The domination game played on diameter 2 graphs

Aequationes mathematicae - Let $$\gamma _g(G)$$ be the game domination number of a graph G. It is proved that if $$\mathrm{diam}(G) = 2$$ , then $$\gamma _g(G) \le \left\lceil \frac{n(G)}{2}...     

Normal bundles of rational curves in projective space

Mathematische Zeitschrift - Let $$b_{\bullet }$$ be a sequence of integers $$1 < b_1 \le b_2 \le \cdots \le b_{n-1}$$ . Let $${\text {M}}_e(b_{\bullet })$$ be the space parameterizing...     

On Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

Journal of Fourier Analysis and Applications - Let f be an element of the subspace $$(L^{q},l^{p})^{\alpha }({\mathbb {R}}^d)$$ ( $$1\le q \le \alpha \le p \le 2 $$ ) of the Wiener amalgam space...     

On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

Mathematical Notes - Let $$f(x)$$ be a function belonging to the Lebesgue class $$L^p({\mathbb R}_+)$$ on the semiaxis $${\mathbb R}_+=[0,+\infty)$$ , $$1\le p\le 2$$ , and let $$\widehat{f}$$ be...     

Fractional Gaussian estimates and holomorphy of semigroups

Let $$\Omega \subset {\mathbb {R}}^N$$ be an arbitrary open set, $$0<s<1$$ and denote by $$(e^{-t(-\Delta )_{{{\mathbb {R}}}^N}^s})_{t\ge 0}$$ the semigroup on $$L^2({{\mathbb {R}}}^N)$$ generated by the fractional Laplace operator. In the first part of the paper, we show that if T is a self-adjoint semigroup on $$L^2(\Omega )$$ satisfying a fractional Gaussian estimate in the sense that $$|T(t)f|\le Me^{-bt(-\Delta )_{{{\mathbb {R}}}^N}^s}|f|$$, $$0\le t \le 1$$, $$f\in L^2(\Omega )$$, for some constants $$M\ge 1$$ and $$b\ge 0$$, then T defines a bounded holomorphic semigroup of angle $$\frac{\pi }{2}$$ that interpolates on $$L^p(\Omega )$$, $$1\le p<\infty $$. Using a duality argument, we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.     

On the complete bounds of $$L_p$$ L p -Schur multipliers

Archiv der Mathematik - We study the class $$\mathcal {M}_p$$ of Schur multipliers on the Schatten-von Neumann class $$\mathcal {S}_p$$ with $$1 \le p \le \infty $$ as well as the class of...     

On the structure of submanifolds in the hyperbolic space

Monatshefte für Mathematik - Let $$M^n$$ be a complete submanifold in the hyperbolic space $$\mathbb {H}^{n+m}$$ . We show the vanishing of the Betti numbers $$\beta _p(M)$$ , $$1 \le p \le...     

On weakening tightness to weak tightness

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property $$wt(X)\le t(X)$$. We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $$wt(X)=\aleph _0<t(X)$$ under $$2^{\aleph _0}=2^{\aleph _1}$$. In particular, this demonstrates the celebrated Balogh’s (Proc Am Math Soc 105(3):755–764, 1989) Theorem does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then $$|X|\le 2^{wt(X)\pi \chi (X)}$$. This refines a theorem of de la Vega (Topol Appl 153:2118–2123, 2006). In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhász and van Mill (Proc Am Math Soc 146(1):429–437, 2018). In this connection we also show $$w(X)\le 2^{wt(X)}$$ for a homogeneous compactum. Third, we show that if X is a $$T_1$$ space, $$wt(X)\le \kappa $$, X is $$\kappa ^+$$-compact, and $$\psi (\overline{D},X)\le 2^\kappa $$ for any $$D\subseteq X$$ satisfying $$|D|\le 2^\kappa $$, then (a) $$d(X)\le 2^\kappa $$ and (b) X has at most $$2^\kappa $$-many $$G_\kappa $$-points. This is a variation of another theorem of Balogh (Topol Proc 27:9–14, 2003). Finally, we show that if X is a regular space, $$\kappa =L(X)wt(X)$$, and $$\lambda $$ is a caliber of X satisfying $$\kappa <\lambda \le \left( 2^{\kappa }\right) ^+$$, then $$d(X)\le 2^{\kappa }$$. This extends of theorem of Arhangel$$'$$skiĭ (Topol Appl 104:13–26, 2000).     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

