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.     

A note on the Schrödinger maximal function

It is shown that control of the Schrödinger maximal function sup_{0 <t<1} ?e^itΔf? for f ∈ H^s(Rⁿ) requires s ≥ n/2(n + 1).     

Approximation by Linear Fractional Transformations of Simple Partial Fractions and Their Differences

Let f be a function and ρ be a simple partial fraction of degree at most n. Under linear-fractional transformations, the difference f ? ρ becomes the difference of another function and a certain simple partial fraction of degree at most n with a quadratic weight. We study applications of this important property. We prove a theorem on uniqueness of interpolating simple partial fraction, generalizing known results, and obtain estimates for the best uniform approximation of certain functions on the real semi-axis ?⁺. For continuous functions of rather common type we first obtain estimates of the best approximation by differences of simple partial fractions on ?⁺. For odd functions we obtain such estimates on the whole axis ?.     

Ergodic theorems with arithmetical weights

We prove that the divisor function d(n) counting the number of divisors of the integer n is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system (X, A, ν, τ) and any f ∈ L ^p (ν), p > 1, the limit

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Sigma _{k = 1}^nd\left( k \right)}}\sum\limits_{k = 1}^n {d\left( k \right)f\left( {{\tau ^k}x} \right)} $$

exists ν-almost everywhere. The proof is based on Bourgain’s method, namely the circle method based on the shift model. Using more elementary ideas we also obtain similar results for other arithmetical functions, like the θ(n) function counting the number of squarefree divisors of n and the generalized Euler totient function J _s (n) = Σ _d|n d ^s μ(n/d), s > 0.

     

Boundedness in a chemotaxis system with nonlinear signal production

This work deals with the zero-Neumann boundary problem to a fully parabolic chemotaxis system with a nonlinear signal production function f(s) fulfilling 0 ≤ f(s) ≤ Ks~α for all s ≥ 0, where K and α are positive parameters. It is shown that whenever 0 α 2/n(where n denotes the spatial dimension) and under suitable assumptions on the initial data,this problem admits a unique global classical solution that is uniformly-in-time bounded in any spatial dimension. The proof is based on some a priori estimate techniques.     

The degree of biholomorphic mappings between special domains in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> preserving 0

Let G _i be a closed Lie subgroup of U(n), Ω _i be a bounded G _i -invariant domain in C ⁿ which contains 0, and \(O{\left( {{\mathbb{C}^n}} \right)^{{G_i}}} = \mathbb{C}\), for i = 1; 2. If f: Ω₁ → Ω₂ is a biholomorphism, and f(0) = 0, then f is a polynomial mapping (see Ning et al. (2017)). In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan’s theorem.     

Isometric embeddings of finite metric spaces

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m, n ∈ N there exists a Cantor set in R^m that isometrically contains all finite metric spaces which can be embedded into R^m, contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on R^m and, in particular, for the Euclidean metric.     

On the sum of the L1 influences of bounded functions

Let f: {-1, 1}ⁿ → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L₁ influence of f can also be bounded as a function of d. Ba?kurs and Bavarian answered this question in the affirmative, providing a bound of O(d³) for general functions and O(d²) for homogeneous functions. We improve on their results by providing a bound of d² for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.     

Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cⁿ, λ ∈ C, in a domain Ω ? Cⁿ which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.     

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.     

One-element differential standard bases with respect to inverse lexicographical orderings

We give a simplified proof of the following fact: for all nonnegative integers n and d the monomial y _n ^d forms a differential standard basis of the ideal [y _n ^d ]. In contrast to Levi’s combinatorial proof, in this proof we use the Gröbner bases technique. Under some assumptions we prove the converse result: if an isobaric polynomial f forms a differential standard basis of [f], then f = y _n ^d .     

Height reducing property of polynomials and self-affine tiles

A monic polynomial \({f(x)\in {\mathbb Z}[x]}\) is said to have the height reducing property (HRP) if there exists a polynomial \({h(x)\in {\mathbb Z}[x]}\) such that

$f(x)h(x)=a_n x^n+a_{n-1}x^{n-1}+\cdots+a_1x\pm q,$

where q = f(0), |a _i | ≤ (|q| ?1), i = 1, . . . , n and a _n > 0. We show that any expanding monic polynomial f(x) has the height reducing property, improving a previous result in Kirat et al. (Discrete Comput Geom 31: 275–286, 2004) for the irreducible case. The proof relies on some techniques developed in the study of self-affine tiles. It is constructive and we formulate a simple tree structure to check for any monic polynomial f(x) to have the HRP and to find h(x). The property is used to study the connectedness of a class of self-affine tiles.

     

On collineation groups of finite projective spaces containing a Singer cycle

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, q^s) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, q^s) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, q^s) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).     

Hirzebruch Functional Equation: Classification of Solutions

The Hirzebruch functional equation is \(\sum\nolimits_{i = 1}^n {\prod\nolimits_{j \ne i} {(1/f({z_j} - {z_i})) = c} } \) with constant c and initial conditions f(0) = 0 and f'(0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n ≤ 6 in the class of meromorphic functions and in the class of series. Previously, such results have been known only for n ≤ 4. The Todd function is the function determining the two-parameter Todd genus (i.e., the χ_a,b-genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N. It gives a solution to the Hirzebruch functional equation for n divisible by N. A series corresponding to a meromorphic function f with parameters in U ? ?^k is a series with parameters in the Zariski closure of U in ?^k, such that for the parameters in U it coincides with the series expansion at zero of f. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for n = 5 corresponds either to the Todd function or to the elliptic function of level 5. (2) Any series solution of the Hirzebruch functional equation for n = 6 corresponds either to the Todd function or to the elliptic function of level 2, 3, or 6. This gives a complete classification of complex genera that are fiber multiplicative with respect to ?P^n?1 for n ≤ 6. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level N for N = 2,..., 6 in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in ?⁴.     

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).     

Very Hyperbolic Polynomials

A real polynomial in one variable is hyperbolic if it has only real roots. A function f is a primitive of order k of a function g if f ^(k) = g. A hyperbolic polynomial is very hyperbolic if it has hyperbolic primitives of all orders. In the paper, we prove a property of the domain of very hyperbolic polynomials and describe this domain in the case of degree 4.     

Holomorphic Minorants of Plurisubharmonic Functions

Let φ be a plurisubharmonic function on a pseudoconvex domain D in an n-dimensional complex space. We show that there exists a nonzero holomorphic function f on D such that some local mean value of φ with logarithmic additional terms majorizes log|f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures.     

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

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.     

Multivariate discriminant and iterated resultant

In this paper,we study the relationship between iterated resultant and multivariate discriminant.We show that,for generic form f(x_n) with even degree d,if the polynomial is squarefreed after each iteration,the multivariate discriminant △(f) is a factor of the squarefreed iterated resultant.In fact,we find a factor Hp(f,[x_1,...,x_n]) of the squarefreed iterated resultant,and prove that the multivariate discriminant △(f) is a factor of Hp(f,[x_1,...,x_n]).Moreover,we conjecture that Hp(f,[x_1,...,x_n]) = △(f) holds for generic form/,and show that it is true for generic trivariate form f(x,y,z).