Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times

For nonnegative integers r, s, let \(^{(r,s)}X_t\) be the Lévy process \(X_t\) with the r largest positive jumps and the s smallest negative jumps up till time t deleted, and let \(^{(r)}\widetilde{X}_t\) be \(X_t\) with the r largest jumps in modulus up till time t deleted. Let \(a_t \in \mathbb {R}\) and \(b_t>0\) be non-stochastic functions in t. We show that the tightness of \(({}^{(r,s)}X_t - a_t)/b_t\) or \(({}^{(r)}{\widetilde{X}}_t - a_t)/b_t\) as \(t\downarrow 0\) implies the tightness of all normed ordered jumps, and hence the tightness of the untrimmed process \((X_t -a_t)/b_t\) at 0. We use this to deduce that the trimmed process \(({}^{(r,s)}X_t - a_t)/b_t\) or \(({}^{(r)}{\widetilde{X}}_t - a_t)/b_t\) converges to N(0, 1) or to a degenerate distribution as \(t\downarrow 0\) if and only if \((X_t-a_t)/b_t \) converges to N(0, 1) or to the same degenerate distribution, as \(t \downarrow 0\).     

Fredholmness of fringe operators over the bidisk

Let M be an invariant subspace of \(H^2\) over the bidisk. Associated with M, we have the fringe operator \(F^M_z\) on \(M\ominus w M\). For \(A\subset H^2\), let [A] denote the smallest invariant subspace containing A. Assume that \(F^M_z\) is Fredholm. If h is a bounded analytic function on \(\mathbb {D}^2\) satisfying \(h(0,0)\not =0\), then \(F^{[h M]}_z\) is Fredholm and \(\mathrm{ind}\,F^{[h M]}_z=\mathrm{ind}\,F^M_z\).     

Contractive determinantal representations of stable polynomials on a matrix polyball

We show that a polynomial p with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that \(p(0)=1\), admits a strictly contractive determinantal representation, i.e., \(p=\det (I-KZ_n)\), where \(n=(n_1,\ldots ,n_k)\) is a k-tuple of nonnegative integers, \(Z_n=\bigoplus _{r=1}^k(Z^{(r)}\otimes I_{n_r})\), \(Z^{(r)}=[z^{(r)}_{ij}]\) are complex matrices, p is a polynomial in the matrix entries \(z^{(r)}_{ij}\), and K is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

A characterization of Inoue surfaces with $$p_g=0$$ and $$K^2=7$$K2=7

Inoue constructed the first examples of smooth minimal complex surfaces of general type with \(p_g=0\) and \(K^2=7\). These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\). For such a surface S, the bicanonical map of S has degree 2 and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components: one is a genus 3 curve with self-intersection number 0 and the other is a genus 2 curve with self-intersection number \(-\,1\). Conversely, assume that S is a smooth minimal complex surface of general type with \(p_g=0\), \(K^2=7\) and having an involution \(\sigma \). We show that, if the divisorial part of the fixed locus of \(\sigma \) consists of two irreducible components \(R_1\) and \(R_2\), with \(g(R_1)=3, R_1^2=0, g(R_2)=2\) and \(R_2^2=-\,1\), then the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\) acts faithfully on S and S is indeed an Inoue surface.     

On the sumsets of exceptional units in $$\mathbb {Z}_n$$

Let R be a commutative ring with \(1\in R\) and \(R^{*}\) be the multiplicative group of its units. In 1969, Nagell introduced the concept of an exceptional unit, namely a unit u such that \(1-u\) is also a unit. Let \({\mathbb {Z}}_n\) be the ring of residue classes modulo n. In this paper, given an integer \(k\ge 2\), we obtain an exact formula for the number of ways to represent each element of \( \mathbb {Z}_n\) as the sum of k exceptional units. This generalizes a recent result of J. W. Sander for the case \(k=2\).     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

The shape of $$ \mathbb {Z}/\ell \mathbb {Z}$$-number fields

Let \(\ell \) be a prime and let \(L/ \mathbb {Q}\) be a Galois number field with Galois group isomorphic to \( \mathbb {Z}/\ell \mathbb {Z}\). We show that the shape of L, see Definition 1.2, is either \(\frac{1}{2}\mathbb {A}_{\ell -1}\) or a fixed sub-lattice depending only on \(\ell \); such a dichotomy in the value of the shape only depends on the type of ramification of L. This work is motivated by a result of Bhargava and Shnidman, and a previous work of the first named author, on the shape of \( \mathbb {Z}/3 \mathbb {Z}\) number fields.     

