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A classical result of K. L. Chung and W. Feller deals with the partial sums S_k arising in a fair coin-tossing game. If N_n is the number of "positive" terms among S_1, S_2, ..., S_n then the quantity 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_{2n+1} = r), r = 0, 1, 2, ..., 2n+1. We get to this ansatz by adaptating the Feynman–Kac methodology. [ABSTRACT FROM AUTHOR]

The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond [Math. Ann. 21 (1883), no. 1, pp. 109–117]). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky [Crelle 80 (1875), pp. 1–32]) that a solution to the heat equation may not be time-analytic at t=0 even if the initial function is real analytic. Recently, it was shown by Dong and Pan [J Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53]; Dong and Zhang [J. Funct. Anal. 279 (2020), no. 4, Paper No. 108563]; Zhang [Proc. Amer. Math. Soc. 148 (2020), no. 4, pp. 1665–1670] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under an essentially optimal growth condition. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any \delta >0, we find a solution to the heat equation on the whole plane, with exponential growth of order 2+\delta, which is nowhere analytic in time. [ABSTRACT FROM AUTHOR]

The chord log-Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-chord measure of a convex body. The chord log-Minkowski problem has been extensively studied by Guo, Xi, and Zhao [Math. Ann. (2023), DOI 10.1007/s00208-023-02721-8]; Lutwak, Xi, Yang, and Zhang [Commun. Pure Appl. Math. (2023), DOI 10.1002/cpa.22190]; Qin [Adv. Math. 427 (2023), Paper No. 109132]. In this paper, we solve the chord log-Minkowski problem when q\in (0,1), without symmetry assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump (possibly degenerate) noise, including stochastic 2D Navier-Stokes equations, stochastic Burgers equations, stochastic singular p-Laplace equations, and stochastic fast diffusion equations. As a further application, we establish the ergodicity of stochastic singular p-Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Compound Poisson processes or Lévy processes with heavy tails. [ABSTRACT FROM AUTHOR]

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

The present paper continues investigation of narrow operators on complex vector lattices started in a recent paper by Dzhusoeva, Huang, Pliev and Sukochev. We answer their questions and strengthen one of their main results by proving that the extension T_{\Bbb {C}}\colon E_{\Bbb {C}}\to F_{\Bbb {C}} of a linear operator T \colon E \to F acting from a Dedekind complete real Riesz space E to a real Banach lattice F to their complexifications is narrow if and only if T is. [ABSTRACT FROM AUTHOR]

In this paper we prove a global spacetime bound for the quintic, nonlinear wave equation in three dimensions. This bound depends on the L_{t}^{\infty } L_{x}^{2} and L_{t}^{\infty } \dot {H}^{2} norms of the solution to the quintic problem. The main motivation for this paper is the use of an interaction Morawetz estimate for the nonlinear wave equation. [ABSTRACT FROM AUTHOR]

Let (M,g) be a compact Riemann surface, h be a positive smooth function on M. It is well known that the functional \begin{equation*} J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^{u}dv_g \end{equation*} achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative h by Yang and the author. Later, Sun and Zhu [ Existence of Kazdan-Warner equation with sign-changing prescribed function , arXiv: 2012.12840 , 2020] showed the Ding-Jost-Li-Wang condition is also sufficient when h changes sign, which was reproved later by Wang and Yang [J. Funct. Anal. 282 (2022), Paper No. 109449] and Li and Xu [Calc. Var. Partial Differential Equations 61 (2022), Paper No. 143] respectively using a flow approach. The aim of this note is to give a new proof of Sun and Zhu's result. Our proof is based on the variational method and the maximum principle. [ABSTRACT FROM AUTHOR]

In \S8.3 of our paper ``Duality and Flat Base Change on Formal Schemes" some important results concerning localization and preservation of coherence by basic duality functors were based on the false statement that any closed formal subscheme of an open subscheme of the completion $\mathscr P$ of a relative projective space is an open subscheme of a closed formal subscheme of $\mathscr P$. In this note, the said results are provided with solid foundations. [ABSTRACT FROM AUTHOR]

Concerns have been voiced about the correctness of certain technical points in DiPerna's paper (Comm. Math. Phys. \textbf{91} (1983), 1--30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. [ABSTRACT FROM AUTHOR]

