On Locally Gabriel Geometric Graphs

Let \(P\) be a set of \(n\) points in the plane. A geometric graph \(G\) on \(P\) is said to be locally Gabriel if for every edge \((u,v)\) in \(G\), the Euclidean disk with the segment joining \(u\) and \(v\) as diameter does not contain any points of \(P\) that are neighbors of \(u\) or \(v\) in \(G\). A locally Gabriel graph(LGG) is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique LGG on a given point set since no edge in a LGG is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. The unit distance graph(UDG), introduced by Erdos, is also a LGG. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of LGG: (i) For any \(n\), there exists LGG with \(\Omega (n^{5/4})\) edges. This improves upon the previous best bound of \(\Omega (n^{1+\frac{1}{\log \log n}})\). (ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of LGG. (iii) For any LGG on any \(n\) point set, there exists an independent set of size \(\Omega (\sqrt{n}\log n)\).     

Symmetry properties of resolving sets and metric bases in hypercubes

In this paper we consider some special characteristics of distances between vertices in the \(n\)-dimensional hypercube graph \(Q_n\) and, as a consequence, the corresponding symmetry properties of its resolving sets. It is illustrated how these properties can be implemented within a simple greedy heuristic in order to find efficiently an upper bound of the so called metric dimension \(\beta (Q_n)\) of \(Q_n\), i.e. the minimal cardinality of a resolving set in \(Q_n\). This heuristic was applied to generate upper bounds of \(\beta (Q_n)\) for \(n\) up to \(22\), which are for \(n\ge 19\) better than the existing ones. Starting from these new bounds, some existing upper bounds for \(23\le n\le 90\) are improved by a dynamic programming procedure.     

Epireflective subcategories of TOP,T$$_$$UNIF,UNIF, closed under epimorphic images,or being algebraic

The epireflective subcategories of \(\mathbf{Top}\), that are closed under epimorphic (or bimorphic) images, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \) and \(\mathbf{Top}\). The epireflective subcategories of \(\mathbf{T_2Unif}\), closed under epimorphic images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is compact \(T_2 \} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{T_2Unif}\). The epireflective subcategories of \(\mathbf{Unif}\), closed under epimorphic (or bimorphic) images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{Unif}\). The epireflective subcategories of \(\mathbf{Top}\), that are algebraic categories, are \(\{ X \mid |X| \le 1 \} \), and \(\{ X \mid X\) is indiscrete\(\} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being varietal, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid X\) is compact \(T_2 \} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being algebraic, are \(\{ X \mid X\) is indiscrete\( \} \), and all epireflective subcategories of \(\{ X \mid X\) is compact \(T_2 \} \). Also we give a sharpened form of a theorem of Kannan-Soundararajan about classes of \(T_3\) spaces, closed for products, closed subspaces and surjective images.     

Network Decontamination with a Single Agent

Faults and viruses often spread in networked environments by propagating from site to neighboring sites. We model this process of network contamination by graphs. Consider a graph \(G=(V,E)\), whose vertex set is contaminated and our goal is to decontaminate the set \(V\) using mobile decontamination agents that traverse along the edge set of \(G\). Temporal immunity, \(\tau (G) \ge 0\), is defined as the time that a decontaminated vertex of \(G\) can remain continuously exposed to some contaminated neighbor without getting infected itself. The immunity number of \(G\), \(\iota _k(G)\), is the least \(\tau (G)\) that is required to decontaminate \(G\) using \(k\) agents. We study immunity number for some classes of graphs corresponding to network topologies and present upper bounds on \(\iota _1(G)\), in some cases, with matching lower bounds. Variations of this problem have been extensively studied in literature, but proposed algorithms have been restricted to monotone strategies, where a vertex, once decontaminated, may not be recontaminated. We exploit nonmonotonicity to give bounds which are strictly better than those derived using monotone strategies.     

Anti-periodic mild solutions to semilinear fractional differential equations

In this paper, by using the \(\alpha \)-resolvent family theory, Banach contraction mapping principle and Schauder’s fixed point theorem, we investigate the existence of anti-periodic mild solutions to the semilinear fractional differential equations \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t)),\ t\in R,1 \le \alpha \le 2 \) and \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t),u'(t)),\ t\in R,1 < \alpha < 2\), where \(A : D(A)\subset X \rightarrow X\) is the infinitesimal generator of an \(\alpha \)-resolvent family defined on a Banach space \(X\) and \(f\) is a suitable function. Furthermore, an example is given to illustrate our results.     

