The Cauchy–Riemann equations in the unit ball of l^{2}

We study the local exactness of the \(\overline{\partial }\) operator in the Hilbert space \(l^2\) for a particular class of \((0,1)\) -forms \(\omega \) of the type \(\omega (z) = \sum _i z_i\omega ^i(z) d\overline{z}_i\) , \(z = (z_i)\) in \(l^2\) . We suppose each function \(\omega ^i\) of class \(C^\infty \) in the closed unit ball of \(l^2\) , of the form \(\omega ^i(z) = \sum _k \omega ^i_k\left( z^k\right) \) , where \(\mathbf N = \bigcup I_k\) is a partition of \(\mathbf N\) , \((\) card \(I_k < +\infty )\) and \(z^k\) is the projection of \(z\) on \(\mathbf C^{I_k}\) . We establish sufficient conditions for exactness of \(\omega \) related to the expansion in Fourier series of the functions \(\omega ^i_k\) .     

Scattered linear sets generated by collineations between pencils of lines

Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\) , and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\) . In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\) -linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n\) with \(A\) and \(B\) . We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\) -linear set of pseudoregulus type with transversal points \(A\) and \(B\) .     

Optimal mass transport and symmetric representations of their cost functions

We consider Monge–Kantorovich problems corresponding to general cost functions \(c(x,y)\) but with symmetry constraints on a Polish space \(X\times X\) . Such couplings naturally generate anti-symmetric Hamiltonians on \(X\times X\) that are \(c\) -convex with respect to one of the variables. In particular, if \(c\) is differentiable with respect to the first variable on an open subset \(X\) in \( \mathbb {R}^d\) , we show that for every probability measure \(\mu \) on \(X\) , there exists a symmetric probability measure \(\pi _0\) on \(X\times X\) with marginals \(\mu \) , and an anti-symmetric Hamiltonian \(H\) such that \(\nabla _2H(y, x)=\nabla _1c(x,y)\) for \( \pi _0\) -almost all \((x,y) \in X \times X.\) If \(\pi _0\) is supported on a graph \((x, Sx)\) , then \(S\) is necessarily a \(\mu \) -measure preserving involution (i.e., \(S^2=I\) ) and \(\nabla _2H(x, Sx)=\nabla _1c(Sx,x)\) for \(\mu \) -almost all \(x \in X.\) For monotone cost functions such as those given by \(c(x,y)=\langle x, u(y)\rangle \) or \(c(x,y)=-|x-u(y)|^2\) where \(u\) is a monotone operator, \(S\) is necessarily the identity yielding a classical result by Krause, namely that \(u(x)=\nabla _2H(x, x)\) where \(H\) is anti-symmetric and concave-convex.     

Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data

This article considers the estimation for bivariate distribution function (d.f.) \(F_0(t, z)\) of survival time \(T\) and covariate variable \(Z\) based on bivariate data where \(T\) is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator \(\hat{F}_n(t,z)\) for \(F_0(t,z)\) , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of \(\hat{F}_n(t,z)\) include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under \(\hat{F}_n(t,z)\) , the conditional d.f. of \(T\) given \(Z\) is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. \(\hat{F}_n(\infty ,z)\) coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, \(\hat{F}_n(t,z)\) coincides with the bivariate empirical d.f. For discrete covariate \(Z\) , the strong consistency and weak convergence of \(\hat{F}_n(t,z)\) are established. Some simulation results are presented.     

Vertex transitive graphs from injective linear mappings

Based on a motivation coming from the study of the metric structure of the category of finite dimensional vector spaces over a finite field \(\mathbb {F}\) , we examine a family of graphs, defined for each pair of integers \(1 \le k \le n\) , with vertex set formed by all injective linear transformations \(\mathbb {F}^k \rightarrow \mathbb {F}^n\) and edges corresponding to pairs of mappings, \(f\) and \(g\) , with \(\lambda (f,g)= \dim \mathrm{Im }(f-g)=1 \) . For \(\mathbb {F}\cong \mathrm{GF }(q)\) , this graph will be denoted by \(\mathrm{INJ }_q(k,n)\) . We show that all such graphs are vertex transitive and Hamiltonian and describe the full automorphism group of each \(\mathrm{INJ }_q (k,n)\) for \(k . Using the properties of line-transitive groups, we completely determine which of the graphs \(\mathrm{INJ }_q (k,n)\) are Cayley and which are not. The Cayley ones consist of three infinite families, corresponding to pairs \((1,n),\,(n-1,n)\) , and \((n,n)\) , with \(n\) and \(q\) arbitrary, and of two sporadic examples \(\mathrm{INJ }_{2} (2,5)\) and \(\mathrm{INJ }_{2}(3,5)\) . Hence, the overwhelming majority of our graphs is not Cayley.     

