Let $B_H=\{B_H(t):t\in\mathbb R\}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$. For the stationary storage process $Q_{B_H}(t)=\sup_{-\infty<s\le t}(B_H(t)-B_H(s)-(t-s))$, $t\ge0$, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(Q_{B_H}(t) > f(t)\, \text{ i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,f_p(t)]}Q_{B_H}(s)>f_p(t))/f_p(t)=\mathscr C(t\log^{1-p} t)^{-1}$, for some $\mathscr C>0$, $\mathbb P(Q_{B_H}(t) > f_p(t)\, \text{ i.o.})= 1_{\{p\ge 0\}}$. Consequently, with $\xi_p (t) = \sup\{s:0\le s\le t, Q_{B_H}(s)\ge f_p(s)\}$, for $p\ge 0$, $\lim_{t\to\infty}\xi_p(t)=\infty$ and $\limsup_{t\to\infty}(\xi_p(t)-t)=0$ a.s. Complementary, we prove an Erd\"os--R\'ev\'esz type law of the iterated logarithm lower bound on $\xi_p(t)$, i.e., $\liminf_{t\to\infty}(\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$; $\liminf_{t\to\infty}\log(\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\in(0,1]$, where $h_p(t)=(1/z_p(t))p\log\log t$.

K. Dębicki K. M. Kosiński

Consider the following statement: $B(t, \Delta t)$: a $t$ years old person NN will survive another $\Delta t$ years, where $t, \Delta t\in \mathbb{R}$ are nonnegative real numbers. We know only that NN is $t$ years old and nothing about the health conditions, gender, race, nationality, etc. We bet that $B(t, \Delta t)$ holds. It seems that our odds are very good, for any $t$ provided $\Delta t$ is small enough, say, $1 / 365$ (that is, one day). However, this is not that obvious and depends on the life-time probabilistic distribution. Let $F(t)$ denote the probability to live at most $t$ years and set $\Phi(t) = 1 - F(t)$. Clearly, $\Phi(t) \rightarrow 0$ as $t \rightarrow \infty$. It is not difficult to verify that $Pr(B(t, \Delta t)) \rightarrow 0$ as $t \rightarrow \infty$, for any fixed $\Delta t$, whenever the convergence of $\Phi$ is fast enough (say, super-exponential). Statistics provide arguments (based on an extrapolation yet) that this is the case. Hence, for an arbitrarily small positive $\Delta t $ and $\epsilon$ there exists a sufficiently large $t$ such that $Pr(B(t, \Delta t)) < \epsilon$, which means that we should not bet... in theory. However, in practice we can bet safely, because for the inequality $Pr(B(t, \Delta t)) < 1/2$ a very large $t$ is required. For example, $\Delta t = 1/365$ may require $t > 125$ years for some typical distributions $F$ considered in the literature. Yet, on Earth there is no person of such age. Thus, our odds are good, either because the chosen testee NN is not old enough, or for technical (or, more precisely, statistical) reasons -- absence of a testee. This situation is similar to the famous St.Petersburg Paradox.

Vladimir Gurvich Mariya Naumova

The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for an epidemic system with multi noises \begin{equation*} \begin{cases} dS(t)=\big[a_1-b_1S(t)-I(t)f(S(t),I(t))\big]dt + \sigma_1 S(t) dB_1(t) -I(t)g(S(t),I(t))dB_3(t),\\ dI(t)=\big[-b_2I(t) + I(t)f(S(t),I(t))\big]dt + \sigma_2I(t) dB_2(t) + I(t)g(S(t),I(t))dB_3(t). \end{cases} \end{equation*} Moreover, the rate of all convergences of the solution are also established. A number of numerical examples are given to illustrate our results

N. H. Du N. N. Nhu

The present paper reports a holographic reconstruction scheme for $f(T,\mathcal T)$ gravity proposed in Harko et al. $\emph{JCAP}\; 12(2014)021$ where $T$ is the torsion scalar and $\mathcal{T}$ is the trace of the energy-momentum tensor considering future event horizon as the enveloping horizon of the universe. We have considered $f(T, \mathcal T)=T + \gamma g(\mathcal T)$ and $f(T,\mathcal T) =\beta \mathcal T + g(T)$ for reconstruction. We observe that the derived $f(T,\mathcal T)$ models can represent phantom or quintessence regimes of the universe which are compatible with the current observational data.