The convolution algebra

For L a complete lattice L and \(\mathfrak {X}=(X,(R_i)_I)\) a relational structure, we introduce the convolution algebra \(L^{\mathfrak {X}}\). This algebra consists of the lattice \(L^X\) equipped with an additional \(n_i\)-ary operation \(f_i\) for each \(n_i+1\)-ary relation \(R_i\) of \(\mathfrak {X}\). For \(\alpha _1,\ldots ,\alpha _{n_i}\in L^X\) and \(x\in X\) we set \(f_i(\alpha _1,\ldots ,\alpha _{n_i})(x)=\bigvee \{\alpha _1(x_1)\wedge \cdots \wedge \alpha _{n_i}(x_{n_i}):(x_1,\ldots ,x_{n_i},x)\in R_i\}\). For the 2-element lattice 2, \(2^\mathfrak {X}\) is the reduct of the familiar complex algebra \(\mathfrak {X}^+\) obtained by removing Boolean complementation from the signature. It is shown that this construction is bifunctorial and behaves well with respect to one-one and onto maps and with respect to products. When L is the reduct of a complete Heyting algebra, the operations of \(L^\mathfrak {X}\) are completely additive in each coordinate and \(L^\mathfrak {X}\) is in the variety generated by \(2^\mathfrak {X}\). Extensions to the construction are made to allow for completely multiplicative operations defined through meets instead of joins, as well as modifications to allow for convolutions of relational structures with partial orderings. Several examples are given.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

Calabi-Yau manifolds with isolated conical singularities

Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\) such that \(-\frac{1}{q}K_{Z}\) is very ample and such that the germ \((X,x)\) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of \(\frac{1}{q}K_{Z}\). We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing \(2\pi c_{1}(L)\) on \(X\) is asymptotic at a polynomial rate near \(x\) to the natural Ricci-flat Kähler cone metric on \(\frac{1}{q}K_{Z}\) constructed using the Calabi ansatz. In particular, our result applies if \((X, \mathcal{O}(1))\) is a nodal quintic threefold in \(\mathbf {P}^{4}\). This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.     

Bounded size bias coupling: a Gamma function bound,and universal Dickman-function behavior

Under the assumption that the distribution of a nonnegative random variable \(X\) admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In particular the upper tail probability is shown to decay at least as fast as the reciprocal of a Gamma function, guaranteeing a moment generating function that converges everywhere. The class of infinitely divisible distributions with finite mean, whose Lévy measure is supported on an interval contained in \([0,c]\) for some \(c < \infty \), forms a special case in which this upper bound is logarithmically sharp. In particular the asymptotic estimate for the Dickman function, that \(\rho (u) \approx u^{-u}\) for large \(u\), is shown to be universal for this class. A special case of our bounds arises when \(X\) is a sum of independent random variables, each admitting a 1-bounded size bias coupling. In this case, our bounds are comparable to Chernoff–Hoeffding bounds; however, ours are broader in scope, sharper for the upper tail, and equal for the lower tail. We discuss bounded and monotone couplings, give a sandwich principle, and show how this gives an easy conceptual proof that any finite positive mean sum of independent Bernoulli random variables admits a 1-bounded coupling with the same conditioned to be nonzero.     

The First Passage Time of a Stable Process Conditioned to Not Overshoot

Consider a stable Lévy process \(X=(X_t,t\ge 0)\) and let \(T_{x}\), for \(x>0\), denote the first passage time of \(X\) above the level \(x\). In this work, we give an alternative proof of the absolute continuity of the law of \(T_{x}\) and we obtain a new expression for its density function. Our constructive approach provides a new insight into the study of the law of \(T_{x}\). The random variable \(T_{x}^{0}\), defined as the limit of \(T_{x}\) when the corresponding overshoot tends to \(0\), plays an important role in obtaining these results. Moreover, we establish a relation between the random variable \(T_{x}^{0}\) and the dual process conditioned to die at \(0\). This relation allows us to link the expression of the density function of the law of \(T_{x}\) presented in this paper to the already known results on this topic.     

Splines,lattice points,and arithmetic matroids

Let X be a \((d\times N)\)-matrix. We consider the variable polytope \(\varPi _X(u) = \{ w \ge 0 : X w = u \}\). It is known that the function \(T_X\) that assigns to a parameter \(u \in \mathbb {R}^d\) the volume of the polytope \(\varPi _X(u)\) is piecewise polynomial. The Brion–Vergne formula implies that the number of lattice points in \(\varPi _X(u)\) can be obtained by applying a certain differential operator to the function \(T_X\). In this article, we slightly improve the Brion–Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e. operators that do not annihilate \(T_X\)) and the space of nice differential operators (i.e. operators that leave \(T_X\) continuous). These two spaces are finite-dimensional homogeneous vector spaces, and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix X. They are closely related to the \(\mathscr {P}\)-spaces studied by Ardila–Postnikov and Holtz–Ron in the context of zonotopal algebra and power ideals.     

