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must hold. By an elementary theorem of geometry, C3 /t2s3M3(s) = 1, and a similar argument shows that tc satisfies this same relation in the case where wx/2 1. Because of the restriction on tc only those values of s may be used for which sM(s) oo as s->1, it is not permissible to let s-*1. Added in proof. See review by E. Reich, Math. Rev. vol. 19 (1958) p. 736.

In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.

In this short paper we prove that, for3≤N≤93 \le N \le 9, the problem−Δu=eu-\Delta u = e^uon the entire Euclidean spaceRN\mathbb {R}^Ndoes not admit any solution stable outside a compact set ofRN\mathbb {R}^N. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.

In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle proposed by Brendle and Schoen.

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short paper we analyse the situation where the underlying non-orientable surfaces have marked points.

In this short paper we establish a (non-trivial) lower bound on the degree two entry h 2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1, r, h 2 , r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h 2 may assume. In fact, we show that limr→∞ f(r) r 2/3 =6 2/3 . In general, we wonder whether our lower bound is sharp for all integers e > 4 and r > 2.

In this short paper, we prove that the group 〈a, b|a = [a, a b ]〉 is hyperlinear. Unlike the nonresidually finite Baumslag-Solitar groups, this group is not residually solvable.

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160

In this short paper we have characterized Suzuki's simple groups SZ(22m+l), m > 1 using only the set 7re(G) of orders of elements in the group G. That is, we have Theorem 2. Let G be a finite group. Then G S (22m+l), m > 1 if and only if ir,(G) = {2, 4, all factors of (22m+l 1), (22m+l 2m+l + 1), and (22m+l + 2m+1 + 1)}. Suzuki's simple groups Sz(22m+l), m > 1 is a family of Zassenhaus groups (Z-groups) of odd degree [9]. In [4] we characterized another family of Zassenhaus groups of odd degree L2(2m) using only the set of orders of elements in the group G. That is, let 7re(G) denote the set of orders of elements in the group G. Then we have proved the following theorem. Theorem 1. Let G be a finite group. Then G L2(2m), m > 2 if and only if 7te(G) = {2, all factors of (2m 1) and 2m + 1)}. In this short paper, we continue this work and obtain the following theorem. Theorem 2. Let G be a finite group. Then G S_(22m+l), m > 1 if and only if 7re(G) = {2, 4, all factors of (22m+1 1), (22m+1 2m+1 + 1), and (22m+1 + 2m+1 + 1)}. Since the simple Z-groups of odd degree consists of L2(2m)(m > 2) and Sz(22m+l), (m > 1), we have Corollary. Let G be a finite group and M a simple Z-groups of odd degree. Then G M if and only if re (G) = 7te(M). Before starting the proof we give a remark about the set 7re(G) in Theorem 2. Since 22m+ I1 4 0 (mod 3) and (22m+I 2m+I + 1) * (22m+I + 2m+I + 1) = 24m+2+ 1 $ 0 (mod 3), 3 ? 7re(G). And 22m+I -1 is prime to 5, but 5 E ire(G) by 24m+2 + 1 = 0 (mod 5). Proof of Theorem 2. We need only prove the sufficiency by [3, XI Theorem 3.10]. Received by the editors June 25, 1990 and, in revised form, October 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 20D05, 20D06. I The author would like to thank the Department of Pure Mathematics of the University of Sydney for the hospitality. ? 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

Let G be a finite, freely acting group of homeomorphisms of the odd-dimensional sphere S 2n−1 . John Oprea has proven that the Gottlieb group of S 2n−1 /G equals Z(G), the centre of G. The purpose of this short paper is to give a considerably shorter, more geometric proof of Oprea's theorem in the important case where G is a linear group

