Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

Topological genericity of nowhere differentiable functions in the disc algebra

In this paper we introduce a class of functions contained in the disc algebra \({\mathcal{A}(D)}\) . We study functions \({f \in \mathcal{A}(D)}\) which have the property that the continuous periodic function \({u = {\rm Re}f|_{\mathbb{T}}}\) , where \({\mathbb{T}}\) is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function \({f \in \mathcal{A}(D)}\) has the above property. Afterwards, we strengthen this result by proving that, generically, for every function \({f \in \mathcal{A}(D)}\) , both continuous periodic functions \({u = {\rm Re}f|_\mathbb{T}}\) and \({\tilde{u} = {\rm Im}f|_\mathbb{T}}\) are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire’s Category Theorem.     

On the minimal volume of simplices enclosing a convex body

Let \({C \subset \mathbb{R}^n}\) be a compact convex body. We prove that there exists an n-simplex \({S\subset \mathbb{R}^n}\) enclosing C such that \({{\rm Vol}(S) \leq n^{n-1} {\rm Vol}(C)}\) .     

Non-Veech surfaces in {\mathcal{H}^{\rm hyp}(4)} are generic

We show that every surface in the component \({\mathcal{H}^{\rm hyp}(4)}\) , that is the moduli space of pairs \({(M,\omega)}\) where M is a genus three hyperelliptic Riemann surface and \({\omega}\) is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its \({{\rm GL}^{+}(2,\mathbb{R})}\) -orbit is either a closed or a dense subset of \({\mathcal{H}^{\rm hyp}(4)}\) . The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmüller curves (closed orbits of surfaces not covering the torus) in \({\mathcal{H}^{\rm hyp}(4)}\) .     

Ascent properties for pairs of modules

Given a flat local ring homomorphism \({R \rightarrow S}\) and two finitely generated R-modules M and N, we describe conditions under which the modules \({{\rm Tor}^{R}_{i}(M,N)}\) and \({{\rm Ext}^{i}_{R}(M,N)}\) have S-module structures that are compatible with their R-module structures.     

Multiplication Operators Defined by a Class of Polynomials on {L_a^2(\mathbb{D}^2)}

In this paper, we consider those multiplication operators M _p on \({L_a^2(\mathbb{D}^2)}\) defined by a class of polynomials p. Also, this paper consider the reducing subspaces of M _p , the von Neumann algebra \({ \mathcal{W}^*(p)}\) generated by M _p , and its commutant \({\mathcal{V}^*(p) = \mathcal{W}^*(p)'.}\) The structure of \({\mathcal{V}^*(p)}\) is completely determined, along with those reducing subspaces of M _p .     

On the Number of Restricted Prime Factors of an Integer

Let \({q\geqq2}\) be an integer and denote S(n) the sum of the digits in base q of the positive integer n. Our main result is to estimate the sum \({\Sigma_{n\leqq x}\tilde{\omega}(n)}\) where \({\tilde{\omega}(n)}\) is the number of distinct prime factors p of n such that \({S(p) \equiv a \,{\rm mod} \,b \,(a \in \mathbb{Z}, b\geqq 2)}\) .     

Finite frames,P-frames and basically disconnected frames

Let \({\mathcal{P}}\) be an ideal of closed quotients of a completely regular frame L and \({\mathcal{R}_{\mathcal{P}}(L)}\) the collection of all functions in the ring \({\mathcal{R}(L)}\) whose support belong to \({\mathcal{P}}\) . We show that \({\mathcal{R}(L)}\) is a Noetherian ring if and only if \({\mathcal{R}(L)}\) is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if \({\mathcal{P}}\) is the ideal of all compact closed quotients of L and L is \({\mathcal{P}}\) -continuous, then \({\mathcal{R}_{\mathcal{P}}(L)}\) is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of \({\mathcal{R}(L)}\) is of the form \({\mathcal{R}_{\mathcal{P}}(L)}\) for some choice of \({\mathcal{P}}\) . We furnish equivalent conditions for \({\mathcal{R}_{\mathcal{P}}(L)}\) to be a prime ideal, a free ideal, and an essential ideal of \({\mathcal{R}(L)}\) separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if \({\mathcal{R}(L)}\) is a coherent ring.     

Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of \({\mathcal{K}(X)}\) -convergence. Moreover, let \({\mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(A)}\) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from \({\mathfrak{M}_{\ell}(C(X, A))}\) onto \({X \times \mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(C(X, A))}\) onto \({X \times \mathfrak{M}(A)}\) . We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication.     

