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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 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]

In this paper, we investigate several Liouville-type theorems related to k-Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S_k[D^2u]\ge h(u,|Du|) in \mathbb {R}^n. The results depend on some qualified growth conditions of h at infinity. A Liouville-type result to subsolutions of a prototype equation S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S_k[D^2u]=f(u)+g(u)|Du|^q for 0\le q

In this paper, we investigate the monotonicity of the functions t \mapsto \frac {\sum _{k=0}^\infty a_k w_k(t)}{\sum _{k=0}^\infty b_k w_k(t)} and x \mapsto \frac {\int _\alpha ^\beta f(t) w(t,x) \mathrm {d} t}{\int _\alpha ^\beta g(t) w(t,x) \mathrm {d} t}, focusing on case where the monotonicity of a_k/b_k and f(t)/g(t) change once. The results presented also provide insights into the monotonicity of the ratios of two power series, two \mathcal {Z}-transforms, two discrete Laplace transforms, two discrete Mellin transforms, two Laplace transforms, and two Mellin transforms. Finally, we employ these monotonicity rules to present several applications in the realm of special functions and stochastic orders. [ABSTRACT FROM AUTHOR]

In this note, we prove the existence of homogeneous gradient solitons for the G_2-Laplacian flow by providing the first known example of this type. This result singles out the G_2-Laplacian flow as the first known geometric flow admitting homogeneous gradient solitons on spaces that are one-dimensional extensions in the sense of Petersen and Wylie [Differential Geom. Appl. 84 (2022), Paper No. 101929, 29]. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic 2. Our aim in this paper is to prove that the extension given by the function field of a singular conic is excellent for bilinear forms. We also give examples showing that in general this extension is not excellent for quadratic forms of arbitrary dimension. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

In this paper we study minimal Legendrian surfaces \Sigma immersed in tangent sphere bundle T_1{\mathbb {S}}^3. We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to one and complete minimal Legendrian surfaces with non-negative Gauss curvature. [ABSTRACT FROM AUTHOR]

Let \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential \Upsilon belonging to the reverse Hölder class B_{Q/2}, where Q is the homogeneous dimension of the stratified Lie group \mathbb {G}. Inspired by Shen's pioneer work and Li's work, we study fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for \mathcal {L}-harmonic functions on \mathbb {G}. [ABSTRACT FROM AUTHOR]

The B-operators (abbreviation for Brownian-type operators) are upper triangular 2\times 2 block matrix operators that satisfy certain algebraic constraints. The purpose of this paper is to characterize the weak, the strong and the uniform stability of B-operators, respectively. This is achieved by giving equivalent conditions for the convergence of powers of a B-operator in each of the corresponding topologies. A more subtle characterization is obtained for B-operators with subnormal (2,2) entry. The issue of the strong stability of the adjoint of a B-operator is also discussed. [ABSTRACT FROM AUTHOR]

This paper classifies the global dynamics of a 4D competitive Lotka-Volterra system (1.3) with two positive parameters k_1,\ k_2 via carrying simplex. It is proved that the interior of the carrying simplex is filled with periodic orbits except equilibria and each interior trajectory is persistent and tends to either a periodic orbit or an equilibrium if k_1/k_2=2. Otherwise, the system admits two 2D carrying simplices \Delta _i on x_1=i for i=0,\ 1, which are filled with periodic orbits surrounding an equilibrium. All interior orbits go in the long run to \Delta _1 if k_1/k_2>2. If 0

In this paper, we study a class of k-Hessian equations, we can deduce an Obata-type formula and a Liouville-type theorem by integration by parts. [ABSTRACT FROM AUTHOR]

In the present paper, we study the shifted hypergeometric function f(z)=z_{2}F_{1}(a,b;c;z) for real parameters with 0

