Leq 2n Research Articles

On $n$-phantom and $n$-Ext-phantom Morphisms

This paper extends many conclusions based on phantom envelopes and Ext-phantom covers of modules, and we find that many important properties still hold after replacing phantom and Ext-phantom with $n$-phantom and $n$-Ext-phantom respectively. In addition, we also obtain some extra results. Specifically, we give a characterization of the weak dimensions of rings in terms of $n$-phantom envelopes and $n$-Ext-phantom covers of modules with the unique mapping property respectively. We show that $\operatorname{wD}(R) \leq 2n$ whenever every right $R$-module has an $n$-phantom envelope with the unique mapping property or every left $R$-module has an $n$-Ext-phantom cover with the unique mapping property over left coherent rings.

Rational curves on general type hypersurfaces

We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree $\frac{3n+1}{2} \leq d \leq 2n - 3$ in $\mathbb{P}^n$ contain lines but no other rational curves.

On Computing Component (Edge) Connectivities of Balanced Hypercubes

Abstract For an integer $\ell \geqslant 2$, the $\ell $-component connectivity (resp. $\ell $-component edge connectivity) of a graph $G$, denoted by $\kappa _{\ell }(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of vertices (resp. edges) whose removal from $G$ results in a disconnected graph with at least $\ell $ components. The two parameters naturally generalize the classical connectivity and edge connectivity of graphs defined in term of the minimum vertex-cut and the minimum edge-cut, respectively. The two kinds of connectivities can help us to measure the robustness of the graph corresponding to a network. In this paper, by exploring algebraic and combinatorial properties of $n$-dimensional balanced hypercubes $BH_n$, we obtain the $\ell $-component (edge) connectivity $\kappa _{\ell }(BH_n)$ ($\lambda _{\ell }(BH_n)$). For $\ell $-component connectivity, we prove that $\kappa _2(BH_n)=\kappa _3(BH_n)=2n$ for $n\geq 2$, $\kappa _4(BH_n)=\kappa _5(BH_n)=4n-2$ for $n\geq 4$, $\kappa _6(BH_n)=\kappa _7(BH_n)=6n-6$ for $n\geq 5$. For $\ell $-component edge connectivity, we prove that $\lambda _3(BH_n)=4n-1$, $\lambda _4(BH_n)=6n-2$ for $n\geq 2$ and $\lambda _5(BH_n)=8n-4$ for $n\geq 3$. Moreover, we also prove $\lambda _\ell (BH_n)\leq 2n(\ell -1)-2\ell +6$ for $4\leq \ell \leq 2n+3$ and the upper bound of $\lambda _\ell (BH_n)$ we obtained is tight for $\ell =4,5$.

The Diameter of Unit Graphs of Rings

Let $R$ be a ring. The unit graph of $R$, denoted by $G(R)$, is the simple graph defined on all elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x+y$ is a unit of $R$. The diameter of a simple graph $G$, denoted by $\operatorname{diam}(G)$, is the longest distance between all pairs of vertices of the graph $G$. In the present paper, we prove that for each integer $n \geq 1$, there exists a ring $R$ such that $n \leq \operatorname{diam}(G(R)) \leq 2n$. We also show that $\operatorname{diam}(G(R)) \in \{ 1,2,3,\infty \}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with $\operatorname{diam}(G(R)) = 1,2,3$ and $\infty$, respectively. This extends [12, Theorem 2 and Corollary 1].

A New Family of MRD Codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb{F_q}^{2n\times2n}$ </tex-math> </inline-formula> With Right and Middle Nuclei <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{q^n}$ </tex-math> </inline-formula>

In this paper, we present a new family of maximum rank distance (MRD for short) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^n}$. We also prove that the MRD codes of minimum distance $2<d<2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.

Canonical Ordering for Graphs on the Cylinder with Applications to Periodic Straight-line Drawings on the Flat Cylinder and Torus

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.

On the DoF of Two-Way &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$2\times 2\times 2$&lt;/tex-math&gt; &lt;/inline-formula&gt; MIMO Relay Networks

This paper studies the degrees of freedom (DoF) of two-way $2\times 2\times 2$ MIMO interference networks, a class of two-way four-unicast networks. We first provide upper and lower bounds on the sum DoF of general two-way $2\times 2\times 2$ MIMO interference networks with any number of antennas in each node. We then investigate the special case where all user nodes have $M$ antennas and relay nodes have $N$ antennas, and provide the simplified bounds on the sum DoF. We show that our proposed achievable rate for this special case is higher than the achievable rates recently proposed in [1] . Moreover, for this special case, we obtain the exact DoF $=4M$ when $N \geq 2M$ , and DoF $=4N$ when $M \leq 2N$ .