The depth of a reflexive polytope

Given arbitrary integers d and r with $$d \ge 4$$ and $$1 \le r \le d + 1$$ , a reflexive polytope $${\mathscr {P}}\subset {\mathbb R}^d$$ of dimension d with $$\mathrm{depth}\,K[{\mathscr {P}}] = r$$ for which its dual polytope $${\mathscr {P}}^\vee $$ is normal will be constructed, where $$K[{\mathscr {P}}]$$ is the toric ring of $${\mathscr {P}}$$ .     

Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight

Mathematical Notes - Let there be given a partition of the closed interval $$[-1,1]$$ by arbitrary nodes $$\{\eta_j\}_{j=0}^N$$ , where $$\lambda_N=\max_{0\le j \le N-1} (\eta_{j+1}-\eta_{j})$$ ....     

Characterization of symmetrically $$\Delta $$ Δ -normed operator ideals which are interpolation spaces between Schatten–von Neumann ideals

Positivity - Let $$\mathcal {E}$$ be a symmetrically $$\Delta $$ -normed ideal in B(H). For $$1\le p&lt;\infty ,$$ $$q\ge 1,$$ we give a necessary and sufficient condition for $$\mathcal {E}$$...     

Quadratic optimization with switching variables: the convex hull for $$n=2$$

Mathematical Programming - We consider quadratic optimization in variables (x,&nbsp;y) where $$0\le x\le y$$ , and $$y\in \{ 0,1 \}^n$$ . Such binary variables are commonly referred to as...     

Sums of the higher divisor function of diagonal homogeneous forms

The Ramanujan Journal - We give an asymptotic formula for $$\sum _{1\le n_1.n_2, \ldots ,n_l\le x^{1/r}}\tau _k(n^r_1+n^r_2+\ldots +n^r_l)$$ , where $$\tau _k(n)$$ represents the k-th divisor...     

On triviality of the reduced Whitehead group over Henselian fields

Let F be a Henselian field of q-cohomological dimension 3, where q is a prime. Let $$\Gamma _F$$ be the totally ordered Abelian value group of F and let D be a central division algebra over F of index a power of q such that the characteristic of the residue field $$\overline{F}$$ is coprime to q. We show that when $$1\le \dim _{{\mathbb {F}}_q}(\Gamma _F/q\Gamma _F)\le 3$$ , the reduced Whitehead group of D is trivial.     

On sums of biquadratic powers with almost equal primes

The Ramanujan Journal - For $$18\le s\le 20,$$ we prove that for any sufficiently large positive integer N satisfying local conditions $$\mathcal N_s$$ the equation $$ N=p_{1}^{4}+p_{2}^{4}+\cdots...     

Dynamically defined sequences with small discrepancy

Monatshefte für Mathematik - We study the problem of constructing sequences $$(x_n)_{n=1}^{\infty }$$ on [0,&nbsp;1] in such a way that $$\begin{aligned} D_N^* = \sup _{0 \le x \le 1}...     

On a generalization of Schur theorem concerning resultants

Periodica Mathematica Hungarica - Let K be a field and put $${\mathcal {A}}:=\{(i,j,k,m)\in \mathbb {N}^{4}:\;i\le j\;\text{ and }\;m\le k\}$$ . For any given $$A\in {\mathcal {A}}$$ we consider...     

Linear independence of certain numbers

Archiv der Mathematik - Let $$k\ge 2$$, $$b\ge 2$$ and $$1\le a_1&lt;\cdots     

Hyperinvariant Closed Ideals for a Finitely Quasinilpotent Collection of Operators

Mathematical Notes - In this paper, it is proved that if $$\mathscr C\ne\{0\}$$ is a collection of continuous operators with modulus on an $$\ell_p$$ -space ( $$1\le p&lt;\infty$$ ) that is...