whose right tail varies regularly; i.e., it has the form P(ξ≥t) = t ^?β L(t) with β > 2 and some slowly varying function L(t). The theorem describes the asymptotic behavior on the whole positive half-axis of the probabilities P(S(n) ∈ [x, x + Δ)) as x → ∞ for a fixed Δ > 0; i.e., in the domain where the normal approximation applies, in the domain where S(n) is approximated by the distribution of its maximum term, as well as at the “junction” of these two domains.     

On extensions of some block designs

There are some results concerning t-designs in which the number of points in the intersection of two blocks takes less than t values. For example, if t = 2, then the design is symmetric (in such a design, v = b or, equivalently, k = r). In 1974, B. Gross described t-(v, k, l) designs that, for some integer s, 0 < s < t, do not contain two blocks intersecting at exactly s points. Below, it is proved that potentially infinite series of designs from the claim of Gross’ theorem are finite. Gross’ theorem is substantially sharpened.     

Superlarge deviations for sums of random variables with arithmetical super-exponential distributions

Local limit theorems are obtained for superlarge deviations of sums S(n) = ξ(1) + ... + ξ(n) of independent identically distributed random variables having an arithmetical distribution with the right-hand tail decreasing faster that that of a Gaussian law. The distribution of ξ has the form ?(ξ = k) = \(e^{ - k^\beta L(k)} \), where β > 2, k ∈ ? (? is the set of all integers), and L(t) is a slowly varying function as t → ∞ which satisfies some regularity conditions. These theorems describing an asymptotic behavior of the probabilities ?(S(n) = k) as k/n → ∞, complement the results on superlarge deviations in [4, 5].     

Loewner matrices and operator convexity

Let f be a function from \({\mathbb{R}_{+}}\) into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form \({\left [\frac{f(p_i) - f(p_j)}{p_i-p_j}\right ]_{\vphantom {X_{X_1}}}}\) are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f (t) = t g(t) for some operator convex function g if and only if these matrices are conditionally positive definite. Elementary proofs are given for the most interesting special cases f (t) = t ^r , and f (t) = t log t. Several consequences are derived.     

A tournament approach to pattern avoiding matrices

We consider the following Turán-type problem: given a fixed tournament H, what is the least integer t = t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t = t(T _n ,H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several results on these problems, our main contributions are the following:

• Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n × n matrix with n(log n) ^O(1) entries equal to 1 contains the pattern M. We show that this conjecture is equivalent to the assertion that t(T _n ,H) = n(log n) ^O(1) if and only if H belongs to a certain (natural) family of tournaments.
• We propose an approach for determining if t(n,H) = n(log n) ^O(1). This approach combines expansion in sparse graphs, together with certain structural characterizations of H-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach–Tardos conjecture.

     

On a result by geronimus

In 1935, Ya.L. Geronimus found the best integral approximation on the period [?π,π) of the function sin(n + 1)t ? 2q sin nt, q ∈ ?, by the subspace of trigonometric polynomials of degree at most n ? 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We propose one more variant of the proof. In the case |q| ≥ 1, we apply the (2π/n)-periodization and the fact that the function | sin nt| is orthogonal to the harmonic cos t on the period. In the case |q| < 1, we use the duality relations for Chebyshev’s theorem (1859) on a rational function least deviating from zero on a closed interval with respect to the uniform metric.     

On the 2-primary part of tame kernels of real quadratic fields

Let \(F = Q\left( {\sqrt p } \right)\), where p = 8t+1 is a prime. In this paper, we prove that a special case of Qin’s conjecture on the possible structure of the 2-primary part of K ₂ O _F up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K ₂ O _F , which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.