Extension of the Set of Complex Hadamard Matrices of Size 8

A complex Hadamard matrix is defined as a matrix H which fulfills two conditions, \(|H_{j,k}|=1\) for all j and k and \(HH^{*}=N \mathbb {I}_N\) where \(\mathbb {I}_N\) is an identity matrix of size N. We explore the set of complex Hadamard matrices \(\mathcal {H}_N\) of size \(N=8\) and present two previously unknown structures: a one-parametric, non-affine family \(T_8^{(1)}\) of complex Hadamard matrices and a single symmetric and isolated matrix \(A_8^{(0)}\).     

A new generalization of Hermite’s reciprocity law

Given a partition \(\lambda \) of n, the Schur functor \({\mathbb {S}}_\lambda \) associates to any complex vector space V, a subspace \({\mathbb {S}}_\lambda (V)\) of \(V^{\otimes n}\). Hermite’s reciprocity law, in terms of the Schur functor, states that \({\mathbb {S}}_{(p)}\left( {\mathbb {S}}_{(q)}({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{(q)}\left( {\mathbb {S}}_{(p)}({\mathbb {C}}^2)\right) . \) We extend this identity to many other identities of the type \({\mathbb {S}}_{\lambda }\left( {\mathbb {S}}_{\delta }({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{\mu }\left( {\mathbb {S}}_{\epsilon }({\mathbb {C}}^2)\right) \).     

$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.     

Simultaneous diophantine approximation in the real and <Emphasis Type="BoldItalic">p</Emphasis>-adic fields with nonmonotonic error function<Superscript>*</Superscript>

In this paper, we show that if the volume sum \( \sum\nolimits_{h = 1}^\infty {{h^{n - 1}}{\Psi^t}(h)} \) converges for a function ψ (not necessarily monotonic), then the set of points \( \left( {x,{w_1}, \ldots, {w_{t - 1}}} \right) \in {\mathbb R} \times {{\mathbb Q}_{{p_1}}} \times \ldots \times {{\mathbb Q}_{{p_{t - 1}}}} \) that simultaneously satisfy the inequalities \( \left| {P(x)} \right| < \Psi (H) {\text{and}} {\left| {P\left( {{w_i}} \right)} \right|_{{p_i}}} < \Phi (H), 1 \leqslant i \leqslant t - 1 \), for infinitely many integer polynomials P has measure zero.     

$${\varvec{}}$$-Jorda homomorphisms ad pseudo $${\varvec{}}$$-Jorda homomorphisms o Baach algebras

Let \(n\in \mathbb {N}\), A and B be Banach algebras and let B be a right A-module. We say that a linear mapping \(\varphi :A\longrightarrow B\) is a pseudo n-Jordan homomorphism if there exists an element \(w\in A\) such that \(\varphi (a^nw)=\varphi (a)^n\cdot w\), for every \(a\in A\) and \(n\ge \) 2. In this paper, among other things, we show that under some conditions if a linear mapping \(\varphi \) is a (pseudo) n-Jordan homomorphism, then it is a (pseudo) \((n + 1)\)-Jordan homomorphism. Additionally, we investigate automatic continuity of surjective pseudo n-Jordan homomorphisms under some conditions.     

Orthogonal-gradings on $$H^*$$-algebras

We study set-gradings on proper \(H^*\)-algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary \(H^*\)-algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary \(H^*\)-algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative \(H^*\)-algebra, or a \({\mathbb Z}_2^3\)-grading of the complex octonions \({\mathbb O}\) or a non-group grading which is a refinement of the Peirce decomposition of \({\mathbb O}\) respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.     

Rank two aCM bundles on general determinantal quartic surfaces in $${\mathbb {P}^{3}}$$

Let \(F\subseteq {\mathbb {P}^{3}}\) be a smooth determinantal quartic surface which is general in the Nöther–Lefschetz sense. In the present paper we give a complete classification of locally free sheaves \({\mathcal E}\) of rank 2 on F such that \(h^1(F,{\mathcal E}(th))=0\) for \(t\in \mathbb {Z}\).     

