Using a Krasnoselskii-type fixed point theorem due to Burton, we discuss the existence of integrable solutions of general quadratic-Urysohn integral equations on a bounded interval $(a,b)$. Uniqueness of the solution is also studied. An example to illustrate our theory is also included.

In this paper, we investigate the boundary value problem for fourth order differential operator equations with unbounded operator coefficients and one $\lambda $-dependent boundary condition. We obtain an asymptotic formula for eigenvalues and a trace formula for the corresponding self-adjoint operator.

In this paper, we obtain the boundedness of minimizers for a class of integral functionals, defined in a weighted anisotropic space.

We study the global structure of positive solutions of the following mean curvature equation in the Minkowski space \[ -\div \bigg (\frac {\nabla u}{\sqrt {1-\vert \nabla u\vert ^2}}\bigg )= \lambda f(x,u), \] on an annular domain with the Robin boundary condition. According to the behavior of $f$ near $0$, we obtain the existence and multiplicity of positive solutions for this problem.

This paper is devoted to the study of the nonexistence and the uniqueness of positive solutions for a class of the following nonlinear coupled Schrodinger systems with saturable nonlinearity \begin{equation} \begin{cases} -\Delta u_{1}+\lambda _{1}u_{1}=\dfrac {u_{1}(\mu _1 u_{1}^2+\beta u_{2}^2)}{1+s(\mu _1 u_{1}^2+\beta u_{2}^2)} &\mbox {in } \mathbb {R}^N, \\-\Delta u_{2}+\lambda _{2}u_{2}=\dfrac {u_{2}(\mu _2 u_{2}^2+\beta u_{1}^2)}{1+s(\mu _2 u_{2}^2+\beta u_{1}^2)} &\mbox {in } \mathbb {R}^N, \\ u_{1}, u_{2}\in H^1(\mathbb {R}^N),\ u_{1}>0,\ u_{2}>0 &\mbox {in } \mathbb {R}^N, \end{cases} \end{equation} where $\lambda _{j}, \mu _{j}, j=1, 2$, are positive constants, $s$ is a positive parameter and $\beta $ is a positive coupling parameter. Moreover, we will show that any positive solution is a priori bounded.

Given a locally compact group $G$ and a unitary representation $\rho :G\to U({\mathcal H})$ on a Hilbert space $\mathcal {H}$, we construct a $C^*$-correspondence ${\mathcal E}(\rho )={\mathcal H}\otimes _{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(\rho )}$. We prove that, for $G$ compact, ${\mathcal O}_{{\mathcal E}(\rho )}$ is strongly Morita equivalent to a graph $C^*$-algebra. If $\lambda $ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(\lambda )}$ is simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is compact abelian, any representation decomposes into characters and determines a skew product graph. We illustrate with several examples, and we compare ${\mathcal E}(\rho )$ with the crossed product $C^*$-correspondence.

Given a Serre class $\mathcal {S}$ of modules, we compare the containment of Koszul homology, Ext modules, Tor modules, local homology and local cohomology in $\mathcal {S}$ up to a given bound $s \geq 0$. As applications, we give a full characterization of Noetherian local homology modules. Furthermore, we establish a comprehensive vanishing result which readily leads to formerly known descriptions of the numerical invariants' width and depth in terms of Koszul homology, local homology and local cohomology. In addition, we recover a few renowned vanishing criteria scattered throughout the literature.

The space of shapes of $n$-gons with marked vertices can be identified with $\mathbb{C} \mathbb{P} ^{n-2}$. The space of shapes of $n$-gons without marked vertices is the quotient of $\mathbb{C} \mathbb{P} ^{n-2}$ by a cyclic group of order $n$ generated by the function which re-enumerates the vertices. In this paper, we prove that the subset corresponding to simple polygons, i.e., without self-intersections, in each case is open and has two homeomorphic, simply connected components.

Let $\phi :\mathcal {C}(S)\to \mathcal {C}(S')$ be a simplicial iso\-mor\-phism between curve graphs of infinite-type surfaces. In this paper, we show that, in this situation, $S$ and $S'$ are homeomorphic and $\phi $ is induced by a homeomorphism $h:S\to S'$.

