On the Hawkes Graphs of Finite Groups

We study the properties and applications of the directed graph, introduced by Hawkes in 1968, of a finite group \( G \). The vertex set of \( \Gamma_{H}(G) \) coincides with \( \pi(G) \) and \( (p,q) \) is an edge if and only if \( q\in\pi(G/O_{p^{\prime},p}(G)) \). In the language of properties of this graph we obtain commutation conditions for all \( p \)-elements with all \( r \)-elements of \( G \), where \( p \) and \( r \) are distinct primes. We estimate the nilpotence length of a solvable finite group in terms of subgraphs of its Hawkes graph. Given an integer \( n>1 \), we find conditions for reconstructing the Hawkes graph of a finite group \( G \) from the Hawkes graphs of its \( n \) pairwise nonconjugate maximal subgroups. Using these results, we obtain some new tests for the membership of a solvable finite group in the well-known saturated formations.

     

Defining Relations for the Carpet Subgroups of Chevalley Groups over Fields

The carpet subgroups admitting a Bruhat decomposition and different from Chevalley groups are exhausted by the groups lying between the Chevalley groups of type \( B_{l} \), \( C_{l} \), \( F_{4} \), or \( G_{2} \) over various imperfect fields of exceptional characteristic 2 or 3, the larger of which is an algebraic extension of the smaller field. Moreover, as regards the types \( B_{l} \) and \( C_{l} \), these subgroups are parametrized by the pairs of additive subgroups one of which may fail to be a field and, for the type \( B_{2} \), even both additive subgroups may fail to be fields. In this paper for the carpet subgroups admitting a Bruhat decomposition we present the relations similar to those well known for Chevalley groups over fields.

     

On the Fourier coefficients of certain Hilbert modular forms

We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).

     

On the Coincidence of the Classes of Finite Groups $ E_{\pi_{x}} $ and $ D_{\pi_{x}} $

Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{??} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.

     

On Categoricity Spectra for Locally Finite Graphs

Under study is the algorithmic complexity of isomorphisms between computable copies of locally finite graphs \( G \) (undirected graphs whose every vertex has finite degree). We obtain the following results: If \( G \) has only finitely many components then \( G \) is \( {\mathbf{d}} \)-computably categorical for every Turing degree \( {\mathbf{d}} \) from the class \( PA({\mathbf{0}}^{\prime}) \). If \( G \) has infinitely many components then \( G \) is \( {\mathbf{0}}^{\prime\prime} \)-computably categorical. We exhibit a series of examples showing that the obtained bounds are sharp.

     

The Defect of a Cauchy Type Problem for Linear Equations with Several Riemann–Liouville Derivatives

We consider the existence and uniqueness of solutions to initial value problems for general linear nonhomogeneous equations with several Riemann–Liouville fractional derivatives in Banach spaces. Considering the equation solved for the highest fractional derivative \( D^{\alpha}_{t} \), we introduce the concept of the defect \( m^{*} \) of a Cauchy type problem which determines the number of the zero initial conditions \( D^{\alpha-m+k}_{t}z(0)=0 \), \( k=0,1,\dots,m^{*}-1 \), necessary for the existence of the finite limits \( D^{\alpha-m+k}_{t}z(t) \) as \( t\to 0+ \) for all \( k=0,1,\dots,m-1 \). We show that the defect \( m^{*} \) is uniquely determined by the set of orders of the Riemann–Liouville fractional derivatives in the equation. Also we prove the unique solvability of the incomplete Cauchy problem \( D^{\alpha-m+k}_{t}z(0)=z_{k} \), \( k=m^{*},m^{*}+1,\dots,m-1 \), for the equation with bounded operator coefficients solved for the highest Riemann–Liouville derivative. The obtained result allowed us to investigate initial problems for a linear nonhomogeneous equation with a degenerate operator at the highest fractional derivative, provided that the operator at the second highest order derivative is 0-bounded with respect to this operator, while the cases are distinguished that the fractional part of the order of the second derivative coincides or does not coincide with the fractional part of the order of the highest derivative. The results for equations in Banach spaces are used for the study of initial boundary value problems for a class of equations with several Riemann–Liouville time derivatives and polynomials in a selfadjoint elliptic differential operator of spatial variables.

