Quadratic Hamilton–Poisson Systems on $\mathfrak{se}(1,1)^{*}_{-}$: The Inhomogeneous Case

The blow-up of smooth solution to the isentropic compressible Navier-Stokes-Poisson (NSP) system on \(\mathbb{R}^{d}\) is studied in this paper. We obtain that if the initial density is compactly supported, the spherically symmetric smooth solution to the NSP system on \(\mathbb{R}^{d}\ (d\geq 2)\) blows up in finite time. In the case \(d=1\), if \(2\mu +\lambda >0\), then the NSP system only exits a zero smooth solution on ? for the compactly supported initial density.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

New nonbinary code bounds based on divisibility arguments

For \(q,n,d \in \mathbb {N}\), let \(A_q(n,d)\) be the maximum size of a code \(C \subseteq [q]^n\) with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds \(A_5(8,6) \le 65\), \(A_4(11,8)\le 60\) and \(A_3(16,11) \le 29\). These in turn imply the new upper bounds \(A_5(9,6) \le 325\), \(A_5(10,6) \le 1625\), \(A_5(11,6) \le 8125\) and \(A_4(12,8) \le 240\). Furthermore, we prove that for \(\mu ,q \in \mathbb {N}\), there is a 1–1-correspondence between symmetric \((\mu ,q)\)-nets (which are certain designs) and codes \(C \subseteq [q]^{\mu q}\) of size \(\mu q^2\) with minimum distance at least \(\mu q - \mu \). We derive the new upper bounds \(A_4(9,6) \le 120\) and \(A_4(10,6) \le 480\) from these ‘symmetric net’ codes.     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

Asymptotic and constructive methods for covering perfect hash families and covering arrays

Covering perfect hash families represent certain covering arrays compactly. Applying two probabilistic methods to covering perfect hash families improves upon the asymptotic upper bound for the minimum number of rows in a covering array with v symbols, k columns, and strength t. One bound can be realized by a randomized polynomial time construction algorithm using column resampling, while the other can be met by a deterministic polynomial time conditional expectation algorithm. Computational results are developed for both techniques. Further, a random extension algorithm further improves on the best known sizes for covering arrays in practice. An extensive set of computations with column resampling and random extension yields explicit constructions when \(k \le 75\) for strength seven, \(k \le 200\) for strength six, \(k \le 600\) for strength five, and \(k \le 2500\) for strength four. When \(v > 3\), almost all known explicit constructions are improved upon. For strength \(t=3\), restrictions on the covering perfect hash family ensure the presence of redundant rows in the covering array, which can be removed. Using restrictions and random extension, computations for \(t=3\) and \(k \le 10{,}000\) again improve upon known explicit constructions in the majority of cases. Computations for strengths three and four demonstrate that a conditional expectation algorithm can produce further improvements at the expense of a larger time and storage investment.     

Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

We consider a continuum percolation model on \(\mathbb {R}^d\), \(d\ge 1\). For \(t,\lambda \in (0,\infty )\) and \(d\in \{1,2,3\}\), the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity \(\lambda >0\). When \(d\ge 4\), the Brownian paths are replaced by Wiener sausages with radius \(r>0\). We establish that, for \(d=1\) and all choices of t, no percolation occurs, whereas for \(d\ge 2\), there is a non-trivial percolation transition in t, provided \(\lambda \) and r are chosen properly. The last statement means that \(\lambda \) has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when \(d\in \{2,3\}\), but finite and dependent on r when \(d\ge 4\)). We further show that for all \(d\ge 2\), the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results.     

Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let \(G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}\) be an abelian group of order \(n=n_1\dots n_m\), where each \(\mathbf{C}_{n_t}\) is cyclic of order \(n_t\). We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in \(G\times Q_8\) relative to the centre \(Z(Q_8)\) and the perfect arrays of size \(n_1\times \dots \times n_m\) over the quaternionic alphabet \(Q_8\cup qQ_8\), where \(q=(1+i+j+k)/2\). In view of this connection, for \(m=2\) we introduce new families of relative difference sets in \(G\times Q_8\), as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

VMO Spaces Associated with Operators with Gaussian Upper Bounds on Product Domains

In this paper, we extend the well-known result “the predual of Hardy space \(H^1\) is VMO” to the product setting, associated with differential operators. Let \(L_i\), \(i = 1, 2\), be the infinitesimal generators of the analytic semigroups \(\{e^{-tL_i}\}\) on \(L^2({\mathbb {R}})\). Assume that the kernels of the semigroups \(\{e^{-tL_i}\}\) satisfy the Gaussian upper bounds. We introduce the VMO spaces VMO\(_{L_1, L_2}(\mathbb {R}\times \mathbb {R})\) associated with operators \(L_1\) and \(L_2\) on the product domain \(\mathbb {R}\times \mathbb {R}\), then show that the dual space of VMO\(_{L_1, L_2}(\mathbb {R}\times \mathbb {R})\) is the Hardy space \(H^1_{L_1^*, L_2^*}(\mathbb {R}\times \mathbb {R})\) associated with the adjoint operators \(L^*_1\) and \(L^*_2\).     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

On Weighted Average Interpolation with Cardinal Splines

Given a sequence of data \(\{ y_{n} \} _{n \in \mathbb{Z}}\) with polynomial growth and an odd number \(d\), Schoenberg proved that there exists a unique cardinal spline \(f\) of degree \(d\) with polynomial growth such that \(f ( n ) =y_{n}\) for all \(n\in \mathbb{Z}\). In this work, we show that this result also holds if we consider weighted average data \(f\ast h ( n ) =y_{n}\), whenever the average function \(h\) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data \(\int_{n-a}^{n+a}f ( x ) dx=y_{n}\) with \(0< a\leq 1/2\). The case of even degree \(d\) is also studied.     

Derivations Vanishing Identities Involving Generalized Derivations and Multilinear Polynomial in Prime Rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a non-zero derivation of R, F and G are two generalized derivations of R such that \(d\{F(u)u-uG^2(u)\}=0\) for all \(u\in f(R)\). Then one of the following holds:

1. (i)
   there exist \(a, b, p\in U\), \(\lambda \in C\) such that \(F(x)=\lambda x+bx+xa^2\), \(G(x)=ax\), \(d(x)=[p, x]\) for all \(x\in R\) with \([p, b]=0\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R;
    
2. (ii)
   there exist \(a, b, p\in U\) such that \(F(x)=ax\), \(G(x)=xb\), \(d(x)=[p,x]\) for all \(x\in R\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R with \([p, a-b^2]=0\);
    
3. (iii)
   there exist \(a\in U\) such that \(F(x)=xa^2\) and \(G(x)=ax\) for all \(x\in R\);
    
4. (iv)
   there exists \(a\in U\) such that \(F(x)=a^2x\) and \(G(x)=xa\) for all \(x\in R\) with \(a^2\in C\);
    
5. (v)
   there exist \(a, p\in U\), \(\lambda , \alpha , \mu \in C\) such that \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) and \(d(x)=[p,x]\) for all \(x\in R\) with \(a^2=\mu -\alpha p\) and \(\alpha p^2+(\lambda -2\mu ) p\in C\);
    
6. (vi)
   there exist \(a\in U\), \(\lambda \in C\) such that R satisfies \(s_4\) and either \(F(x)=\lambda x+xa^2\), \(G(x)=ax\) or \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) for all \(x\in R\).
    

     

Nodal surfaces with obstructed deformations

In this text we show that the deformation space of a nodal surface X of degree d is smooth and of the expected dimension if \(d\le 7\) or \(d\ge 8\) and X has at most \(4d-5\) nodes (The case \(d\le 7\) was previously covered by Alexandru Dimca using different techniques). For \(d\ge 8\) we give explicit examples of nodal surfaces with \(4d-4\) nodes, for which the tangent space to the deformation space has larger dimension than expected.     

Weighted Harmonic Bloch Spaces on the Ball

We study the family of weighted harmonic Bloch spaces \(b_\alpha , \alpha \in {\mathbb {R}}\), on the unit ball of \({\mathbb {R}}^n\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We consider a class of integral operators related to harmonic Bergman projection and determine precisely when they are bounded on \(L^\infty _\alpha \). We define projections from \(L^\infty _\alpha \) to \(b_\alpha \) and as a consequence obtain integral representations. We solve the Gleason problem and provide atomic decomposition for all \(b_\alpha , \alpha \in {\mathbb {R}}\). Finally we give an oscillatory characterization of \(b_\alpha \) when \(\alpha >-1\).     