Let p p be an odd prime. h 1 ( p ) {h_1}(p) is the first factor of the class number of field Q ( ζ p ) Q({\zeta _p}) . We proved that \[ h 1 ( p ) ⩽ { 2 p ( p − 1 8 ( 2 l / 2 + 1 ) 4 / l ) ( p − 1 ) / 4 , if l is even, 2 p ( p − 1 8 ( 2 l − 1 ) 2 / l ) ( p − 1 ) / 4 , if l is odd . {h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right . \] From that we obtain h 1 ( p ) ⩽ 2 p ( ( p − 1 ) / 31.997158 … ) ( p − 1 ) / 4 {h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}} which is better than Carlitz’s and Metsänkyla’s results. For the fields Q ( ζ 2 n ) Q({\zeta _{{2^n}}}) and Q ( ζ p n ) ( n ⩾ 2 ) Q({\zeta _{{p^n}}})(n \geqslant 2) , we get the similar results.

This paper provides a constructive method using unitary diagonalizable elements to obtain all hermitian matrices A A in M n ( C ) M_n(\Bbb C) such that ‖ A ‖ = min B ∈ B ‖ A + B ‖ , \begin{equation*} \|A\|=\min _{B\in \mathcal {B}}\|A+B\|, \end{equation*} where B \mathcal {B} is a C*-subalgebra of M n ( C ) M_n(\Bbb C) , ‖ ⋅ ‖ \|\cdot \| denotes the operator norm. Such an A A is called B \mathcal {B} -minimal. Moreover, for a C*-subalgebra B \mathcal {B} determined by a conditional expectation from M n ( C ) M_n(\Bbb C) onto it, this paper constructs ⨁ i = 1 k B \bigoplus _{i=1}^k\mathcal {B} -minimal hermitian matrices in M k n ( C ) M_{kn}(\Bbb C) through B \mathcal {B} -minimal hermitian matrices in M n ( C ) M_n(\Bbb C) , and gets a dominated condition that the matrix A ^ = diag ⁡ ( A 1 , A 2 , ⋯ , A k ) \hat {A}\!=\!\operatorname {diag}(A_1,A_2,\cdots , A_k) is ⨁ i = 1 k B \bigoplus _{i=1}^k\mathcal {B} -minimal if and only if ‖ A ^ ‖ ≤ ‖ A s ‖ \|\hat {A}\|\leq \|A_s\| for some s ∈ { 1 , 2 , ⋯ , k } s\in \{1,2,\cdots ,k\} and A s A_s is B \mathcal {B} -minimal, where A i ( 1 ≤ i ≤ k ) A_i(1\leq i\leq k) are hermitian matrices in M n ( C ) M_n(\Bbb C) .

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.

In the last letter to Hardy, Ramanujan [Collected Papers, Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962] introduced seventeen functions defined by q q -series convergent for | q | > 1 |q|>1 with a complex variable q 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 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 ) g_3(x,q) (resp. g 2 ( x , q ) 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 q -series identities, we find four functions, and express them in terms of Appell–Lerch sums. For example, 1 + ( x q − 1 − x − 1 q ) ∑ n = 0 ∞ ( − 1 ; q ) 2 n q n ( x q − 1 , x − 1 q ; q 2 ) n + 1 = 2 m ( x , q 2 , q ) . \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 ) 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 q -series identities and some properties of Appell–Lerch sums, we derive four radial limit results related to these functions.

In this paper we give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space $F(M,k)$ for a topological manifold $M$. Our technique is to compute the Spanier-Whitehead dual of $\Sigma^\infty_+ F(M,k)$ and use the results of Spivak and Wall on normal spherical fibrations to deduce that the Spanier-Whitehead dual is a proper homotopy invariant. This stable invariance was recently proved by Knudsen using factorization homology. Aside from being elementary, our proof has the advantage that it readily extends to ``generalized configuration spaces'' which have recently undergone study., Comment: Minor changes to structure of paper; to appear in "Proceedings of the American Mathematical Society"

In this paper, we deal with the first and second widths of the real projective space R P n \mathbb {RP}^{n} , for n n ranging from 4 4 to 7 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.

