We prove a law of the iterated logarithm for sums of the form ${\sum }_{k=1}^N{a}_{k}f\left(n_{k}x\right)$ where the $n_k$ satisfy a Hadamard gap condition. Here we assume that f is a Dini continuous function on ℝⁿ which has the property that for every cube Q of sidelength 1 with corners in the lattice ℤⁿ, f vanishes on ∂Q and has mean value zero on Q.

For a fixed positive integer $k$ and $G=(V,E)$ a connected graph of order $n$, whose minimum vertex degree is at least $k$, a set $S\subseteq V$ is a total $k$-dominating set, also known as a $k$-tuple total dominating set, if every vertex $v\in V$ has at least $k$ neighbors in $S$. The minimum size of a total $k$-dominating set for $G$ is called the total $k$-domination number of $G$, denoted by ${\gamma }_{kt}\left(G\right)$. The total $k$-domination problem is to determine a minimum total $k$-dominating set of $G$. Since the exact problem is in general quite difficult...

Let $S_n^{\left(2\right)}$ denote the iterated partial sums. That is, $S_n^{\left(2\right)}=S_1+S_2+\cdots +S_n$, where $S_i=X_1+X_2+\cdots +X_i$. Assuming $X_1,X_2,...,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}lt;0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$ with $c\le 6\sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{\left(2\right)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0lt;\gamma lt;1/4$ there exist integrable,...

Let $k\ge 2$ and let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence defined by $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}$$ for $n\ge 2$ with initial conditions $$P_{-(k-2)}^{\left(k\right)}=P_{-(k-3)}^{\left(k\right)}=\cdots =P_{-1}^{\left(k\right)}=P_0^{\left(k\right)}=0,P_1^{\left(k\right)}=1.$$ In this study, we handle the equation $P_n^{\left(k\right)}=y^m$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\ge 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_n^{\left(k\right)}=y^m$ with $2\le y\le 1000$ has only one solution given by $P_7^{\left(2\right)}={13}^2.$

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.

The sequence of balancing numbers $\left(B_n\right)$ is defined by the recurrence relation $B_n=6B_{n-1}-B_{n-2}$ for $n\ge 2$ with initial conditions $B_0=0$ and $B_1=1.$ $B_n$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_n$ is repdigit in the base $b$ and has at least two digits, then $(n,b)=(2,5),(3,6)$. Namely, $B_2=6={\left(11\right)}_5$ and $B_3=35={\left(55\right)}_6.$

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_{\epsilon }\left(v\right)=\epsilon {\int }_{\Omega }|{\nabla }^2{v|}^{2}dx+{\epsilon }^{-1}{\int }_{\Omega }{\left(1-\right|\nabla v|}^2{)}^{2}dx$ over $v\in H^2\left(\Omega \right)$, where $\epsilon >0$ is a small parameter. Given $v\in W^{1,\infty }\left(\Omega \right)$ such that $\nabla v\in BV$ and $\left|\nabla v\right|=1$ a.e., we construct a family $\left\{v_{\epsilon }\right\}$ satisfying: $v_{\epsilon }\to v$ in $W^{1,p}\left(\Omega \right)$ and $E_{\epsilon }\left(v_{\epsilon }\right)\to \frac{1}{3}{\int }_{J_{\nabla v}}{\left|{\nabla }^{+}v-{\nabla }^{-}v\right|}^{3}d{\mathscr{H}}^{N-1}$ as $\epsilon$ goes to 0.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

For integers $m>r\ge 0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d,h)={\left(d_{n,k}\right)}_{n,k\in \mathbb{N}}$ as $d_{mn+r,(m-1)n+r}$, with $n=0,1,2,\cdots$, and the $(m,r)$-central coefficient triangle of $G$ as $$G^{(m,r)}={\left(d_{mn+r,(m-1)n+k+r}\right)}_{n,k\in \mathbb{N}}.$$ It is known that the $(m,r)$-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G=(d,h)$ with $h\left(0\right)=0$ and $d\left(0\right),h^{\text{'}}\left(0\right)\ne 0$, we obtain the generating function of its $(m,r)$-central coefficients and give an explicit representation for the $(m,r)$-central Riordan array $G^{(m,r)}$ in terms of the Riordan array $G$....

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

Given a Tychonoff space $X$ and an infinite cardinal $\kappa$, we prove that exponential $\kappa$-domination in $X$ is equivalent to exponential $\kappa$-cofinality of $C_p\left(X\right)$. On the other hand, exponential $\kappa$-cofinality of $X$ is equivalent to exponential $\kappa$-domination in $C_p\left(X\right)$. We show that every exponentially $\kappa$-cofinal space $X$ has a ${\kappa }^{+}$-small diagonal; besides, if $X$ is $\kappa$-stable, then $nw\left(X\right)\le \kappa$. In particular, any compact exponentially $\kappa$-cofinal space has weight not exceeding $\kappa$. We also establish that any exponentially $\kappa$-cofinal...

Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_1,P_2,P_3)$ with the property ${\overline{\varphi }}_3=1$. In fact, our method generalizes to provide $n$-systems with ${\overline{\varphi }}_n=1$, for arbitrary $n\ge 3$. We visualize our constructions with graphics. We further present explicit examples of numbers ${\xi }_1,...,{\xi }_{n-1}$ that induce the $n$-systems in question.

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

We prove that for a normed linear space $X$, if $f_1:X\to ℝ$ is continuous and semiconvex with modulus $\omega$, $f_2:X\to ℝ$ is continuous and semiconcave with modulus $\omega$ and $f_1\le f_2$, then there exists $f\in C^{1,\omega }\left(X\right)$ such that $f_1\le f\le f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega \left(t\right)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C_{loc}^{1,\omega }$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

For $X,Y\in {𝐌}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X{\prec }_{\mathrm{gut}}Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation ${\sim }_{\mathrm{gut}}$ as follows. $X{\sim }_{\mathrm{gut}}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of ${\prec }_{\mathrm{gut}}$ on ${ℝ}^n$ and strong linear preservers of this relation on ${𝐌}_{n,m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ${\sim }_{\mathrm{gut}}$ on ${ℝ}^n$ and ${𝐌}_{n,m}$.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.