Many previous works used the admissibility of function classes to characterize nonuniform exponential dichotomy (for short, NEDs) by employing Lyapunov norms. Recently, L. Zhou and W. Zhang [J. Funct. Anal. 271 (2016), pp. 1087–1129] characterized NEDs without Lyapunov norms. They utilized two admissible pairs of function classes and an assumption on certain subspaces to describe NEDs under the prerequisite of bounded growth, which plays an essential role in their arguments. However, in this paper, we remove the condition of bounded growth when characterizing NEDs. Neither bounded growth nor Lyapunov norms are used to describe NEDs in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the first and second widths of the real projective space \mathbb {RP}^{n}, for n ranging from 4 to 7, and for this we used some tools from the Almgren-Pitts min-max theory. In a recent paper, Ramirez-Luna computed the first width of the real projective spaces, and, at the same time, we obtained optimal sweepouts realizing the first and second widths of those spaces. [ABSTRACT FROM AUTHOR]

In the last letter to Hardy, Ramanujan [ Collected Papers , Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962] introduced seventeen functions defined by q-series convergent for |q|<1 with a complex variable q, and called these functions "mock theta functions". Subsequently, mock theta functions were widely studied in the literature. In the survey of B. Gordon and R. J. McIntosh [ A survey of classical mock theta functions , Partitions, q-series, and modular forms, Dev. Math., vol. 23, Springer, New York, 2012, pp. 95–144], they showed that the odd (resp. even) order mock theta functions are related to the function g_3(x,q) (resp. g_2(x,q)). These two functions are usually called "universal mock theta functions". D. R. Hickerson and E. T. Mortenson [Proc. Lond. Math. Soc. (3) 109 (2014), pp. 382–422] expressed all the classical mock theta functions and the two universal mock theta functions in terms of Appell–Lerch sums. In this paper, based on some q-series identities, we find four functions, and express them in terms of Appell–Lerch sums. For example, \begin{equation*} 1+(xq^{-1}-x^{-1}q)\sum _{n=0}^{\infty }\frac {(-1;q)_{2n}q^{n}}{(xq^{-1},x^{-1}q;q^2)_{n+1}}=2m(x,q^2,q). \end{equation*} Then we establish some identities related to these functions and the universal mock theta function g_2(x,q). These relations imply that all the classical mock theta functions can be expressed in terms of these four functions. Furthermore, by means of q-series identities and some properties of Appell–Lerch sums, we derive four radial limit results related to these functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider k-curvature equations \sigma _k(\kappa [M_u])=f(x,u,\nabla u) subject to (k+1)-convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these k-curvature equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove a vanishing theorem concerning the periods of cuspidal automorphic sheaves for \operatorname {GL}_{m+n} along the Levi subgroup \operatorname {GL}_{m}\times \operatorname {GL}_{n} for m \neq n. [ABSTRACT FROM AUTHOR]

In this paper we provide a detailed topological and measure-theoretic study of Lebesgue measure-preserving continuous circle maps that are composed with independent rotations on each of the sides. In particular, we analyze the stability of the locally eventually onto and measure-theoretic mixing properties. [ABSTRACT FROM AUTHOR]

The upper and lower solution method is a recently developed and currently the most effective tool for determining the sign of bistable traveling wave. However, for systems with nonlocal diffusion terms, it is extremely challenging to find upper and lower solutions. In this paper, we develop a new idea for constructing the upper and lower solutions to establish the explicit conditions for obtaining positive or negative wave speed for a Lotka-Volterra competitive system with bistable nonlinearity. The theoretical results are demonstrated by directly integrating the considered system. This method can be used to improve or correct the related results in the known references. [ABSTRACT FROM AUTHOR]

In this paper, we prove embedding theorems for the Möbius invariant space \mathcal {Q}_p on the open unit ball of \mathbb {C}^n into logarithmic tent spaces in the Bergman metric. [ABSTRACT FROM AUTHOR]

In this paper, given a prescribed measure on \mathbb {S}^1 whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar L_p dual Minkowski problem when 0

In this paper, using the method of blow-up analysis, we establish a trace Trudinger-Moser inequality involving L^p-norm and obtain the corresponding extremal on a compact Riemann surface with a smooth boundary. The result generalizes those of Li-Liu [Math. Z. 250 (2005), pp. 363–686] and Zhang [Commun. Pure Appl. Anal. 20 (2021), pp. 1721–1735]. [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]