     

Proportionally modular numerical semigroups generated by arithmetic progressions

A numerical semigroup is a submonoid of \({{\mathbb {Z}}}_{\ge 0}\) whose complement in \({{\mathbb {Z}}}_{\ge 0}\) is finite. For any set of positive integers a, b, c, the numerical semigroup S(a, b, c) formed by the set of solutions of the inequality \(ax \bmod {b} \le cx\) is said to be proportionally modular. For any interval \([\alpha ,\beta ]\), \(S\big ([\alpha ,\beta ]\big )\) is the submonoid of \({{\mathbb {Z}}}_{\ge 0}\) obtained by intersecting the submonoid of \({{\mathbb {Q}}}_{\ge 0}\) generated by \([\alpha ,\beta ]\) with \({{\mathbb {Z}}}_{\ge 0}\). For the numerical semigroup S generated by a given arithmetic progression, we characterize a, b, c and \(\alpha ,\beta \) such that both S(a, b, c) and \(S\big ([\alpha ,\beta ]\big )\) equal S.

     

Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let \(p(\cdot ):\ {{\mathbb {R}}}^n\rightarrow (0,\infty ]\) be a variable exponent function satisfying the globally log-Hölder continuous condition, \(q\in (0,\infty ]\) and A be a general expansive matrix on \({\mathbb {R}}^n\). Let \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) by means of the Littlewood–Paley g-function or the Littlewood–Paley \(g_\lambda ^*\)-function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space \(L^{p(\cdot ),q}({\mathbb {R}}^n)\). Moreover, the finite atomic characterization of \(H_A^{p(\cdot ),q}({{\mathbb {R}}}^n)\) is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from \(H^{p(\cdot ),q}_A({\mathbb {R}}^n)\) to \(L^{p(\cdot ),q}({\mathbb {R}}^n)\) and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.

     

Extended Relativistic Toda Lattice,L-Orthogonal Polynomials and Associated Lax Pair

When a measure \(\varPsi(x)\) on the real line is subjected to the modification \(d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)\), then the coefficients of the recurrence relation of the orthogonal polynomials in \(x\) with respect to the measure \(\varPsi^{(t)}(x)\) are known to satisfy the so-called Toda lattice formulas as functions of \(t\). In this paper we consider a modification of the form \(e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}\) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either \(\mathfrak{p}=0\) or \(\mathfrak{q}=0\). However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.

     

Finite Groups with Generalized Subnormal Schmidt Subgroups

Given a prime \( p \) and a partition \( \sigma=\{\{p\},\{p\}^{\prime}\} \) of the set of all primes, we describe the structure of the nonnilpotent finite groups whose every Schmidt subgroup is \( \sigma \)-subnormal.

     

A Menon-type identity with Dirichlet characters in residually finite Dedekind domains

This paper studies Menon–Sury’s identity in a general case, i.e., the Menon–Sury’s identity involving Dirichlet characters in residually finite Dedekind domains. By using the filtration of the ring \({\mathfrak {D}}/{\mathfrak {n}}\) and its unit group \(U({\mathfrak {D}}/{\mathfrak {n}})\), we explicitly compute the following two summations:

$$\begin{aligned} \sum _{\begin{array}{c} a\in U({\mathfrak {D}}/{\mathfrak {n}}) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a-1,b_1, b_2, \ldots , b_r \rangle +{\mathfrak {n}})\chi (a) \end{aligned}$$

and

$$\begin{aligned} \sum _{\begin{array}{c} a_{1},\ldots , a_{s}\in U({\mathfrak {D}}/{\mathfrak {n}}) \\ b_1, \ldots , b_r\in {\mathfrak {D}}/{\mathfrak {n}} \end{array}} N(\langle a_{1}-1,\ldots , a_{s}-1,b_1, b_2, \ldots , b_r \rangle +{\mathfrak {n}})\chi _{1}(a_1) \cdots \chi _{s}(a_s), \end{aligned}$$

where \({\mathfrak {D}}\) is a residually finite Dedekind domain and \({\mathfrak {n}}\) is a nonzero ideal of \({\mathfrak {D}}\), \(N({\mathfrak {n}})\) is the cardinality of quotient ring \({\mathfrak {D}}/{\mathfrak {n}}\), \(\chi _{i}~(1\le i\le s)\) are Dirichlet characters mod \({\mathfrak {n}}\) with conductor \({\mathfrak {d}}_i\).