Let F q \mathbb {F}_q be a finite field with q = p n q=p^n elements. In this paper, we study the number of F q \mathbb {F}_q -rational points on the affine hypersurface X \mathcal X given by a 1 x 1 d 1 + ⋯ + a s x s d s = b a_1 x_1^{d_1}+\dots +a_s x_s^{d_s}=b , where b ∈ F q ∗ b\in \mathbb {F}_q^* . A classic well-known result of Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of X \mathcal X with respect to Weil’s bound.

In this paper we study properties of the triangular projection P n \mathscr {P}_n on the space of n × n n\times n matrices. The projection P n \mathscr {P}_n annihilates the entries of an n × n n\times n matrix below the main diagonal and leaves the remaining entries unchanged. We estimate the p p -norms of P n \mathscr {P}_n as an operator on the Schatten–von Neumann class S p {\boldsymbol S}_p for 0 > p > 1 0>p>1 . The main result of the paper shows that for p ∈ ( 0 , 1 ) p\in (0,1) , the p p -norms of P n \mathscr {P}_n on S p {\boldsymbol S}_p behave as n → ∞ n\to \infty as n 1 / p − 1 n^{1/p-1} . This solves a problem posed by B. S. Kashin. Among other results of this paper we mention the result that describes the behaviour of the S p {\boldsymbol S}_p -quasinorms of the n × n n\times n matrices whose entries above the diagonal are equal to 1 while the entries below the diagonal are equal to 0.

Let Λ ( f ) = K [ x ] [ y ; f d d x ] \Lambda (f) = K[x][y; f\frac {d}{dx} ] be an Ore extension of a polynomial algebra K [ x ] K[x] over a field K K of characteristic zero where f ∈ K [ x ] f\in K[x] . For a given polynomial f f , the automorphism group of the algebra Λ ( f ) \Lambda (f) is explicitly described. The polynomial case Λ ( 0 ) = K [ x , y ] \Lambda (0) = K[x,y] and the case of the Weyl algebra A 1 = K [ x ] [ y ; d d x ] A_1= K[x][y; \frac {d}{dx} ] were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras Λ ( f ) \Lambda (f) and Λ ( g ) \Lambda (g) are isomorphic iff g ( x ) = λ f ( α x + β ) g(x) = \lambda f(\alpha x+\beta ) for some λ , α ∈ K ∖ { 0 } \lambda , \alpha \in K\backslash \{ 0\} and β ∈ K \beta \in K . Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras Λ ( f ) \Lambda (f) . In this paper we complete the picture, i.e. given the polynomial f f we have the explicit description of the automorphism group of Λ ( f ) \Lambda (f) . The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest).

This paper serves as a corrigenda for the article P. W. Eloe and J. Henderson, “Two-point boundary value problems for ordinary differential equations, uniqueness implies existence”, Proc. Amer. Math. Soc. 148 (2020), 4377–4387. In particular, the proof the authors give in that paper of Theorem 3.3 is incorrect, and so, that alleged theorem remains a conjecture. In this corrigenda, the authors state and prove a correct theorem.