In this paper, we make the first attempt to figure out the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field B is C_{t}^{\frac {\alpha }2} provided it belongs to L_{t}^{\infty } C_{x}^{\alpha } for any \alpha >0, its energy is C_{t}^{\frac {2\alpha }{2-\alpha }} as long as B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <1 and its magnetic helicity is C_{t}^{\frac {2\alpha +1}{2-\alpha }} supposing B belongs to L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0<\alpha <\frac 12, which are quite different from the classical incompressible Euler equations. [ABSTRACT FROM AUTHOR]

In the present paper, we deepen the works of L. Abatangelo, V. Felli, L. Hillairet and C. Léna on the asymptotic estimates of the eigenvalue variation under removal of segments from the domain in \mathbb {R}^2. We get a sharp asymptotic estimate when the eigenvalue is simple and the removed segment is tangent to a nodal line of the associated eigenfunction. Moreover, we extend their results to the case when the eigenvalue is not simple. [ABSTRACT FROM AUTHOR]

In this paper, we investigate applications of the ordinary derivative operator, instead of the q-derivative operator, to the theory of q-series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the q-binomial theorem, Ramanujan's {}_1\psi _1 formula, the quintuple product identity, Gasper's q-Clausen product formula, and Rogers' {}_6\phi _5 formula, etc. Among these results is a finite form of the Rogers-Ramanujan identity and a short way to Eisenstein's theorem on Lambert series. [ABSTRACT FROM AUTHOR]

In this paper, we study long time behavior of solution to the quantum Zakharov system in two dimensions. We construct a small global solution to the quantum Zakharov system which scatters to a given free solution by using space-time resonance method developed by Gustafson-Nakanishi-Tsai [Commun. Contemp. Math. 11 (2009), pp. 657–707] and Germain-Masmoudi-Shatah [Int. Math. Res. Not. IMRN 3 (2009), 414–432; J. Math. Pures Appl. (9) 97 (2012), pp. 505–543] etc. [ABSTRACT FROM AUTHOR]

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseteq \mathbb {P}_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a\leq b. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were first described in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids. In this paper, based partly on the author's thesis, we use a feature of geometry in positive characteristic to give new methods of producing geproci half grids and non-half grids. [ABSTRACT FROM AUTHOR]

We continue the program first initiated by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and develop a modification of the technique introduced in that paper to study the spectral asymptotics, namely the Riesz means and eigenvalue counting functions, of functional difference operators H_0 = \mathcal {F}^{-1} M_{\cosh (\xi)} \mathcal {F} with potentials of the form W(x) = \left \lvert {x} \right \rvert ^pe^{\left \lvert {x} \right \rvert ^\beta } for either \beta = 0 and p > 0 or \beta \in (0, 2] and p \geq 0. We provide a new method for studying general potentials which includes the potentials studied by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and [J. Math. Phys. 60 (2019), p. 103505]. The proof involves dilating the variance of the gaussian defining the coherent state transform in a controlled manner preserving the expected asymptotics. [ABSTRACT FROM AUTHOR]

In this paper we consider 0-th order pseudodifferential operators on the circle. We show that inside any interval disjoint from the critical values of the principal symbol, the spectrum is absolutely continuous with possibly finitely many embedded eigenvalues. We also give an example of embedded eigenvalues. [ABSTRACT FROM AUTHOR]

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable L² dependence on initial data of Lax 1- or n-shock solutions of an n X n system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freistühler and Zum-brun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of exis-tence, at least for small BV or Hs perturbations. [ABSTRACT FROM AUTHOR]

We provide a corrected proof of one of the two main results (Theorem 3.2) of the paper "The space of persistence diagrams on n points coarsely embeds into Hilbert space" [Proc. Amer. Math. Soc. 149 (2021), pp. 2693–2703]. [ABSTRACT FROM AUTHOR]

