Composition operators acting on Besov spaces on the real line

We study the composition operator \(T_f(g):= f\circ g\) on Besov spaces \(B_{{p},{q}}^{s}(\mathbb{R })\) . In case \(1 < p< +\infty ,\, 0< q \le +\infty \) and \(s>1+ (1/p)\) , we will prove that the operator \(T_f\) maps \(B_{{p},{q}}^{s}(\mathbb{R })\) to itself if, and only if, \(f(0)=0\) and \(f\) belongs locally to \(B_{{p},{q}}^{s}(\mathbb{R })\) . For the case \(p=q\) , i.e., in case of Slobodeckij spaces, we can extend our results from the real line to \(\mathbb{R }^n\) .     

Filtered Hopf algebras and counting geodesic chords

We prove lower bounds on the growth of certain filtered Hopf algebras by means of a Poincaré–Birkhoff–Witt type theorem for ordered products of primitive elements. When applied to the loop space homology algebra endowed with a natural length-filtration, these bounds lead to lower bounds for the number of geodesic paths between two points. Specifically, given a closed manifold  \(M\) whose universal covering space is not homotopy equivalent to a finite complex and whose fundamental group has polynomial growth, for any Riemannian metric on  \(M\) , any pair of non-conjugate points \(p,q \in M\) , and every component  \({\mathcal C}\) of the space of paths from  \(p\) to  \(q\) , the number of geodesics in  \({\mathcal C}\) of length at most  \(T\) grows at least like \(e^{\sqrt{T}}\) . Using Floer homology, we extend this lower bound to Reeb chords on the spherisation of  \(M\) , and give a lower bound for the volume growth of the Reeb flow.     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster

In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on \(\mathbb{Z }^d\) . More precisely, we count \(Z_N\) , the number of self-avoiding paths of length \(N\) on the infinite cluster starting from the origin (which we condition to be in the cluster). We are interested in estimating the upper growth rate of \(Z_N\) , \(\limsup _{N\rightarrow \infty } Z_N^{1/N}\) , which we call the connective constant of the dilute lattice. After proving that this connective constant is a.s. non-random, we focus on the two-dimensional case and show that for every percolation parameter \(p\in (1/2,1)\) , almost surely, \(Z_N\) grows exponentially slower than its expected value. In other words, we prove that \(\limsup _{N\rightarrow \infty } (Z_N)^{1/N}{<}\lim _{N\rightarrow \infty } \mathbb{E }[Z_N]^{1/N}\) , where the expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walks on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on the specifics of percolation on \(\mathbb{Z }^2\) , so our result can be extended to a large family of two-dimensional models including general self-avoiding walks in a random environment.     

Unimodality via Kronecker products

We present new proofs and generalizations of unimodality of the \(q\) -binomial coefficients  \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q\) as polynomials in  \(q\) . We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker coefficients of representations of  \(S_n\) . Other applications of this approach include strict unimodality of the diagonal \(q\) -binomial coefficients and unimodality of certain partition statistics.     

Growth in solvable subgroups of {{\mathrm{GL}}}_r({\mathbb {Z}}/p{\mathbb {Z}})

Let \(K={\mathbb {Z}}/p{\mathbb {Z}}\) and let \(A\) be a subset of \({{\mathrm{GL}}}_r(K)\) such that \(\langle A \rangle \) is solvable. We reduce the study of the growth of \(A\) under the group operation to the nilpotent setting. Fix a positive number \(C\ge 1\) ; we prove that either \(A\) grows (meaning \(|A_3|\ge C|A|\) ), or else there are groups \(U_R\) and \(S\) , with \(U_R\unlhd S \unlhd \langle A\rangle \) , such that \(S/U_R\) is nilpotent, \(A_k\cap S\) is large and \(U_R\subseteq A_k\) , where \(k\) depends only on the rank \(r\) of \({{\mathrm{GL}}}_r(K)\) . Here \(A_k = \{x_1 x_2 \cdots x_k : x_i \in A \cup A^{-1} \cup \{1\}\}\) . When combined with recent work by Pyber and Szabó, the main result of this paper implies that it is possible to draw the same conclusions without supposing that \(\langle A \rangle \) is solvable.     