A function g : R n → R g\colon \mathbb {R}^n\to \mathbb {R} is linearly continuous provided its restriction g ↾ ℓ g\restriction \ell to every straight line ℓ ⊂ R n \ell \subset \mathbb {R}^n is continuous. It is known that the set D ( g ) D(g) of points of discontinuity of any linearly continuous g : R n → R g\colon \mathbb {R}^n\to \mathbb {R} is a countable union of isometric copies of (the graphs of) f ↾ P f\restriction P , where f : R n − 1 → R f\colon \mathbb {R}^{n-1}\to \mathbb {R} is Lipschitz and P ⊂ R n − 1 P\subset \mathbb {R}^{n-1} is compact nowhere dense. On the other hand, for every twice continuously differentiable function f : R → R f\colon \mathbb {R}\to \mathbb {R} and every nowhere dense perfect P ⊂ R P\subset \mathbb {R} there is a linearly continuous g : R 2 → R g\colon \mathbb {R}^2\to \mathbb {R} with D ( g ) = f ↾ P D(g)=f\restriction P . The goal of this paper is to show that this last statement fails, if we do not assume that f f is continuous. More specifically, we show that this failure occurs for every continuously differentiable function f : R → R f\colon \mathbb {R}\to \mathbb {R} with nowhere monotone derivative, which includes twice differentiable functions f f with such property. This generalizes a recent result of professor Luděk Zajíček [On sets of discontinuities of functions continuous on all lines, arxiv.org/abs/2201.00772v1, 2022] and fully solves a problem from a 2013 paper of the first author and Timothy Glatzer [Real Anal. Exchange 38 (2012/13), pp. 377–389].

The main result of this paper is that, under PFA, for every regular space X X with F ( X ) = ω F(X) = \omega we have | X | ≤ w ( X ) ω |X| \le w(X)^\omega ; in particular, w ( X ) ≤ c w(X) \le \mathfrak {c} implies | X | ≤ c |X| \le \mathfrak {c} . This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . In fact, this space X X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X X intersect, which is much stronger than F ( X ) = ω F(X) = \omega . Moreover, any non-empty open set in X X also has size 2 c 2^\mathfrak {c} , and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points.

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition., Comment: 15 pages, v2 corrects a few minor points. The final version of the paper will appear in the Proceedings of the American Mathematical Society

In this paper the authors introduce an analogue of the nilpotent cone, N {\mathcal N} , for a classical Lie superalgebra, g {\mathfrak g} , that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g 0 ¯ ⊕ g 1 ¯ {\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1} with Lie G 0 ¯ = g 0 ¯ \text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0} , it is shown that there are finitely many G 0 ¯ G_{\bar 0} -orbits on N {\mathcal N} . Later the authors prove that the Duflo-Serganova commuting variety, X {\mathcal X} , is contained in N {\mathcal N} for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.

This paper deals with the mathematical analysis of solutions for a new class of Choquard equations. The main features of the problem studied in this paper are the following: (i) the equation is driven by a differential operator with variable exponent; (ii) the Choquard term contains a nonstandard potential with double variable growth; and (iii) the lack of compactness of the reaction, which is generated by a critical nonlinearity. The main result establishes the existence of infinitely many solutions in the case of high perturbations of the source term. The proof combines variational and analytic methods, including the Hardy-Littlewood-Sobolev inequality for variable exponents and the concentration-compactness principle for problems with variable growth.

In this paper, we consider a class of almost periodically forced neutral delayed equation, which arises from population model with delays. A threshold parameter in terms of basic reproduction ratio R 0 R_0 is introduced into this neutral system. We derive the strongly subhomonogenous property of skew-product semiflow generated by the linearized neutral system under the assumptions of non-neutral case. We show that the positive almost periodic solution is globally stable by applying the approach of monotone skew-product semiflow. Finally, as a classical example, we illustrate the asymptotic behavior of Nicholson model with neutral type delays by using of the new theoretical results. The dynamical behaviors of neutral delayed equation forced by almost periods in this paper cover automatically some known ones of the non-neutral cases.

In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial time t = 0 t=0 . Indeed, the solution satisfies the similar decay estimate to that of the corresponding heat equation if it is large and to that of the modified wave equation if it is small. In our previous paper, we obtained the scattering result and its asymptotic order for the data in the energy space H 1 × L 2 H^1\times L^2 when the coefficients are in the wave regime. In fact, the threshold of the coefficients relies on the spatial decay of the initial data. Namely, it varies depending on r r when the initial data is in L r L^r ( 1 ≤ r > 2 1\leq r > 2 ). In the present paper, we will show the scattering result and the asymptotic order in the wave regime for L r L^r -data, which is wider than the wave regime for the data in the energy space. Moreover, we give an improvement of the asymptotic order obtained in our previous paper for the data in the energy space.

