An analog of Rudin’s theorem for continuous radial positive definite functions of several variables

Let Open image in new window be the class of radial real-valued functions of m variables with support in the unit ball \(\mathbb{B}\) of the space ? ^m that are continuous on the whole space ? ^m and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class Open image in new window can be presented as the sum \(\sum {f_k \tilde *f_k } \) of at most countably many self-convolutions of real-valued functions f _k with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions f _k are complex-valued.     

On Taylor’s formula for functions of several variables

Elementary courses in mathematical analysis often mention some trick that is used to construct the remainder of Taylor’s formula in integral form. The trick is based on the fact that, differentiating the difference $f(x) - f(t) - f'(t)\frac{{(x - t)}} {{1!}} - \cdots - f^{(r - 1)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ between the function and its degree r ? 1 Taylor polynomial at t with respect to t, we obtain $ - f^{(r)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ , so that all derivatives of orders below r disappear. The author observed previously a similar effect for functions of several variables. Differentiating the difference between the function and its degree r ? 1 Taylor polynomial at t with respect to its components, we are left with terms involving only order r derivatives. We apply this fact here to estimate the remainder of Taylor’s formula for functions of several variables along a rectifiable curve.     

A new bound for an extension of Mason’s theorem for functions of several variables

In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Masons theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) where , such that k out of the n-polynomials are constant, and under certain conditions for has no non-constant solution. Received: 20 March 2003     

On a Turán type problem of Erdös

Let L ^k be the graph formed by the lowest three levels of the Boolean lattice B _k , i.e.,V(L ^k )={0, 1,...,k, 12, 13,..., (k–1)k} and 0is connected toi for all 1ik, andij is connected toi andj (1i<jk).It is proved that if a graph G overn vertices has at leastk ^3/2 n ^3/2 edges, then it contains a copy of L ^k .Research supported in part by the Hungarian National Science Foundation under Grant No. 1812     

Generalizations of Young’s Theorem to real functions of several variables

In 1907 W. H. Young classified the real-valued Baire one functions on the line which have the Darboux (intermediate-value) property as those which are bilaterally approachable. Here we investigate generalizations of this theorem to the setting of real-valued Baire one functions of several variables which possess various “Darboux-like” properties.      

A version of Brauer’s theorem for integer central functions

In this article we prove an effective version of the classical Brauer’s Theorem for integer class functions on finite groups.      

Notes on a problem of Turán

Let z1,z2, ... ,znbe complex numbers, and write S= z ^j _{1 + ... +} z ^j _n for their power sums. Let R _n= min_{z 1,z2,...,zn} max_1&le;j&le;n &verbar;Sj&verbar; where the minimum is taken under the condition that max_1&le;t&le;n &verbar;z_t&verbar; = 1 Improving a result of Komlós, Sárközy and Szemerédi (see [KSSz]) we prove here that Rn &lt;1 -(1 - ") log log n /log n We also discuss a related extremal problem which occurred naturally in our earlier proof ([B1]) of the fact that Rn &gt;&half;     

Turán type inequalities for Krätzel functions

Complete monotonicity, Laguerre and Turán type inequalities are established for the so-called Krätzel function $Z_{\rho }^{\nu }$, defined by$Z_{\rho }^{\nu }(u)=\underset{0}{\overset{\infty }{\int }}{t}^{\nu ?1}{e}^{?t^{\rho }?\frac{u}{t}}\mathrm{d}t,$ where $u>0$ and $\rho ,\nu \in \mathbb{R}$. Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Krätzel function.     

Exact solution of the hypergraph Turán problem for k-uniform linear paths

A k-uniform linear path of length ?, denoted by ? _? ^(k) , is a family of k-sets {F ₁,...,F _? such that |F _i ∩ F _i+1|=1 for each i and F _i ∩ F _bj = \(\not 0\) whenever |i?j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex _k (n, H), is the maximum number of edges in a k-uniform hypergraph \(\mathcal{F}\) on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex _k (n, P _? ^(k) exactly for all fixed ? ≥1, k≥4, and sufficiently large n. We show that $ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$ . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that $ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$ , and describe the unique extremal family. Stability results on these bounds and some related results are also established.     

Turán’s inequality for the Kummer function of the phase shift of two parameters

Direct and inverse Turán’s inequalities are proved for the confluent hypergeometric function (the Kummer function) viewed as a function of the phase shift of the upper and lower parameters. The inverse Turán inequality is derived from a stronger result on the log-convexity of a function of sufficiently general form, a particular case of which is the Kummer function. Two conjectures on the log-concavity of the Kummer function are formulated. The paper continues the previous research on the log-convexity and log-concavity of hypergeometric functions of parameters conducted by a number of authors. Bibliography: 18 titles.     

On Waring’s problem for several algebraic forms

We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a grove, which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict.     