Ines G. Salako Abdul Jawad Surajit Chattopadhyay

To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle t, we attach two "universal algebras" A(H,t) and U(H,t). The algebra A(H,t) is obtained by twisting the multiplication of H with the most general two-cocycle u formally cohomologous to t. The cocycle u takes values in the field of rational functions on H. By construction, A(H,t) is a cleft H-Galois extension of a "big" commutative algebra B(H,t). Any "form" of H[t] can be obtained from A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] is simple, then A(H,t) is an Azumaya algebra with center B(H,t). The algebra U(H,t) is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H[t] are satisfied. We construct an embedding of U(H,t) into A(H,t); this embedding maps the center Z(H,t) of U(H,t) into B(H,t) when the algebra H[t] is simple. In this case, under an additional assumption, A(H,t) is isomorphic to B(H,t) \otimes_{Z(H,t)} U(H,t), thus turning A(H,t) into a central localization of U(H,t). We work out these constructions in full detail for the four-dimensional Sweedler algebra.

Eli Aljadeff Christian Kassel

Let $(R, S,_R\negthinspace M_S,_S\negthinspace N_R, f, g)$ be a general Morita context, and let $T=[{cc} R &_RM_S_SN_R & S]$ be the ring associated with this context. Similarly, let $T'=[{cc} R' & M' N' & S']$ be another Morita context ring. We study the set ${Iso}(T,T')$ of ring isomorphisms from $T$ to $T'$. Our interest in this problem is motivated by: (i) the problem to determine the automorphism group of the ring $T$, and (ii) the recovery of the non-diagonal tiles problem for this type of generalized matrix rings. We introduce two classes of isomorphisms from $T$ to $T'$, the disjoint union of which is denoted by ${Iso}_0(T,T')$. We describe ${Iso}_0(T,T')$ by using the $\Z$-graded ring structure of $T$ and $T'$. Our main result characterizes ${Iso}_0(T,T')$ as the set consisting of all semigraded isomorphisms and all anti-semigraded isomorphisms from $T$ to $T'$, provided that the rings $R'$ and $S'$ are indecomposable and at least one of $M'$ and $N'$ is nonzero; in particular ${Iso}_0(T,T')$ contains all graded isomorphisms and all anti-graded isomorphisms from $T$ to $T'$. We also present a situation where ${Iso}_0(T,T')={Iso}(T,T')$. This is in the case where $R,S,R'$ and $S'$ are rings having only trivial idempotents and all the Morita maps are zero. In particular, this shows that the group of automorphisms of $T$ is completely determined.

C. Boboc S. Dascalescu L. van Wyk

A Straight-Line Program (SLP) for a stirng $T$ is a context-free grammar in Chomsky normal form that derives $T$ only, which can be seen as a compressed form of $T$. Kida et al.\ introduced collage systems [Theor. Comput. Sci., 2003] to generalize SLPs by adding repetition rules and truncation rules. The smallest size $c(T)$ of collage systems for $T$ has gained attention to see how these generalized rules improve the compression ability of SLPs. Navarro et al. [IEEE Trans. Inf. Theory, 2021] showed that $c(T) \in O(z(T))$ and there is a string family with $c(T) \in \Omega(b(T) \log |T|)$, where $z(T)$ is the number of Lempel-Ziv parsing of $T$ and $b(T)$ is the smallest size of bidirectional schemes for $T$. They also introduced a subclass of collage systems, called internal collage systems, and proved that its smallest size $\hat{c}(T)$ for $T$ is at least $b(T)$. While $c(T) \le \hat{c}(T)$ is obvious, it is unknown how large $\hat{c}(T)$ is compared to $c(T)$. In this paper, we prove that $\hat{c}(T) = \Theta(c(T))$ by showing that any collage system of size $m$ can be transformed into an internal collage system of size $O(m)$ in $O(m^2)$ time. Thanks to this result, we can focus on internal collage systems to study the asymptotic behavior of $c(T)$, which helps to suppress excess use of truncation rules. As a direct application, we get $b(T) = O(c(T))$, which answers an open question posed in [Navarro et al., IEEE Trans. Inf. Theory, 2021]. We also give a MAX-SAT formulation to compute $\hat{c}(T)$ for a given $T$.

Soichiro Migita Kyotaro Uehata Tomohiro I

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $H$ is a Hilbert space, $t\in \R_+:=[0,\infty)$, $A(t)$ is a linear dissipative operator: Re$(A(t)u,u)\le -\gamma(t)(u,u)$, %$\gamma(t)\ge 0$, $F(t,u)$ is a nonlinear operator, $|F(t,u)|\le c_0|u|^p$, $p>1$, $c_0,p$ are positive constants, $|b(t)|\le \beta(t),$ $\beta(t)\ge 0$ is a continuous function. Sufficient conditions are given for the solution $u(t)$ to problem (*) to exist for all $t\ge0$, to be bounded uniformly on $\R_+$, and a bound on $|u(t)|$ is given. This bound implies the relation $\lim_{t\to \infty}|u(t)|=0$ under suitable conditions on $\gamma(t)$ and $\beta(t)$.