This paper is devoted to the isochronicity of two kinds of planar piecewise polynomial potential systems separated by x x -axis and y y -axis, respectively. By means of the expansion of the period function near the infinity, we obtain in this paper that for these two cases, the piecewise potential system has an isochronous center at the origin if and only if the subsystems are linear. It is however not true in the analytic scenario; one can find in the paper by F. Mañosas and P. J. Torres [Proc. Amer. Math. Soc. 133 (2005), pp. 3027–3035] that the isochronicity of the center of the piecewise analytic potential system cannot imply the linearity of the two subsystems.

We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality., The final version this paper will be published in Proc. AMS

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer.

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.

In this paper we study the b b -ary expansions of the square roots of the function defined by the recurrence f b ( n ) = b f b ( n − 1 ) + n f_b(n)=b f_b(n-1)+n with initial value f ( 0 ) = 0 f(0)=0 taken at odd positive integers n n , of which the special case b = 10 b=10 is often referred to as the “schizophrenic” or “mock-rational” numbers. Defined by Darling in 2004 2004 and studied in more detail by Brown in 2009 2009 , these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 b\geq 2 by formally defining the schizophrenic pattern present in the b b -ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.

Let S n S_n be partial sums of an i.i.d. sequence { X i } \{X_i\} . We assume that E X 1 > 0 \mathbb {E} X_1 >0 and P [ X 1 > 0 ] > 0 \mathbb {P}[X_1>0]>0 . In this paper we study the first passage time τ u = inf { n : S n > u } . \begin{equation*} \tau _u = \inf \{n:\; S_n > u\}. \end{equation*} The classical Cramér’s estimate of the ruin probability says that P [ τ u > ∞ ] ∼ C e − α 0 u as u → ∞ , \begin{equation*} \mathbb {P}[\tau _u>\infty ] \sim C e^{-\alpha _0 u}\qquad \text {as } u\to \infty , \end{equation*} for some parameter α 0 \alpha _0 . The aim of the paper is to describe precise large deviations of the first crossing by S n S_n a linear boundary. More precisely for a fixed parameter ρ \rho we study asymptotic behavior of P [ τ u = ⌊ u / ρ ⌋ ] \mathbb {P}\big [\tau _u = \lfloor u/\rho \rfloor \big ] as u u tends to infinity.

In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new properties of Gossez's skew linear operator and its adjoint. While most of this paper uses elementary functional analysis, we correlate our results with those obtained by using the Stone-Cech compactification of the integers.

In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear perturbations of a smooth mapping from a submanifold of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ yield a transverse mapping with respect to a given modular submanifold. Moreover, applications of this result are given., Comment: 11 pages, to appear in Proceedings of the American Mathematical Society

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\infty,1}^1} \max\big\{\|A_1-A_2\|_{\boldsymbol S_p},\|B_1-B_2\|_{\boldsymbol S_p},\|C_1-C_2\|_{\boldsymbol S_p}\big\} \end{align*} holds for an arbitrary function $f$ in $B_{\infty,1}^1({\Bbb R}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,\,B_1,\,C_1,\,A_2,\,B_2$ and $C_2$., 14 pages. arXiv admin note: substantial text overlap with arXiv:1606.08961

In this paper, we investigate asymptotic behaviors of Racah polynomials with fixed parameters and scaled variable as the polynomial degree tends to infinity. We start from the difference equation satisfied by the polynomials and derive an asymptotic formula in the outer region via ratio asymptotics. Next, we find the asymptotic formulas in the oscillatory region via a simple matching principle. Unlike the varying parameter case considered in a previous paper, the zeros of Racah polynomials with fixed parameters may not always be real. For this unusual case, we also provide a standard method to determine the oscillatory curve which attracts the zeros of Racah polynomials when the degree becomes large.

In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].

This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.