On the Intersection of the Normalizers of the \boldsymbol{\cal{F}}-Residuals of Subgroups of a Finite Group

In this paper we give criteria for a finite group to belong to a formation. As applications, recent theorems of Li, Shen, Shi and Qian are generalized. Let G  be a finite group, $\cal F$ a formation and p  a prime. Let $D_{\mathcal {F}}(G)$ be the intersection of the normalizers of the $\cal F$ -residuals of all subgroups of G, and let $D_{\mathcal {F}}^{p}(G)$ be the intersection of the normalizers of $(H^{\cal F}O_{p'}(G))$ for all subgroups H of G. We then define $D_{\mathcal F}^{0}(G)=D_{\mathcal F, p}^{~0}(G)=1$ and $D_{\mathcal F}^{i+1}(G)/D_{\mathcal F}^{i}(G)=D_{\mathcal F}(G/D_{\mathcal F}^{i}(G))$ , $D_{\mathcal F, p}^{i+1}(G)/D_{\mathcal F, p}^{~i}(G)=D_{\mathcal F, p}(G/D_{\mathcal F, p}^{~i}(G))$ . Let $D_{\mathcal {F}}^{\infty}(G)$ and $D_{\mathcal {F}, p}^{~\infty}(G)$ denote the terminal member of the ascending series of $D_{\mathcal F}^{i}(G)$ and $D_{\mathcal F, p}^{~i}(G)$ respectively. In this paper we prove that under certain hypotheses, the the $\cal F$ -residual $G^{\cal F}$ is nilpotent (respectively,p-nilpotent) if and only if $G=D_{\mathcal {F}}^{\infty}(G)$ (respectively, $G=D_{\mathcal {F}, p}^{~\infty}(G)$ ). Further more, if the formation $\cal F$ is either the class of all nilpotent groups or the class of all abelian groups, then $G^{\cal F}$ is p-nilpotent if and only if and only if every cyclic subgroup of G order p and 4 (if p?=?2) is contained in $D_{\mathcal {F}, p}^{~\infty}(G)$ .     

The Degree of Weakly Discretely Generated Spaces

A space X is discretely generated at a point \({x \in X}\) if for any \({A \subseteqq X}\) with \({x \in \textsf{cl}(A)}\) , there exists a discrete set \({D \subseteqq A}\) such that \({x \in \textsf{cl}(D)}\) . The space X is discretely generated if it is discretely generated at every point \({x \in X}\) . We say that X is weakly discretely generated if for any non-closed set \({A \subseteqq X}\) , there exists a discrete set \({D \subseteqq A}\) such that \({\textsf{cl}(D) \setminus A \neq \emptyset}\) . New results about these properties in the classes of pseudocompact and ?ech-complete spaces are obtained and a theorem of Ivanov and Osipov concerning the ordinal function idc is generalized to the class of ?ech-complete spaces.     

Big elements in irreducible linear groups

Let V be a linear space over a field K of dimension n > 1, and let \({G \leq {\rm GL}(V)}\) be an irreducible linear group. In this paper we prove that the group G contains an element g such that rank \({(g - \alpha E_{n}) \geq \frac{n}{2}}\) for every \({\alpha \in K}\) , where E _n is the identity operator on V. This estimate is sharp for any \({n = 2^{m}}\) . The existence of such an element implies that the conjugacy class of G in GL(V) intersects the big Bruhat cell \({B\dot{w}_{0}B}\) of GL(V) non-trivially (here B is a fixed Borel subgroup of G). The latter fact is equivalent to the existence of a complete flag \({\mathfrak{F}}\) such that the flags \({g(\mathfrak{F}), \mathfrak{F}}\) are in general position for some g ∈ G.     

Toeplitz Operators on Fock Spaces {F^{p}(\varphi)}

Given \({\varphi\in \verb"C"^2(\textbf{C}^n)}\) satisfying \({dd^{c}\varphi\simeq \omega_0}\) , 0 < p < ∞, let \({F^p(\varphi)}\) be the generalized Fock space of all holomorphic functions f on \({{\mathbf C}^n}\) for which the Fock norm $$\|f\|_{p, \varphi}=\left(\,\int_{{\mathbf C}^n} \left|f(z)\right|^{p}e^ {-p\varphi(z)}dv(z)\right)^{\frac{1}{p}} < \infty. $$ While \({\varphi(z)=\frac{1}{2}|z|^2}\) , \({F^{p}(\varphi)}\) is the classical Fock space F ^p . In this paper, for all possible 0 < p,q < ∞ we characterize those positive Borel measures μ on \({{\mathbf C}^n}\) for which the induced Toeplitz operators T _μ are bounded (or compact) from one generalized Fock spaces \({F^p(\varphi)}\) to another \({F^q(\varphi)}\) . With symbols \({g\in BMO}\) , we obtain Zorborska’s criterion for boundedness (or compactness) of Toeplitz operators T _g on F ^p , our work extends the known results on F ². Toeplitz operators on p-th Fock space with 0 < p < 1 have not been studied before, even in the simplest case that \({\varphi(z)=\frac{1}{2}|z|^2}\) . Our analysis shows a significant difference between Bergman spaces and Fock spaces.     

