On an Extremal Problem for Poset Dimension

Let f(n) be the largest integer such that every poset on n elements has a 2-dimensional subposet on f(n) elements. What is the asymptotics of f(n)? It is easy to see that f(n) = n ^1/2. We improve the best known upper bound and show f(n) = O (n ^2/3). For higher dimensions, we show \(f_{d}(n)=\O \left (n^{\frac {d}{d + 1}}\right )\), where f _d (n) is the largest integer such that every poset on n elements has a d-dimensional subposet on f _d (n) elements.     

A discrete version of Koldobsky’s slicing inequality

Let #K be a number of integer lattice points contained in a set K. In this paper we prove that for each d ∈ N there exists a constant C(d) depending on d only, such that for any origin-symmetric convex body K ? R ^d containing d linearly independent lattice points

$$\# K \leqslant C\left( d \right)\max \left( {\# \left( {K \cap H} \right)} \right)vo{l_d}{\left( K \right)^{\frac{{d - m}}{d}}},$$

where the maximum is taken over all m-dimensional subspaces of R ^d . We also prove that C(d) can be chosen asymptotically of order O(1) ^d d ^d?m . In particular, we have order O(1) ^d for hyperplane slices. Additionally, we show that if K is an unconditional convex body then C(d) can be chosen asymptotically of order O(d) ^d?m .

     

Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation

We consider the Cauchy problem for the nonlinear differential equation

$$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$

where ? > 0 is a small parameter, f(x, u) ∈ C ^∞ ([0, d] × ?), R ₀ > 0, and the following conditions are satisfied: f(x, u) = x ? u ^p + O(x ² + |xu| + |u|^p+1) as x, u → 0, where p ∈ ? \ {1} f(x, 0) > 0 for x > 0; f _u ²(x, u) < 0 for (x, u) ∈ [0, d] × (0, + ∞); Σ ₀ ^+∞ f _u ²(x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval [0, d].

     

Composition limits and separating examples for some boolean function complexity measures

Block sensitivity (bs(f)), certificate complexity (C(f)) and fractional certificate complexity (C*(f)) are three fundamental combinatorial measures of complexity of a boolean function f. It has long been known that bs(f) ≤ C*(f) ≤ C(f) = O(bs(f)²). We provide an infinite family of examples for which C(f) grows quadratically in C*(f) (and also bs(f)) giving optimal separations between these measures. Previously the biggest separation known was \(C(f) = C*(f)^{\log _{4,5} 5}\). We also give a family of examples for which C*(f)= Ω (bs(f)^3/2).These examples are obtained by composing boolean functions in various ways. Here the composition fog of f with g is obtained by substituting for each variable of f a copy of g on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure s(f). The measures s(f), C(f) and C*(f) behave nicely under composition: they are submultiplicative (where measure m is submultiplicative if m(fog) ≤ m(f)m(g)) with equality holding under some fairly general conditions. The measure bs(f) is qualitatively different: it is not submultiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure m at function f, m ^lim(f) to be the limit as k grows of m(f ^(k))^1/k , where f ^(k) is the iterated composition of f with itself k-times. For any function f we show that bs ^lim(f) = (C*)^lim(f) and characterize s ^lim(f); (C*)^lim(f), and C ^lim(f) in terms of the largest eigenvalue of a certain set of 2×2 matrices associated with f.     

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 some arithmetic properties of Mahler functions

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer ? ≥ 2 such that f(x), f(x^?),..., \(f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})\) are linearly dependent over ?(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(x^?) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.     

New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n ? m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) ? w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).     

On the positive definiteness of some functions related to the Schoenberg problem

For a broad class of functions f: [0,+∞) → ?, we prove that the function f(ρ ^λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ? 0 and α(E, ρ) is the Schoenberg constant.     

Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

Holomorphic injective extensions of functions in <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) and algebra generators

We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta K ? C and f ∈ P(K) with the property that f ∈ P(K) is a homeomorphism of K onto its image, but for which f ^?1 ? P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|_G of the Gelfand-transform f* of an injective function f ∈ P(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.     

Quantum j-invariant in positive characteristic II: formulas and values at the quadratics

In this sequel to Demangos and Gendron (Arch Math 107:23–35, 2016), the multi-valued quantum j-invariant in positive characteristic is studied at quadratic elements. For every quadratic f, an explicit expression for each of the values of j^qt(f) is given as a limit of rational functions of f. It is proved that the number of values of j^qt(f) is finite.     

On Elliptic Extensions in the Disk

Given two arbitrary functions f ⁽⁰⁾, f ⁽¹⁾ on the boundary of the unit disk D in \({\mathbb R}^2\), it is shown that there exists a second order uniformly elliptic operator L and a function v in L ^p , with L ^p second derivatives (1?p?Lv?=?0 a.e. in D and with v?=?f ⁽⁰⁾ and \(\frac{ \partial v}{\partial n} = f^{(1)}\) on \(\partial{D}\). A similar extension property was proved in Cavazzoni (2003) for any pair of functions f ⁽⁰⁾, f ⁽¹⁾ that are analytic; a result is obtained under weaker regularity assumptions, e.g. with \(\frac{\partial f^{(0)}}{\partial \theta}\) and f ⁽¹⁾ Hölder continuous with exponent \(\eta > \frac{1}{2}\).     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

The barycenter property for robust and generic diffeomorphisms

Let f:M~d→M~d(d≥2) be a diffeomorphism on a compact C~∞ manifold on M.If a diffeomorphism f belongs to the C~1-interior of the set of all diffeomorphisms having the barycenter property,then f is Ω-stable.Moreover,if a generic diffeomorphism f has the barycenter property,then f is Ω-stable.We also apply our results to volume preserving diffeomorphisms.     

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.     

Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Hölder functions on proper subintervals of ? are Lip? _α (Mf)≤(1+α)^?1Lip? _α (f), α∈(0,1]. On ?, the best bound for Lipschitz functions is \(\operatorname{Lip} ( Mf) \le (\sqrt{2} -1)\operatorname{Lip}( f)\). In higher dimensions, we determine the asymptotic behavior, as d→∞, of the norm of the maximal operator associated with cross-polytopes, Euclidean balls, and cubes, that is, ? _p balls for p=1,2,∞. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Hölder functions, the operator norm of the maximal operator is uniformly bounded by 2^?α/q , where q is the conjugate exponent of p=1,2, and as d→∞ the norms approach this bound. When p=∞, best constants are the same as when p=1.     

Uniform,on the entire axis,convergence of the spectral expansion for Schrödinger operator with a potential-distribution

A uniform, on ?, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ? by the differential operation (?d/dx)² + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W ₂ ^?1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ?.     

Some relations among multiplicative and <Emphasis Type="Italic">q</Emphasis>-additive functions

We give all solutions of the equation f(n) = g(n) + h(n) for every n ∈ ?, where f is a completely multiplicative, g is a 2-additive, and h is a 3-additive function. We also determine all completely multiplicative functions f and all q-additive functions g for which f(n) = g ²(n) for every n ∈ ?.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

Value sharing problems for differential and difference polynomials of meromorphic functions in a non-Archimedean field

In this paper we discuss the uniqueness problem for differential and difference polynomials of the form (f ^nm (z)f ^nd (qz + c))^(k) for meromorphic functions in a non-Archimedean field.