     

Functional Matrix Multipliers for Parseval Gabor Multi-frame Generators

We consider the problem of characterizing the bounded linear operator multipliers on \(L^{2}(\mathbb{R})\) that map Gabor frame generators to Gabor frame generators. We prove that a functional matrix \(M(t)=[f_{ij}(t)]_{m \times m}\) (where \(f_{ij}\in L^{\infty}(\mathbb{R})\)) is a multiplier for Parseval Gabor multi-frame generators with parameters \(a, b >0\) if and only if \(M(t)\) is unitary and \(M^{*}(t)M(t+\frac{1}{b})= \lambda(t)I\) for some unimodular \(a\)-periodic function \(\lambda(t)\). As a special case (\(m =1\)) this recovers the characterization of functional multipliers for Parseval Gabor frames with single function generators.

     

Non-separable Lattices,Gabor Orthonormal Bases and Tilings

Let \(K\subset {\mathbb {R}}^d\) be a bounded set with positive Lebesgue measure. Let \(\Lambda =M({\mathbb {Z}}^{2d})\) be a lattice in \({\mathbb {R}}^{2d}\) with density dens\((\Lambda )=1\). It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis for \(L^2({\mathbb {R}}^d)\) if and only if K tiles both by \(A({\mathbb {Z}}^d)\) and \(B^{-t}({\mathbb {Z}}^d)\). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis, then K can be written as a finite union of fundamental domains of \(A({{\mathbb {Z}}}^d)\) and at the same time, as a finite union of fundamental domains of \(B^{-t}({{\mathbb {Z}}}^d)\). If \(A^tB\) is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domain is also possible. We also provide a constructive way for forming a Gabor window function for a given upper triangular lattice. Our study is related to a Fuglede’s type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.

     

Two-valued number pyramids

Number pyramids are common in elementary school mathematics. Trying to express the value of the top block in terms of the values at the base leads to the binomial coefficients. It also seems natural to ask for the maximal number of odd numbers in a number pyramid of a given size. The answer is easy to state, but the proof is nontrivial: A \(k\) step number pyramid can have at most \(\left\lfloor\frac{k(k+1)+1}{3}\right\rfloor\) odd numbers, which equals two thirds of the number of blocks rounded to the nearest integer. All maximal and almost maximal solutions are given explicitly. To this end, we rephrase the question in terms of colored tilings. In the outlook we present relations to other—mostly geometric—subjects and problems.

     

The Dual Kaczmarz Algorithm

The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector \(x\) in a (separable) Hilbert space from the inner-products \(\{\langle x, \phi _{n} \rangle \}\). The Kaczmarz algorithm defines a sequence of approximations from the sequence \(\{\langle x, \phi _{n} \rangle \}\); these approximations only converge to \(x\) when \(\{\phi _{n}\}\) is effective. We dualize the Kaczmarz algorithm so that \(x\) can be obtained from \(\{\langle x, \phi _{n} \rangle \}\) by using a second sequence \(\{\psi _{n}\}\) in the reconstruction. This allows for the recovery of \(x\) even when the sequence \(\{\phi _{n}\}\) is not effective; in particular, our dualization yields a reconstruction when the sequence \(\{\phi _{n}\}\) is almost effective. We also obtain some partial results characterizing when the sequence of approximations from \(\{\langle x, \phi _{n} \rangle \}\) using \(\{\psi _{n}\}\) converges to \(x\), in which case \(\{(\phi _{n}, \psi _{n})\}\) is called an effective pair.

     

Weighted $$L^2$$ L 2 -norms of Gegenbauer polynomials

We study integrals of the form

$$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$

where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).

     

Analysis of a Free Boundary Problem Modeling the Growth of Spherically Symmetric Tumors with Angiogenesis

This paper is concerned with a free boundary problem modeling the growth of a spherically symmetric tumor with angiogenesis. The unknown nutrient concentration \(\sigma =\sigma (r,t)\) occupies the unknown tumor region \(r< R(t)\) and satisfies a nonlinear reaction diffusion equation, and the unknown tumor radius \(R=R(t)\) satisfies a nonlinear integro-differential equation. Unlike existing literatures on this topic where Dirichlet boundary condition for \(\sigma \) is imposed, in this paper the model uses the Robin boundary condition for \(\sigma \). We prove existence and uniqueness of a global in-time classical solution (\(\sigma (r,t),R(t)\)) for arbitrary \(c>0\) and establish asymptotic stability of the unique stationary solution (\(\sigma _{s}(r),R_{s}\)) for sufficiently small \(c\), where \(c\) is a positive constant reflecting the ratio between nutrient diffusion scale and the tumor cell-doubling scale.