A hybrid inequality of Erdös–Turán–Koksma for digital sequences

For bases $\mathbf{b}=(b_1, \ldots , b_s)$ of $s$ not necessarily distinct integers $b_i\ge 2$ , we prove a version of the inequality of Erdös–Turán–Koksma for the hybrid function system composed of the Walsh functions in base $\mathbf{b}^{(1)}=(b_1, \ldots , b_{s_1})$ and, as second component, the $\mathbf{b}^{(2)}$ -adic functions, $\mathbf{b}^{(2)}=(b_{s_1+1}, \ldots , b_s)$ , with $s=s_1+s_2$ , $s_1$ and $s_2$ not both equal to 0. Further, we point out why this choice of a hybrid function system covers all possible cases of sequences that employ addition of digit vectors as their main construction principle.     

A Note on Turán Numbers for Even Wheels

The Turán number ex(n, G) is the maximum number of edges in any n-vertex graph that does not contain a subgraph isomorphic to G. We consider a very special case of the Simonovits’s theorem (Simonovits in: Theory of graphs, Academic Press, New York, 1968) which determine an asymptotic result for Turán numbers for graphs with some properties. In the paper we present a more precise result for even wheels. We provide the exact value for Turán number ex(n, W _2k ) for n ≥ 6k ? 10 and k ≥ 3. In addition, we show that ${ex(n,W_6)= \lfloor\frac{n^2}{3}\rfloor}$ for all n ≥ 6. These numbers can be useful to calculate some Ramsey numbers.     

Turán numbers for odd wheels

The Turán number $\text{ex}(n,G)$ is the maximum number of edges in any $n$-vertex graph that does not contain a subgraph isomorphic to $G$. A wheel$W_n$ is a graph on $n$ vertices obtained from a $C_{n?1}$ by adding one vertex $w$ and making $w$ adjacent to all vertices of the $C_{n?1}$. We obtain two exact values for small wheels:

$\text{ex}(n,W_5)=?\frac{n^2}{4}+\frac{n}{2}?,$

$\text{ex}(n,W_7)=?\frac{n^2}{4}+\frac{n}{2}+1?.$

Given that $\text{ex}(n,W_6)$ is already known, this paper completes the spectrum for all wheels up to 7 vertices. In addition, we present the construction which gives us the lower bound $\text{ex}(n,W_{2k+1})>?\frac{n^2}{4}?+?\frac{n}{2}?$ in general case.     

Handelman’s Positivstellensatz for polynomial matrices positive definite on polyhedra

In this paper we give a matrix version of Handelman’s Positivstellensatz (Handelman in Pac J Math 132:35–62, 1988), representing polynomial matrices which are positive definite on convex, compact polyhedra. Moreover, we propose also a procedure to find such a representation. As a corollary of Handelman’s theorem, we give a special case of Schmüdgen’s Positivstellensatz for polynomial matrices positive definite on convex, compact polyhedra.     

Hardy’s theorem and rotations of several complex variables

In this paper, Hardy’s theorem and rotations characterized by complex Gaussians in the complex plane due to Hogan and Lakey are extended to complex spaces of several variables. We point out that conditions under which a function on the n-dimensional real Euclidean space has an analytic extension to the complex space. Moreover, we prove that the function is a rotation of a multiple of real Gaussians through some angle if the extension satisfies certain assumptions.     

On some Turán-type inequalities for q-special functions

Intrinsic inequalities involving Turán-type inequalities for some q-special functions are established. A special interest is granted to q-Dunkl kernel. The results presented here would provide extensions of those given in the classical case.     

A Ramsey–Turán theory for tilings in graphs

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted ${\alpha }_r(G)$, is the largest size of a $K_r$-free set of vertices in $G$. In this article, we discuss Ramsey–Turán-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. Our results unify and generalise previous results of Balogh–Molla–Sharifzadeh [Random Struct. Algoritm. 49 (2016), no. 4, 669–693], Nenadov–Pehova [SIAM J. Discret. Math. 34 (2020), no. 2, 1001–1010] and Balogh–McDowell–Molla–Mycroft [Comb. Probab. Comput. 27 (2018), no. 4, 449–474] on the subject.     

Explicit evaluations of Turán expressions

Summary Positive representations for [P _n ^(λ) (x)]²−P _n ^−1/(λ) (x)P _n ^+1/(λ) (x) and for analogous expressions involving orthogonal polynomials are obtained. This is an excerpt from the author's doctoral dissertation, written under the direction of ProfessorW. Seidel, to whom the author is grateful for his encouragement and assistance.     

High-precision Gauss–Turán quadrature rules for Laguerre and Hermite weight functions

Procedures and corresponding Matlab software are presented for generating Gauss–Turán quadrature rules for the Laguerre and Hermite weight functions to arbitrarily high accuracy. The focus is on the solution of certain systems of nonlinear equations for implicitly defined recurrence coefficients. This is accomplished by the Newton–Kantorovich method, using initial approximations that are sufficiently accurate to be capable of producing n-point quadrature formulae for n as large as 42 in the case of the Laguerre weight function, and 90 in the case of the Hermite weight function.