Quadratic properties of least-squares solutions of linear matrix equations with statistical applications

Assume that a quadratic matrix-valued function \(\psi (X) = Q - X^{\prime }PX\) is given and let \(\mathcal{S} = \left\{ X\in {\mathbb R}^{n \times m} \, | \, \mathrm{trace}[\,(AX - B)^{\prime }(AX - B)\,] = \min \right\} \) be the set of all least-squares solutions of the linear matrix equation \(AX = B\). In this paper, we first establish explicit formulas for calculating the maximum and minimum ranks and inertias of \(\psi (X)\) subject to \(X \in {\mathcal S}\), and then derive from the formulas the analytic solutions of the two optimization problems \(\psi (X) =\max \) and \(\psi (X)= \min \) subject to \(X \in \mathcal{S}\) in the Löwner partial ordering. As applications, we present a variety of results on equalities and inequalities of the ordinary least squares estimators of unknown parameter vectors in general linear models.     

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve \(E/\mathbb {Q}\), which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval \((X, X + cX^{\frac{1}{4}})\) for all \(X \gg 0\) and for some \(c > 0\). We use this fact to produce non-CM cuspidal eigenforms f of level \(N>1\) and weight \(k > 2\) such that \(i_f(n) \ll n^{\frac{1}{4}}\) for all \(n \gg 0\).     

On the structure of modules of vector-valued modular forms

If \(\rho \) denotes a finite-dimensional complex representation of \(\mathbf {SL}_{2}(\mathbf {Z})\), then it is known that the module \(M(\rho )\) of vector-valued modular forms for \(\rho \) is free and of finite rank over the ring M of scalar modular forms of level one. This paper initiates a general study of the structure of \(M(\rho )\). Among our results are absolute upper and lower bounds, depending only on the dimension of \(\rho \), on the weights of generators for \(M(\rho )\), as well as upper bounds on the multiplicities of weights of generators of \(M(\rho )\). We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free basis for \(M(\rho )\) in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series \(E_6\) of weight six.     

Averaged Wave Operators and Complex-symmetric Operators

We study the behaviour of sequences \(U_2^n X U_1^{-n}\), where \(U_1, U_2\) are unitary operators, whose spectral measures are singular with respect to the Lebesgue measure, and the commutator \(XU_1-U_2X\) is small in a sense. The conjecture about the weak averaged convergence of the difference \(U_2^n X U_1^{-n}-U_2^{-n} X U_1^n\) to the zero operator is discussed and its connection with complex-symmetric operators is established in a general situation. For a model case where \(U_1=U_2\) is the unitary operator of multiplication by z on \(L^2(\mu )\), sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.     

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain \(D \subset \mathbb {R}^d\) with \(d \ge 2\), for every (measurable) uniformly elliptic tensor field a and for almost every point \(y \in D\), there exists a unique Green’s function centred in y associated to the vectorial operator \(-\nabla \cdot a\nabla \) in D. This result implies the existence of the fundamental solution for elliptic systems when \(d>2\), i.e. the Green function for \(-\nabla \cdot a\nabla \) in \(\mathbb {R}^d\). In the second part, we introduce a shift-invariant ensemble \(\langle \cdot \rangle \) over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for \(\langle |G(\cdot ; x,y)|\rangle \), \(\langle |\nabla _x G(\cdot ; x,y)|\rangle \) and \(\langle |\nabla _x\nabla _y G(\cdot ; x,y)|\rangle \). These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Error Bounds for Consistent Reconstruction: Random Polytopes and Coverage Processes

Consistent reconstruction is a method for producing an estimate \(\widetilde{x} \in {\mathbb {R}}^d\) of a signal \(x\in {\mathbb {R}}^d\) if one is given a collection of \(N\) noisy linear measurements \(q_n = \langle x, \varphi _n \rangle + \epsilon _n\), \(1 \le n \le N\), that have been corrupted by i.i.d. uniform noise \(\{\epsilon _n\}_{n=1}^N\). We prove mean-squared error bounds for consistent reconstruction when the measurement vectors \(\{\varphi _n\}_{n=1}^N\subset {\mathbb {R}}^d\) are drawn independently at random from a suitable distribution on the unit-sphere \({\mathbb {S}}^{d-1}\). Our main results prove that the mean-squared error (MSE) for consistent reconstruction is of the optimal order \({\mathbb {E}}\Vert x - \widetilde{x}\Vert ^2 \le K\delta ^2/N^2\) under general conditions on the measurement vectors. We also prove refined MSE bounds when the measurement vectors are i.i.d. uniformly distributed on the unit-sphere \({\mathbb {S}}^{d-1}\) and, in particular, show that in this case, the constant \(K\) is dominated by \(d^3\), the cube of the ambient dimension. The proofs involve an analysis of random polytopes using coverage processes on the sphere.     

On nodal Enriques surfaces and quartic double solids

We consider the class of singular double coverings \(X \rightarrow {\mathbb {P}}^3\) ramified in the degeneration locus \(D\) of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface \(D,\) one can associate an Enriques surface \(S\) which is the factor of the blowup of \(D\) by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface \(S\) is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of \(X\).