Intrinsically p-biharmonic maps

For a compact Riemannian manifold \(N\) , a domain \(\Omega \subset \mathbb {R}^m\) and for \(p\in (1, \infty )\) , we introduce an intrinsic version \(E_p\) of the \(p\) -biharmonic energy functional for maps \(u : \Omega \rightarrow N\) . This requires finding a definition for the intrinsic Hessian of maps \(u : \Omega \rightarrow N\) whose first derivatives are merely \(p\) -integrable. We prove, by means of the direct method, existence of minimizers of \(E_p\) within the corresponding intrinsic Sobolev space, and we derive a monotonicity formula. Finally, we also consider more general functionals defined in terms of polyconvex functions.     

Isomorphic Steiner symmetrization of p-convex sets

In this paper we show that given a \(p\) -convex set \(K \subset \mathbb{R }^n\) , there exist \(5n\) Steiner symmetrizations that transform it into an isomorphic Euclidean ball. That is, if \(|K| = |D_n| = \kappa _n\) , we may symmetrize it, using \(5n\) Steiner symmetrizations, into a set \(K'\) such that \(c_p D_n \subset K' \subset C_p D_n\) , where \(c_p\) and \(C_p\) are constants dependent on \(p\) only.     

Proper circulant weighing matrices of weight p^2

A circulant weighing matrix \(CW(v,n)\) is a circulant matrix \(M\) of order \(v\) with \(0,\pm 1\) entries such that \(MM^T=nI_v\) . In this paper, we study proper circulant matrices with \(n=p^2\) where \(p\) is an odd prime divisor of \(v\) . For \(p\ge 5\) , it turns out that to search for such circulant matrices leads us to two group ring equations and by studying these two equations, we manage to prove that no proper \(CW(pw,p^2)\) exists when \(p\equiv 3\pmod {4}\) or \(p=5\) .     

On the non-existence of maximal difference matrices of deficiency 1

A \(k\times u\lambda \) matrix \(M=[d_{ij}]\) with entries from a group \(U\) of order \(u\) is called a \((u,k,\lambda )\) -difference matrix over \(U\) if the list of quotients \(d_{i\ell }{d_{j\ell }}^{-1}, 1 \le \ell \le u\lambda ,\) contains each element of \(U\) exactly \(\lambda \) times for all \(i\ne j.\) Jungnickel has shown that \(k \le u\lambda \) and it is conjectured that the equality holds only if \(U\) is a \(p\) -group for a prime \(p.\) On the other hand, Winterhof has shown that some known results on the non-existence of \((u,u\lambda ,\lambda )\) -difference matrices are extended to \((u,u\lambda -1,\lambda )\) -difference matrices. This fact suggests us that there is a close connection between these two cases. In this article we show that any \((u,u\lambda -1,\lambda )\) -difference matrix over an abelian \(p\) -group can be extended to a \((u,u\lambda ,\lambda )\) -difference matrix.     

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.     

New M-ary sequences with low autocorrelation from interleaved technique

Let \(p\) and \(q\) be two odd primes with \(p=Mf+1\) and \(M\) is even. A new construction of \(M\) -ary sequences of period \(pq\) with low periodic autocorrelation is presented in this paper based on interleaving the \(M\) -ary power residue sequence of period \(p\) according to the quadratic residue with respect to \(q\) . This construction can generate the well-known twin-prime sequence and generalized cyclotomy sequence of order two if \(M=2\) . For \(M=4\) , a new class of quaternary sequences of period \(pq\) with maximal nontrivial autocorrelation value being either \(\sqrt{5}\) or \(3\) is obtained. This achieves the best known results for such kind of quaternary sequences.     