In this paper we prove a group theoretic analogue of the well known local nilpotence theorem for sandwich Lie algebras due to Kostrikin and Zel'manov [Trudy Mat. Inst. Steklov. 183 (1990), pp. 106–111, 225]. We introduce the notion of a strong left 3-Engel element of a group G and show that these are always in the locally nilpotent radical of G. This generalises a previous result of Jabara and Traustason [Proc. Amer. Math. Soc. 147 (2019), pp. 1921–1927] that showed that a left 3-Engel element a of a group G is in the locally nilpotent radical of G whenever a is of odd order. [ABSTRACT FROM AUTHOR]

Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]

In this paper, we present two sufficient conditions keeping the total energy and cross-helicity conservation in spaces \underline {B}^{\alpha }_{3,VMO}(\Omega) and B^{\beta }_{3,\infty }(\Omega) for the ideal magnetohydrodynamic equations in a bounded domain. This generalizes the sufficient criteria for conserved quantities recently obtained by Bardos, Gwiazda, Świerczewska Gwiazda, Titi and Wiedemann [Proc. A. 475 (2019), 18 pp.] and by Wang and Zuo [J. Differential Equations 268 (2020), pp. 4079–4101]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a Cartan calculus on the homology of free loop spaces which is introduced by Kuribayashi, Wakatsuki, Yamaguchi and the author [ Cartan calculi on the free loop spaces , Preprint, arXiv:2207.05941, 2022]. In particular, it is proved that the Cartan calculus can be described by the loop product and the loop bracket in string topology. Moreover, by using the descriptions, we show that the loop product behaves well with respect to the Hodge decomposition of the homology of free loop spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce k-Young tableaux and their g-indices. We first present certain expansions of (c(x)D)^n in terms of inversion sequences as well as k-Young tableaux, where c(x) is a smooth function in the indeterminate x and D is the derivative with respect to x. By studying the connections between k-Young tableaux and standard Young tableaux, we then present combinatorial interpretations of Eulerian polynomials, second-order Eulerian polynomials, and André polynomials in terms of standard Young tableaux. [ABSTRACT FROM AUTHOR]

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [ABSTRACT FROM AUTHOR]

Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions f having Hausdorff dimension of the Julia set \operatorname {HD} (\mathcal {J}_f)<2 and hyperbolic dimension \mathrm {HypDim}(f)<\mathrm {HD}(\mathcal {J}_f). [ABSTRACT FROM AUTHOR]

Extending previous results in the literature, random colored substitution networks and degree dimension are defined in this paper. The scale-freeness of these networks is proved by introducing a new definition for degree dimension that is associated with Lyapunov exponents. The random colored substitution network hence turns out to be a simple, powerful and promising model to generate random scale-free networks. [ABSTRACT FROM AUTHOR]

A subset A of a vector space X is called \alpha-lineable whenever A contains, except for the null vector, a subspace of dimension \alpha. If X has a topology, then A is \alpha-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of (\alpha,\beta)-lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of (\alpha,\beta)-lineability/spaceability and, as applications, we extend recent results of different authors. [ABSTRACT FROM AUTHOR]

Let (A,\mathfrak {m}) be an excellent normal domain of dimension two containing a field k \cong A/\mathfrak {m}. An \mathfrak {m}-primary ideal I is a p_g-ideal if the Rees algebra A[It] is a Cohen-Macaulay normal domain. If k is algebraically closed then Okuma, Watanabe and Yoshida proved that A has p_g-ideals and furthermore product of two p_g-ideals is a p_g ideal. Previously we showed that if k has characteristic zero then A has p_g-ideals. In this paper we prove that if k is perfect field of positive characteristic then also A has p_g ideals. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

Mean dimension may decrease after taking the natural extension. In this paper we show that mean dimension is preserved by natural extension for an endomorphism on a compact metrizable abelian group. As an application, we obtain that the mean dimension of an algebraic cellular automaton coincides with the mean dimension of its natural extension, which strengthens a result of Burguet and Shi [Israel J. Math. (to appear).] with a different proof. [ABSTRACT FROM AUTHOR]