Restrictions on classical distance-regular graphs

Let \(\varGamma \) be a distance-regular graph with diameter \(d \ge 2\). It is said to have classical parameters \((d, b, \alpha , \beta )\) when its intersection array \(\{b_0,b_1,\dots ,b_{d-1};c_1,c_2,\dots ,c_d\}\) satisfies

$$\begin{aligned} b_i= & {} ([d]_b - [i]_b)(\beta - \alpha [i]_b) \qquad \text {and} \qquad c_{i+1} = [i+1]_b (1 + \alpha [i]_b)\\&\quad (0 \le i \le d-1), \end{aligned}$$

where \([i]_b := 1 + b + \cdots + b^{i-1}\). Apart from the well-known families, there are many sets of classical parameters for which the existence of a corresponding graph is still open. It turns out that in most such cases we have either \(\alpha = b-1\) or \(\alpha = b\). For these two cases, we derive bounds on the parameter \(\beta \), which give us complete classifications when \(b = -2\). Distance-regular graphs with classical parameters are antipodal iff \(b=1\) and \(\beta =1+\alpha [d-1]_b\). If we drop the condition \(b=1\), it turns out that one obtains either bipartite or tight graphs. For the latter graphs, we find closed formulas for the parameters of the CAB partitions and the distance partition corresponding to an edge. Finally, we find a two-parameter family of feasible intersection arrays for tight distance-regular graphs with classical parameters \((d,b,b-1,b^{d-1})\) (primitive iff \(b \ne 1\)) and apply our results to show that it is realized only by d-cubes (\(b = 1\)).

     

On Reachable Families of the Loewner Differential Equation in Several Complex Variables

Graham, Hamada, Kohr and Kohr studied the normalized time \(T\) reachable families \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) of the Loewner differential equation, which are generated by the Carathéodory mappings with values in a subfamily \(\Omega \) of the Carathéodory family \({\mathcal {N}}_A\) for the Euclidean unit ball \({\mathbb {B}}^n\), where \(A\) is a linear operator with \(k_+(A)<2m(A)\) (\(k_+(A)\) is the Lyapunov index of \(A\) and \(m(A)=\min \{\mathfrak {R}\left\langle Az,z\right\rangle \big |z\in {\mathbb {C}}^n,\Vert z\Vert =1\}\)). They obtained some compactness and density results, as generalizations of related results due to Roth, and conjectured that if \(\Omega \) is compact and convex, then \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) is compact and \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},ex\,\Omega )\) is dense in \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\), where \(ex\,\Omega \) denotes the corresponding set of extreme points and \(T\in [0,\infty ]\). We confirm this, by embedding the Carathéodory mappings in a suitable Bochner space.     

Standard Jordan partitions with three parameters

We extend previous work on standard two-parameter Jordan partitions by Barry (Commun Algebra 43:4231–4246, 2015) to three parameters. Let \(J_r\) denote an \(r \times r\) matrix with minimal polynomial \((t-1)^r\) over a field F of characteristic p. For positive integers \(n_1\), \(n_2\), and \(n_3\) satisfying \(n_1 \le n_2 \le n_3\), the Jordan canonical form of the \(n_1 n_2 n_3 \times n_1 n_2 n_3\) matrix \(J_{n_1} \otimes J_{n_2} \otimes J_{n_3}\) has the form \(J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _m}\) where \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _m>0\) and \(\sum _{i=1}^m \lambda _i=n_1 n_2 n_3\). The partition \(\lambda (n_1,n_2,n_3:p)=(\lambda _1, \lambda _2,\ldots , \lambda _m)\) of \(n_1 n_2 n_3\), which depends on \(n_1\), \(n_2\), \(n_3\), and p, will be called a Jordan partition. We will define what we mean by a standard Jordan partition and give necessary and sufficient conditions for its existence.     

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and càdlàg functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).