We apply a result of Kani relating genera and Hasse-Witt invariants of Galois extensions to a family of abelian $p$-extensions. Our formulas generalize the case of elementary abelian $p$-extensions found by Garcia and Stichtenoth.

This note concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of $p$ in the binomial distribution with parameters $n$, $p$, namely, what moment order produces the best Chebyshev estimate of $p$? If $S_n(p)$ has a binomial distribution with parameters $n$, $p$, then it is readily observed that ${argmax }_{0\le p\le 1}{\mathbb E}S_n^2(p) = {argmax }_{0\le p\le 1}np(1-p)= \frac{1}{2}$, and ${\mathbb E}S_n^2(\frac{1}{2}) = \frac{n}{4}$. Bhattacharya observed that, while the second moment Chebyshev sample size for a 95 percent confidence estimate within $\pm 5$ percentage points is $n = 2000$, the fourth moment yields the substantially reduced polling requirement of $n = 775$. Why stop at the fourth moment? Is the argmax achieved at $p = \frac{1}{2}$ for higher order moments, and, if so, does it help in computing $\mathbb {E}S_n^{2m}(\frac{1}{2})$? As captured by the title of this note, answers to these questions lead to a simple rule of thumb for the best choice of moments in terms of an effective sample size for Chebyshev concentration inequalities.

This note is about hyperkahler fourfolds $X$ admitting a non-symplectic involution $\iota $. The Bloch-Beilinson conjectures predict the way $\iota $ should act on certain pieces of the Chow groups of $X$. The main result of this note is a verification of this prediction for Fano varieties of lines on certain cubic fourfolds. This has some interesting consequences for the Chow ring of the quotient $X/\iota $.

We define a class of so-called thinnable ideals $\mathcal {I}$ on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence $(x_n)$ taking values in a separable metric space and a thinnable ideal $\mathcal {I}$, it is shown that the set of $\mathcal {I}$-cluster points of $(x_n)$ is equal to the set of $\mathcal {I}$-cluster points of almost all of its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergence, which improves the main result in Miller 1995 (MR1260176).

Given a finite group $G,$ let $e(G)$ be the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa $, where $\kappa $ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, $\kappa $ equals 2.752394. If $G$ is a permutation group of degree $n,$ then either $G={Sym} (3)$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor +\kappa ^*$ with $\kappa ^* \sim 1.606695.$ These results improve weaker bounds recently obtained by Lucchini.

The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute values of zeros of its matching polynomial. In LiH:2016, the main result, Theorem 3.4, is in error. In this paper, the correct result is given.

Let $-\Delta _\Omega ^N$ be the Neumann Laplacian operator restricted to a twisted waveguide $\Omega $. Our first goal is to find the effective operator when $\Omega $ is ``squeezed.'' However, since, in this process, there are divergent eigenvalues, we consider $-\Delta _\Omega ^N$ acting in specific subspaces of the initial Hilbert space. The strategy is interesting since we find different effective operators in each situation. In the case where $\Omega $ is periodic and sufficiently thin, we also obtain information regarding the absolutely continuous spectrum of $-\Delta _\Omega ^N$ (restricted to such subspaces) and the existence and location of band gaps in its structure.

In this paper, by using the variational method and the theory of genus, we obtain the existence of multiple solutions to a fractional $p$-Kirchhoff problem with subcritical and critical Hardy-Sobolev exponent.

In this article, we study the weighted composition operators preserving the class $\mathcal {P}_{\alpha }$ of analytic functions subordinate to $({1+\alpha z})/({1-z})$ for $|\alpha |\leq 1$, $\alpha \neq -1$. Some of its consequences and examples for some special cases are presented.

We prove two main results concerning mesoprimary decomposition of monoid congruences, as introduced by Kahle and Miller. First, we identify which associated prime congruences appear in every mesoprimary decomposition, thereby completing the theory of mesoprimary decomposition of monoid congruences as a more faithful analogue of primary decomposition. Second, we answer a question posed by Kahle and Miller by characterizing which finite posets arise as the set of associated prime congruences of monoid congruences.

In this paper, along the spirit of Greene, Nijenhuis and Wilf's probabilistic method for the classical hook-length formula for standard Young tableaux, we present a probabilistic proof of the hook-length formula for standard immaculate tableaux, which arose in the study of non-commutative symmetric functions.