In this paper we are concerned with a class of time-varying discounted Markov decision models ${ℳ}_n$ with unbounded costs $c_n$ and state-action dependent discount factors. Specifically we study controlled systems whose state process evolves according to the equation $x_{n+1}=G_n(x_n,a_n,{\xi }_n),n=0,1,...$, with state-action dependent discount factors of the form ${\alpha }_n(x_n,a_n)$, where $a_n$ and ${\xi }_n$ are the control and the random disturbance at time $n$, respectively. Assuming that the sequences of functions $\left\{{\alpha }_n\right\}$,$\left\{c_n\right\}$ and $\left\{G_n\right\}$ converge, in certain sense, to ${\alpha }_{\infty }$,...

We deal with the so-called Ahlfors regular sets (also known as $s$-regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: Under which conditions does the limit ${lim}_{\epsilon \to 0+}{\epsilon }^{s}N(\epsilon ,K)$ exist, where $K$ is an $s$-regular set and $N(\epsilon ,K)$ is for instance the $\epsilon$-packing number of $K$?

For a Fuchsian system $dY/dx=({\sum }_{j=}^p\left(A_j\right)/(x-t_j))Y$, (F) $t₁,t₂,...,t_p$ being distinct points in ℂ and $A₁,A₂,...,A_p\in M(n\times n;ℂ)$, the number α of accessory parameters is determined by the spectral types $s\left(A₀\right),s\left(A₁\right),...,s\left(A_p\right)$, where $A₀=-{\sum }_{j=1}^p{A}_j$. We call the set $z=(z₁,z₂,...,z_{\alpha })$ of α parameters a regular coordinate if all entries of the $A_j$ are rational functions in z. It is not yet known that, for any irreducibly realizable set of spectral types, a regular coordinate does exist. In this paper we study a process of obtaining a new regular coordinate from a given one by a coalescence of eigenvalues...

Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\overline{\delta }\left(S\right)$ denote the upper asymptotic density of $S$. We consider the problem of finding $$\mu \left(M\right):=\underset{S}{sup}\overline{\delta }\left(S\right),$$ where the supremum is taken over all sets $S$ satisfying that for each $a,b\in S$, $a-b\notin M.$ In this paper we discuss the values and bounds of $\mu \left(M\right)$ where $M=\{a,b,a+nb\}$ for all even integers and for all sufficiently large odd integers $n$ with $a<b$ and $gcd(a,b)=1.$

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new...

We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of $P_{\kappa }\left(X\right)$, where κ ⊆ X. Suppose μ < κ, ${\mu }^{<\mu }=\mu$, and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in ${\left[H\left(\lambda \right)\right]}^{\mu ⁺}$ which are internally club but not internally approachable.

Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x,y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)=c_k{H}^3+O\left(H^{9/4+\epsilon }\right)$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O\left(H^{7/3+\epsilon }\right)$ obtained by G.-L. Zhou, Y. Ding (2022).

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

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 $X=X^B-X^A$.

Given a coarse space $(X,ℰ)$ with the bornology $ℬ$ of bounded subsets, we extend the coarse structure $ℰ$ from $X\times X$ to the natural coarse structure on $(ℬ\setminus \{\varnothing \left\}\right)\times (ℬ\setminus \{\varnothing \left\}\right)$ and say that a macro-uniform mapping $f:(ℬ\setminus \{\varnothing \left\}\right)\to X$ (or $f:{\left[X\right]}^2\to X$) is a selector (or 2-selector) of $(X,ℰ)$ if $f\left(A\right)\in A$ for each $A\in ℬ\setminus \left\{\varnothing \right\}$ ($A\in {\left[X\right]}^2$, respectively). We prove that a discrete coarse space $(X,ℰ)$ admits a selector if and only if $(X,ℰ)$ admits a 2-selector if and only if there exists a linear order “$\le$" on $X$ such that the family of intervals $\left\{\right[a,b]:a,b\in X,a\le b\}$ is a base for the bornology $ℬ$.

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

Let $Q={\left(q_n\right)}_{n=1}^{\infty }$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties. ...