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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as $$t\rightarrow T$$ (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]).

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

In this paper author was proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space. Note that the proof of multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent don't contained any mistakes. But at the proving of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in this paper. This result is assigned as Theorem 5 in noted paper. In other words, sufficient conditions for general weights ensuring the validity of the two-weight strong type inequalities for some sublinear operator was found. In this theorem the inequality (9) isn't true. In this note we give the details of the correct argument. We presume that the reader is familiar with the contents and notation of our original paper. At the heart of our correction is the following Theorem which replaces Theorem 5.

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.