Additive mappings satisfying algebraic conditions in rings

Let \(R\) be any \((n+1)!\) -torsion free ring and \(F,D: R\rightarrow R\) be additive mappings satisfying \(F(x^{n+1})=(\alpha (x))^nF(x)+\sum \nolimits _{i=1}^n (\alpha (x))^{n-i}(\beta (x))^iD(x)\) for all \(x\in R\) , where \(n\) is a fixed integer and \(\alpha \) , \(\beta \) are automorphisms of \(R\) . Then, \(D\) is Jordan left \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan left \((\alpha , \beta )\) -derivation on \(R\) and if additive mappings \(F\) and \(D\) satisfying \(F(x^{n+1})=F(x)(\alpha (x))^n+\sum \nolimits _{i=1}^n (\beta (x))^iD(x)(\alpha (x))^{n-i}\) for all \(x\in R\) . Then, \(D\) is Jordan \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan \((\alpha , \beta )\) -derivation on \(R\) . At last some immediate consequences of the above theorems have been given.     

Tree metrics and edge-disjoint S-paths

Given an undirected graph \(G=(V,E)\) with a terminal set \(S \subseteq V\) , a weight function on terminal pairs, and an edge-cost \(a: E \rightarrow \mathbf{Z}_+\) , the \(\mu \) -weighted minimum-cost edge-disjoint \(S\) -paths problem ( \(\mu \) -CEDP) is to maximize \(\sum \nolimits _{P \in \mathcal{P}} \mu (s_P,t_P) - a(P)\) over all edge-disjoint sets \(\mathcal{P}\) of \(S\) -paths, where \(s_P,t_P\) denote the ends of \(P\) and \(a(P)\) is the sum of edge-cost \(a(e)\) over edges \(e\) in \(P\) . Our main result is a complete characterization of terminal weights \(\mu \) for which \(\mu \) -CEDP is tractable and admits a combinatorial min–max theorem. We prove that if \(\mu \) is a tree metric, then \(\mu \) -CEDP is solvable in polynomial time and has a combinatorial min–max formula, which extends Mader’s edge-disjoint \(S\) -paths theorem and its minimum-cost generalization by Karzanov. Our min–max theorem includes the dual half-integrality, which was earlier conjectured by Karzanov for a special case. We also prove that \(\mu \) -EDP, which is \(\mu \) -CEDP with \(a = 0\) , is NP-hard if \(\mu \) is not a truncated tree metric, where a truncated tree metric is a weight function represented as pairwise distances between balls in a tree. On the other hand, \(\mu \) -CEDP for a truncated tree metric \(\mu \) reduces to \(\mu '\) -CEDP for a tree metric \(\mu '\) . Thus our result is best possible unless P = NP. As an application, we obtain a good approximation algorithm for \(\mu \) -EDP with “near” tree metric \(\mu \) by utilizing results from the theory of low-distortion embedding.     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Salem numbers in dynamics on Kähler threefolds and complex tori

Let \(X\) be a compact Kähler manifold of dimension \(k\!\le \! 4\) and \(f{:}X\!\rightarrow \! X\) a pseudo-automorphism. If the first dynamical degree \(\lambda _1(f)\) is a Salem number, we show that either \(\lambda _1(f)=\lambda _{k-1}(f)\) or \(\lambda _1(f)^2=\lambda _{k-2}(f)\) . In particular, if \({\dim }(X)=3\) then \(\lambda _1(f)=\lambda _2(f)\) . We use this to show that if \(X\) is a complex 3-torus and \(f\) is an automorphism of \(X\) with \(\lambda _1(f)>1\) , then \(f\) has a non-trivial equivariant holomorphic fibration if and only if \(\lambda _1(f)\) is a Salem number. If \(X\) is a complex 3-torus having an automorphism \(f\) with \(\lambda _1(f)=\lambda _2(f)>1\) but is not a Salem number, then the Picard number of \(X\) must be 0, 3 or 9, and all these cases can be realized.     

Uniqueness of generalized p-area minimizers and integrability of a horizontal normal in the Heisenberg group

We study the uniqueness of generalized \(p\) -minimal surfaces in the Heisenberg group. The generalized \(p\) -area of a graph defined by \(u\) reads \(\int |\nabla u+\vec {F}|+Hu.\) If \(u\) and \(v\) are two minimizers for the generalized \(p\) -area satisfying the same Dirichlet boundary condition, then we can only get \(N_{\vec {F}}(u) = N_{\vec {F}}(v)\) (on the nonsingular set) where \(N_{\vec {F}}(w) := \frac{\nabla w+\vec {F}}{|\nabla w+\vec {F}|}.\) To conclude \(u = v\) (or \(\nabla u = \nabla v)\) , it is not straightforward as in the Riemannian case, but requires some special argument in general. In this paper, we prove that \(N_{\vec {F}}(u) = N_{ \vec {F}}(v)\) implies \(\nabla u = \nabla v\) in dimension \(\ge \) 3 under some rank condition on derivatives of \(\vec {F}\) or the nonintegrability condition of contact form associated to \(u\) or \(v\) . Note that in dimension 2 ( \(n=1),\) the above statement is no longer true. Inspired by an equation for the horizontal normal \(N_{\vec {F}}(u),\) we study the integrability for a unit vector to be the horizontal normal of a graph. We find a Codazzi-like equation together with this equation to form an integrability condition.     