A. G. Ramm

A fundamental problem in analysis is to decide whether a smooth solution exists for the Navier-Stokes equations in three dimensions. In this paper we shall study this problem. The Navier-Stokes equations are given by: $u_{it}(x,t)-\rho\triangle u_i(x,t)-u_j(x,t) u_{ix_j}(x,t)+p_{x_i}(x,t)=f_i(x,t)$ , $div\textbf{u}(x,t)=0$ with initial conditions $\textbf{u}|_{(t=0)\bigcup\partial\Omega}=0$. We introduce the unknown vector-function: $\big(w_i(x,t)\big)_{i=1,2,3}: u_{it}(x,t)-\rho\triangle u_i(x,t)-\frac{dp(x,t)}{dx_i}=w_i(x,t)$ with initial conditions: $u_i(x,0)=0,$ $u_i(x,t)\mid_{\partial\Omega}=0$. The solution $u_i(x,t)$ of this problem is given by: $u_i(x,t) = \int_0^t \int_\Omega G(x,t;\xi,\tau)~\Big(w_i(\xi,\tau) + \frac{dp(\xi,\tau)}{d\xi_i}\Big)d\xi d\tau$ where $G(x,t;\xi,\tau)$ is the Green function. We consider the following N-Stokes-2 problem: find a solution $\textbf{w}(x,t)\in \textbf{L}_2(Q_t), p(x,t): p_{x_i}(x,t)\in L_2(Q_t)$ of the system of equations: $w_i(x,t)-G\Big(w_j(x,t)+\frac{dp(x,t)}{dx_j}\Big)\cdot G_{x_j}\Big(w_i(x,t)+\frac{dp(x,t)}{dx_i}\Big)=f_i(x,t)$ satisfying almost everywhere on $Q_t.$ Where the v-function $\textbf{p}_{x_i}(x,t)$ is defined by the v-function $\textbf{w}_i(x,t)$. Using the following estimates for the Green function: $|G(x,t;\xi ,\tau)| \leq\frac{c}{(t-\tau)^{\mu}\cdot |x-\xi|^{3-2\mu}}; |G_{x}(x,t;\xi,\tau)|\leq\frac{c}{(t-\tau)^{\mu}\cdot|x-\xi|^{3-(2\mu-1)}}(1/2<\mu<1),$ from this system of equations we obtain: $w(t)<f(t)+b\Big(\int_0^{t}\frac{w(\tau)}{(t-\tau)^{\mu}} d\tau\Big)^2$; $w(t)=\|\textbf{w}(x,t)\|_{L_2(\Omega)}, f(t)=\|\textbf{f}(x,t)\|_{L_2(\Omega)}.$ Further, using the replacement of the unknown function by \textbf{Riccati}, from this inequality we obtain the a priori estimate. By the Leray-Schauder's method and this a priori estimate the existence and uniqueness of the solution is proved.

Argyngazy Bazarbekov

Consider the first passage percolation model on ${\bf Z}^d$ for $d\geq 2$. In this model we assign independently to each edge the value zero with probability $p$ and the value one with probability $1-p$. We denote by $T({\bf 0}, v)$ the passage time from the origin to $v$ for $v\in {\bf R}^d$ and $$B(t)=\{v\in {\bf R}^d: T({\bf 0}, v)\leq t\}{and} G(t)=\{v\in {\bf R}^d: ET({\bf 0}, v)\leq t\}.$$ It is well known that if $p < p_c$, there exists a compact shape $B_d\subset {\bf R}^d$ such that for all $\epsilon >0$ $$t B_d(1-\epsilon) \subset {B(t)} \subset tB_d(1+\epsilon){and} G(t)(1-{\epsilon}) \subset {B(t)} \subset G(t)(1+{\epsilon}) {eventually w.p.1.}$$ We denote the fluctuations of $B(t)$ from $tB_d$ and $G(t)$ by &&F(B(t), tB_d)=\inf \{l:tB_d(1-{l\over t})\subset B(t)\subset tB_d(1+{l\over t})\} && F(B(t), G(t))=\inf\{l:G(t)(1-{l\over t})\subset B(t)\subset G(t)(1+{l\over t})\}. The means of the fluctuations $E[F(B(t), tB_d]$ and $E[F(B(t), G(t))]$ have been conjectured ranging from divergence to non-divergence for large $d\geq 2$ by physicists. In this paper, we show that for all $d\geq 2$ with a high probability, the fluctuations $F(B(t), G(t))$ and $F(B(t), tB_d)$ diverge with a rate of at least $C \log t$ for some constant $C$. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting.

Yu Zhang