On the number of distinct prime factors of nj+a^hk

Let \(\omega (n)\) denote the number of distinct prime factors of \(n\) . Then for any given \(K\ge 2\) , small \(\epsilon >0\) and sufficiently large (only depending on \(K\) and \(\epsilon \) ) \(x\) , there exist at least \(x^{1-\epsilon }\) integers \(n\in [x,(1+K^{-1})x]\) such that \(\omega (nj\pm a^hk)\ge (\log \log \log x)^{\frac{1}{3}-\epsilon }\) for all \(2\le a\le K\) , \(1\le j,k\le K\) and \(0\le h\le K\log x\) .     

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

Fully inert subgroups of free Abelian groups

A subgroup \(H\) of an Abelian group \(G\) is called fully inert if \((\phi H + H)/H\) is finite for every \(\phi \in \mathrm{End}(G)\) . Fully inert subgroups of free Abelian groups are characterized. It is proved that \(H\) is fully inert in the free group \(G\) if and only if it is commensurable with \(n G\) for some \(n \ge 0\) , that is, \((H + nG)/H\) and \((H + nG)/nG\) are both finite. From this fact we derive a more structural characterization of fully inert subgroups \(H\) of free groups \(G\) , in terms of the Ulm–Kaplansky invariants of \(G/H\) and the Hill–Megibben invariants of the exact sequence \(0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0\) .     

Minimum total coloring of planar graph

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is familiar to us by the name of “total coloring”. Total coloring is a coloring of \(V\cup {E}\) such that no two adjacent or incident elements receive the same color. In other words, total chromatic number of \(G\) is the minimum number of disjoint vertex independent sets covering a total graph of \(G\) . Here, let \(G\) be a planar graph with \(\varDelta \ge 8\) . We proved that if for every vertex \(v\in V\) , there exists two integers \(i_{v},j_{v} \in \{3,4,5,6,7,8\}\) such that \(v\) is not incident with intersecting \(i_v\) -cycles and \(j_v\) -cycles, then the vertex chromatic number of total graph of \(G\) is \(\varDelta +1\) , i.e., the total chromatic number of \(G\) is \(\varDelta +1\) .     

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.     

Construction of relative difference sets and Hadamard groups

‘There exist normal \((2m,2,2m,m)\) relative difference sets and thus Hadamard groups of order \(4m\) for all \(m\) of the form $$\begin{aligned} m= x2^{a+t+u+w+\delta -\epsilon +1}6^b 9^c 10^d 22^e 26^f \prod _{i=1}^s p_i^{4a_i} \prod _{i=1}^t q_i^2 \prod _{i=1}^u \left( (r_i+1)/2)r_i^{v_i}\right) \prod _{i=1}^w s_i \end{aligned}$$ under the following conditions: \(a,b,c,d,e,f,s,t,u,w\) are nonnegative integers, \(a_1,\ldots ,a_r\) and \(v_1,\ldots ,v_u\) are positive integers, \(p_1,\ldots ,p_s\) are odd primes, \(q_1,\ldots ,q_t\) and \(r_1,\ldots ,r_u\) are prime powers with \(q_i\equiv 1\ (\mathrm{mod}\ 4)\) and \(r_i\equiv 1\ (\mathrm{mod}\ 4)\) for all \(i, s_1,\ldots ,s_w\) are integers with \(1\le s_i \le 33\) or \(s_i\in \{39,43\}\) for all \(i, x\) is a positive integer such that \(2x-1\) or \(4x-1\) is a prime power. Moreover, \(\delta =1\) if \(x>1\) and \(c+s>0, \delta =0\) otherwise, \(\epsilon =1\) if \(x=1, c+s=0\) , and \(t+u+w>0, \epsilon =0\) otherwise. We also obtain some necessary conditions for the existence of \((2m,2,2m,m)\) relative difference sets in partial semidirect products of \(\mathbb{Z }_4\) with abelian groups, and provide a table cases for which \(m\le 100\) and the existence of such relative difference sets is open.     

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 \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)