A construction of nonabelian simple {\acute{\mathrm{e}}}tale fundamental groups

Under the assumption of the generalized Riemann hypothesis (GRH), we show that there is a real quadratic field \(K\) such that the \({\acute{\mathrm{e}}}\) tale fundamental group \(\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)\) of the spectrum of the ring of integers \(\mathcal {O}_K\) of \(K\) is isomorphic to \(A_5\) . The proof uses standard methods involving Odlyzko bounds, as well as the proof of Serre’s modularity conjecture. To the best of the author’s knowledge, this is the first example of a number field \(K\) for which \(\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)\) is finite, nonabelian and simple under the assumption of the GRH.     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

Recognition of \text {PSL}(2,p) by order and some information on its character degrees where p is a prime

Let \(G\) be a finite group and \(\text {cd}(G)\) be the set of irreducible character degrees of \(G\) . In this paper we prove that if \(p\) is a prime number, then the simple group \(\text {PSL}(2,p)\) is uniquely determined by its order and some information about its character degrees. In fact we prove that if \(G\) is a finite group such that (i) \(|G|=|\text {PSL}(2,p)|\) , (ii) \(p\in \text {cd}(G)\) , (iii) \(\text {cd}(G)\) has an even integer, and (iv) there does not exist any element \(a\in \text {cd}(G)\) such that \(2p\mid a\) , then \(G\cong \text {PSL}(2,p)\) . As a consequence of our result we get that \(\text {PSL}(2,p)\) is uniquely determined by its order and the largest and the second largest character degrees.     

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form \(\varPhi (x)=\phi (|x_1|, \dots , |x_m|), x_i\in \mathbf R^{n_i}\) , in terms of the Fourier transform of the function \(\phi \) on \(\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}\) , where \(r_i\) is either \(1\) or \(2\) .     

Compactness and sequential completeness in some spaces of operators

Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\) , endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) . Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\) -continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\) -compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \) ; \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and ?ech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.     

Shellability of the higher pinched Veronese posets

The pinched Veronese poset \({\mathcal {V}}^{\bullet }_n\) is the poset with ground set consisting of all nonnegative integer vectors of length \(n\) such that the sum of their coordinates is divisible by \(n\) with exception of the vector \((1,\ldots ,1)\) . For two vectors \(\mathbf {a}\) and \(\mathbf {b}\) in \({\mathcal {V}}^{\bullet }_n\) , we have \(\mathbf {a}\preceq \mathbf {b}\) if and only if \(\mathbf {b}- \mathbf {a}\) belongs to the ground set of \({\mathcal {V}}^{\bullet }_n\) . We show that every interval in \({\mathcal {V}}^{\bullet }_n\) is shellable for \(n \ge 4\) . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in \({\mathcal {V}}^{\bullet }_n\) has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for \(n \ge 4\) . (This also follows from a result by Conca, Herzog, Trung, and Valla.)     

Spherical functions associated with the three-dimensional sphere

In this paper, we determine all irreducible spherical functions \(\Phi \) of any \(K \) -type associated with the pair \((G,K)=(\mathrm{SO }(4),\mathrm{SO }(3))\) . This is accomplished by associating with \(\Phi \) a vector-valued function \(H=H(u)\) of a real variable \(u\) , which is analytic at \(u=0\) and whose components are solutions of two coupled systems of ordinary differential equations. By an appropriate conjugation involving Hahn polynomials, we uncouple one of the systems. Then, this is taken to an uncoupled system of hypergeometric equations, leading to a vector-valued solution \(P=P(u)\) , whose entries are Gegenbauer’s polynomials. Afterward, we identify those simultaneous solutions and use the representation theory of \(\mathrm{SO }(4)\) to characterize all irreducible spherical functions. The functions \(P=P(u)\) corresponding to the irreducible spherical functions of a fixed \(K\) -type \(\pi _\ell \) are appropriately packaged into a sequence of matrix-valued polynomials \((P_w)_{w\ge 0}\) of size \((\ell +1)\times (\ell +1)\) . Finally, we prove that \(\widetilde{P}_w={P_0}^{-1}P_w\) is a sequence of matrix orthogonal polynomials with respect to a weight matrix \(W\) . Moreover, we show that \(W\) admits a second-order symmetric hypergeometric operator \(\widetilde{D}\) and a first-order symmetric differential operator \(\widetilde{E}\) .     

A difference version of Nori’s theorem

We consider (Frobenius) difference equations over \((\mathbb {F}\!_q(s,t), \phi _q)\) where \(\phi _q\) fixes \(t\) and acts on \(\mathbb {F}\!_q(s)\) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group \(\mathcal {G}\) defined over \(\mathbb {F}\!_q\) can be realized as a difference Galois group over \((\mathbb {F} \! _{q^i} (s,t),\phi _{q^i})\) for some \(i \in \mathbb {N}\) . The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori’s theorem which states that \(\mathcal {G}(\mathbb {F}\!_q)\) occurs as a (finite) Galois group over \(\mathbb {F}\!_q(s)\) .