New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access network for 2-dimensional image transmission. There is a one-to-one correspondence between an \((m, n, w, \lambda )\)-OOSPC and a \((\lambda +1)\)-(mn, w, 1) packing design admitting an automorphism group isomorphic to \(\mathbb {Z}_m\times \mathbb {Z}_n\). In 2010, Sawa gave a construction of an (m, n, 4, 2)-OOSPC from a one-factor of Köhler graph of \(\mathbb {Z}_m\times \mathbb {Z}_n\) which contains a unique element of order 2. In this paper, we study the existence of one-factor of Köhler graph of \(\mathbb {Z}_m\times \mathbb {Z}_n\) having three elements of order 2. It is proved that there is a one-factor in the Köhler graph of \(\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}\) relative to the Sylow 2-subgroup if there is an S-cyclic Steiner quadruple system of order 2p, where \(p\equiv 5\pmod {12}\) is a prime and \(1\le \epsilon ,\epsilon '\le 2\). Using this one-factor, we construct a strictly \(\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}\)-invariant regular \(G^*(p,2^{\epsilon +\epsilon '},4,3)\) relative to the Sylow 2-subgroup. By using the known S-cyclic SQS(2p) and a recursive construction for strictly \(\mathbb {Z}_{m}\times \mathbb {Z}_{n}\)-invariant regular G-designs, we construct more strictly \(\mathbb {Z}_{m}\times \mathbb {Z}_{n}\)-invariant 3-(mn, 4, 1) packing designs. Consequently, there is an optimal \((2^{\epsilon }m,2^{\epsilon '}n,4,2)\)-OOSPC for any \(\epsilon ,\epsilon '\in \{0,1,2\}\) with \(\epsilon +\epsilon '>0\) and an optimal (6m, 6n, 4, 2)-OOSPC where m, n are odd integers whose all prime divisors from the set \(\{p\equiv 5\pmod {12}:p\) is a prime, \(p<\)1,500,000}.     

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.     

Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

Let \(F(X,Y)=\sum \nolimits _{i=0}^sa_iX^{r_i}Y^{r-r_i}\in {\mathbb {Z}}[X,Y]\) be a form of degree \(r=r_s\ge 3\), irreducible over \({\mathbb {Q}}\) and having at most \(s+1\) non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality

$$\begin{aligned} |F(X,Y)|\le h \end{aligned}$$

is \(\ll s^2h^{2/r}(1+\log h^{1/r})\). They conjectured that \(s^2\) may be replaced by s. Let

$$\begin{aligned} \Psi = \max _{0\le i\le s} \max \left( \sum _{w=0}^{i-1} \frac{1}{r_i-r_w},\sum _{w= i+1}^{s}\frac{1}{r_w-r_i}\right) . \end{aligned}$$

Then we show that \(s^2\) may be replaced by \(\max (s\log ^3s, se^{\Psi })\). We also show that if \(|a_0|=|a_s|\) and \(|a_i|\le |a_0|\) for \(1\le i\le s-1\), then \(s^2\) may be replaced by \(s\log ^{3/2}s\). In particular, this is true if \(a_i\in \{-1,1\}\).

     

The primitive idempotents and weight distributions of irreducible constacyclic codes

Let \({\mathbb {F}}_q\) be a finite field with q elements such that \(l^v||(q^t-1)\) and \(\gcd (l,q(q-1))=1\), where l, t are primes and v is a positive integer. In this paper, we give all primitive idempotents in a ring \(\mathbb F_q[x]/\langle x^{l^m}-a\rangle \) for \(a\in {\mathbb {F}}_q^*\). Specially for \(t=2\), we give the weight distributions of all irreducible constacyclic codes and their dual codes of length \(l^m\) over \({\mathbb {F}}_q\).     

On the peripheral spectrum of positive elements

Let A be an ordered Banach algebra with a unit \(\mathbf{e}\) and a cone \(A^+\). An element p of A is said to be an order idempotent if \(p^2 = p\) and \(0 \le p\le \mathbf{e}\). An element \(a\in A^+\) is said to be irreducible if the relation \((\mathbf{e}-p)ap = 0\), where p is an order idempotent, implies \(p = 0\) or \(p = \mathbf{e}\). For an arbitrary element a of A the peripheral spectrum \(\sigma _\mathrm{per}(a)\) of a is the set \(\sigma _\mathrm{per}(a) = \{\lambda \in \sigma (a):|\lambda | = r(a)\}\), where \(\sigma (a)\) is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which \(\sigma _\mathrm{per}(a)\) contains or coincides with \(r(a)H_m\), where \(H_m\) is the group of all \(m^\mathrm{th}\) roots of unity, and the spectrum \(\sigma (a)\) is invariant under rotation by the angle \(\frac{2\pi }{m}\) for some \(m\in {\mathbb N}\), are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., \(\sigma _\mathrm{per}(a) = \{r(a)\}\), are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities \(0 \le b < a\) imply \(r(b) < r(a)\) are studied. The closedness of the center \(A_\mathbf{e}\), i.e., of the order ideal generated by \(\mathbf{e}\) in A, is proved.