The spectral rigidity of complex projective spaces, revisited

A classical question in spectral geometry is, for each pair of nonnegative integers $(p,n)$ such that $p\leq 2n$, if the eigenvalues of Laplacian on $p$-forms of a compact K\"{a}hler manifold are the same as those of $\mathbb{C}P^n$ equipped with the Fubini-Study metric, then whether or not this K\"{a}hler manifold is holomorphically isometric to $\mathbb{C}P^n$. For every positive even number $p$, we affirmatively solve this problem in all dimensions $n$ with at most two possible exceptions. We also clarify in this paper some gaps in previous literature concerned with this question, among which one is related to the volume estimate of Fano K\"{a}hler-Einstein manifolds.

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt.

Secrecy in MIMO Networks with No Eavesdropper CSIT

We consider two fundamental multi-user channel models: the multiple-input multiple-output (MIMO) wiretap channel with one helper (WTH) and the MIMO multiple access wiretap (MAC-WT) channel. In each case, the eavesdropper has $K$ antennas while the remaining terminals have $N$ antennas each. We consider a fast fading channel where the channel state information (CSI) of the legitimate receiver is available at the transmitters but no CSI at the transmitters (CSIT) is available for the eavesdropper’s channel. We determine the optimal sum secure degrees of freedom (s.d.o.f.) for each channel model for the regime $K\leq N$ , and show that in this regime, the MAC-WT channel reduces to the WTH in the absence of eavesdropper CSIT. For the regime $N\leq K\leq 2N$ , we obtain the optimal linear s.d.o.f., and show that the MAC-WT channel and the WTH have the same optimal s.d.o.f. when restricted to linear encoding strategies. In the absence of any such restrictions, we provide an upper bound for the sum s.d.o.f. of the MAC-WT channel in the regime $N\leq K\leq 2N$ . Our results show that unlike in the single-input single-output case, there is loss of s.d.o.f. for even the WTH due to lack of eavesdropper CSIT when $K\geq N$ .

Toward Wojda's conjecture on digraph packing

Given a positive integer \(m\leq n/2\), Wojda conjectured in 1985 that if \(D_1\) and \(D_2\) are digraphs of order \(n\) such that \(|A(D_1)|\leq n-m\) and \(|A(D_2)|\leq 2n-\lfloor n/m\rfloor-1\) then \(D_1\) and \(D_2\) pack. The cases when \(m=1\) or \(m = n/2\) follow from known results. Here we prove the conjecture for \(m\geq\sqrt{8n}+418275\).

Elliptic Equations with Weight and Combined Nonlinearities

Abstract We consider the equation - div ⁡ ( a ⁢ ( x ) ⁢ ∇ ⁡ u ) = b ⁢ ( x ) ⁢ | u | q - 2 ⁢ u + c ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , u ∈ H 0 1 ⁢ ( Ω ) , $-\operatorname{div}(a(x)\nabla u)=b(x)|u|^{q-2}u+c(x)|u|^{p-2}u,\quad u\in H_{% 0}^{1}(\Omega),$ where Ω ⊂ ℝ N ${\Omega\subset\mathbb{R}^{N}}$ is a bounded smooth domain and N ≥ 4 ${N\geq 4}$ . The functions a, b and c satisfy some hypotheses which provide a variational structure for the problem. For 1 &lt; q &lt; 2 &lt; p ≤ 2 ⁢ N / ( N - 2 ) ${1&lt;q&lt;2&lt;p\leq 2N/(N-2)}$ we obtain the existence of two nonzero solutions if the function b has small Lebesgue norm. The proof is based on minimization arguments and the Mountain Pass Theorem.

Free and properly discontinuous actions of discrete groups on homotopy circles

Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $\Sigma(2n)$ which has the homotopy type of the $2n$- sphere $\mathbb{S}^{2n}$. Then, this action induces an action of the group $G$ on the top cohomology of $\Sigma(2n)$. For the family of virtually cyclic groups, we classify all groups which act on $\Sigma(2n)$, the homotopy type of all possible orbit spaces and all actions on the top cohomology as well. \par Under the hypothesis that $\mbox{dim}\,\Sigma(2n)\leq 2n+1$, we study the groups with the virtual cohomological dimension $\mbox{vcd}\,G<\infty$ which act as above on $\Sigma(2n)$. It turns out that they consist of free groups and certain semi-direct products $F\rtimes \mathbb{Z}_2$ with $F$ a free group. For those groups $G$ and a given action of $G$ on $\mbox{Aut}(\mathbb{Z})$, we present an algebraic criterion equivalent to the realizability of an action $G$ on $\Sigma(2n)$ which induces the given action on its top cohomology. Then, we obtain a classification of those groups together with actions on the top cohomology of $\Sigma(2n)$.