Variants of the Kakeya problem over an algebraically closed field

First, we study constructible subsets of \({\mathbb{A}^n_k}\) which contain a line in any direction. We classify the smallest such subsets in \({\mathbb{A}^3}\) of the type \({R \cup \{g \neq 0\},}\) where \({g \in k[x_1,\ldots, x_n]}\) is irreducible of degree d and \({R \subset V(g)}\) is closed. Next, we study subvarieties \({X \subset \mathbb{A}^N}\) for which the set of directions of lines contained in X has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

Complete bounded null curves immersed in {\mathbb {C}^3} and {\rm {SL}(2,\mathbb {C})}

We construct a simply connected complete bounded mean curvature one surface in the hyperbolic 3-space ${\mathcal {H}^3}$ . Such a surface in ${\mathcal {H}^3}$ can be lifted as a complete bounded null curve in ${\rm {SL}(2,\mathbb {C})}$ . Using a transformation between null curves in ${\mathbb {C}^3}$ and null curves in ${\rm {SL}(2,\mathbb {C})}$ , we are able to produce the first examples of complete bounded null curves in ${\mathbb {C}^3}$ . As an application, we can show the existence of a complete bounded minimal surface in ${\mathbb {R}^3}$ whose conjugate minimal surface is also bounded. Moreover, we can show the existence of a complete bounded immersed complex submanifold in ${\mathbb {C}^2}$ .     

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Divide and Conquer Roadmap for Algebraic Sets

Let \(\mathrm{R}\) be a real closed field and \(\hbox {D}\subset \mathrm{R}\) an ordered domain. We describe an algorithm that given as input a polynomial \(P \in \hbox {D}[ X_{1} , \ldots ,X_{{ k}} ]\) and a finite set, \(\mathcal {A}= \{ p_{1} , \ldots ,p_{m} \}\) , of points contained in \(V= {\mathrm{{Zer}}} ( P, \mathrm{R}^{{ k}})\) described by real univariate representations, computes a roadmap of \(V\) containing \(\mathcal {A}\) . The complexity of the algorithm, measured by the number of arithmetic operations in \(\hbox {D}\) , is bounded by \(\big ( \sum _{i=1}^{m} D^{O ( \log ^{2} ( k ) )}_{i} +1 \big ) ( k^{\log ( k )} d )^{O ( k\log ^{2} ( k ))}\) , where \(d= \deg ( P )\) and \(D_{i}\) is the degree of the real univariate representation describing the point \(p_{i}\) . The best previous algorithm for this problem had complexity card \(( \mathcal {A} )^{O ( 1 )} d^{O ( k^{3/2} )}\) (Basu et al., ArXiv, 2012), where it is assumed that the degrees of the polynomials appearing in the representations of the points in \(\mathcal {A}\) are bounded by \(d^{O ( k )}\) . As an application of our result we prove that for any real algebraic subset \(V\) of \(\mathbb {R}^{k}\) defined by a polynomial of degree \(d\) , any connected component \(C\) of \(V\) contained in the unit ball, and any two points of \(C\) , there exists a semi-algebraic path connecting them in \(C\) , of length at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) , consisting of at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) curve segments of degrees bounded by \(( k ^{\log ( k )} d )^{O ( k \log ( k) )}\) . While it was known previously, by a result of D’Acunto and Kurdyka (Bull Lond Math Soc 38(6):951–965, 2006), that there always exists a path of length \(( O ( d ) )^{k-1}\) connecting two such points, there was no upper bound on the complexity of such a path.     

The error term in the Sato–Tate conjecture

Let \({f(z) = \sum_{n=1}^\infty a(n)e^{2\pi i nz} \in S_k^{\mathrm{new}}(\Gamma_0(N))}\) be a newform of even weight \({k \geq 2}\) that does not have complex multiplication. Then \({a(n) \in \mathbb{R}}\) for all n; so for any prime p, there exists \({\theta_p \in [0, \pi]}\) such that \({a(p) = 2p^{(k-1)/2} {\rm cos} (\theta_p)}\) . Let \({\pi(x) = \#\{p \leq x\}}\) . For a given subinterval \({[\alpha, \beta]\subset[0, \pi]}\) , the now-proven Sato–Tate conjecture tells us that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} \sim \mu_{ST} ([\alpha, \beta])\pi(x),\quad \mu_{ST} ([\alpha, \beta]) = \int\limits_{\alpha}^\beta \frac{2}{\pi}{\rm sin}^2(\theta) d\theta. $$ Let \({\epsilon > 0}\) . Assuming that the symmetric power L-functions of f are automorphic, we prove that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} = \mu_{ST} ([\alpha, \beta])\pi(x) + O\left(\frac{x}{(\log x)^{9/8-\epsilon}} \right), $$ where the implied constant is effectively computable and depends only on k,N, and \({\epsilon}\) .     

Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$(-i\nabla - A(x))^2{u} = \mu|u|^{q-2}u + |u|^{2^*-2}u\, {\rm in}\, \Omega,\quad u=0\, {\rm on}\, \partial\, \Omega$$ where \({\Omega \subset \mathbb{R}^N(N \geq 4)}\) is a bounded domain with smooth boundary, \({A: \overline{\Omega} \rightarrow \mathbb{R}^N}\) is a continuous magnetic potential and \({2 \leq q < 2^* = \frac{2N}{N-2}}\) . Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of Ω.