The Ramsey Numbers of Paths Versus Wheels: a Complete Solution

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. We denote by $P_n$ the path on $n$ vertices and $W_m$ the wheel on $m+1$ vertices. Chen et al. and Zhang determined the values of $R(P_n,W_m)$ when $m\leq n+1$ and when $n+2\leq m\leq 2n$, respectively. In this paper we determine all the values of $R(P_n,W_m)$ for the left case $m\geq 2n+1$. Together with Chen et al.'s and Zhang's results, we give a complete solution to the problem of determining the Ramsey numbers of paths versus wheels.

$$L^p$$ L p -Integrability, Dimensions of Supports of Fourier Transforms and Applications

It is proved that there does not exist any non zero function in $L^p(\R^n)$ with $1\leq p\leq 2n/\alpha$ if its Fourier transform is supported by a set of finite packing $\alpha$-measure where $0<\alpha<n$. It is shown that the assertion fails for $p>2n/\alpha$. The result is applied to prove $L^p$ Wiener-Tauberian theorems for $\R^n$ and M(2).

Singularities of affine equidistants: projections and contacts

Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine $\lambda$-equidistants of $n$-dimensional closed submanifolds of $\mathbb R^q$, for $q\leq 2n$, whenever $(2n,q)$ is a pair of nice dimensions.

Reverse Hölder inequalities and higher integrability for subcritical parabolic equations

We prove a local higher integrability result for the gradient of solutions to singular, parabolic equations of $p$-Laplacian type. To this end, we show that solutions satisfy a reverse H\"older inequality on intrinsic cylinders, whose geometry depends on the $L^r$-norm of the solution. The exponent $r \geq 2$ allows us to derive estimates in the subcritical range $1 < p \leq 2N/(N+2)$.

An upper bound on the total outer-independent domination number of a tree

A total outer-independent dominating set of a graph \(G=(V(G),E(G))\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) has a neighbor in \(D\), and the set \(V(G) \setminus D\) is independent. The total outer-independent domination number of a graph \(G\), denoted by \(\gamma_t^{oi}(G)\), is the minimum cardinality of a total outer-independent dominating set of \(G\). We prove that for every tree \(T\) of order \(n \geq 4\), with \(l\) leaves and \(s\) support vertices we have \(\gamma_t^{oi}(T) \leq (2n + s - l)/3\), and we characterize the trees attaining this upper bound.

The algebra of integro-differential operators on a polynomial algebra

We prove that the algebra $\mI_n:=K\langle x_1, ..., x_n, \frac{\der}{\der x_1},...,\frac{\der}{\der x_n}, \int_1, ..., \int_n\rangle $ of integro-differential operators on a polynomial algebra is a prime, central, catenary, self-dual, non-Noetherian algebra of classical Krull dimension $n$ and of Gelfand-Kirillov dimension $2n$. Its weak homological dimension is $n$, and $n\leq \gldim (\mI_n)\leq 2n$. All the ideals of $\mI_n$ are found explicitly, there are only finitely many of them ($\leq 2^{2^n}$), they commute ($\ga \gb = \gb\ga$) and are idempotent ideals ($\ga^2= \ga$). The number of ideals of $\mI_n$ is equal to the {\em Dedekind number} $\gd_n$. An analogue of Hilbert's Syzygy Theorem is proved for $\mI_n$. The group of units of the algebra $\mI_n$ is described (it is a huge group). A canonical form is found for each integro-differential operators (by proving that the algebra $\mI_n$ is a generalized Weyl algebra). All the mentioned results hold for the Jacobian algebra $\mA_n$ (but $\GK (\mA_n) =3n$, note that $\mI_n\subset \mA_n$). It is proved that the algebras $\mI_n$ and $\mA_n$ are ideal equivalent.

Schatten p-norm inequalities related to a characterization of inner product spaces

Let $A_1, ... A_n$ be operators acting on a separable complex Hilbert space such that $\sum_{i=1}^n A_i=0$. It is shown that if $A_1, ... A_n$ belong to a Schatten $p$-class, for some $p>0$, then 2^{p/2}n^{p-1} \sum_{i=1}^n \|A_i\|^p_p \leq \sum_{i,j=1}^n\|A_i\pm A_j\|^p_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. Moreover, \sum_{i,j=1}^n\|A_i\pm A_j\|^2_p \leq 2n^{2/p} \sum_{i=1}^n \|A_i\|^2_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch.