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Let a , Q ∈ Q be given and consider the set G (a , Q) = { a Q i : i ∈ N } of terms of geometric progression with 0th term equal to a and the quotient Q. Let f ∈ Q (x , y) and V f be the set of finite values of f. We consider the problem of existence of a , Q ∈ Q such that G (a , Q) ⊂ V f . In the first part of the paper we describe certain classes of rational functions for which our problem has a positive solution. In the second, experimental, part of the paper we study the stated problem for the rational function f (x , y) = (y 2 − x 3) / x. We relate the problem to the existence of rational points on certain elliptic curves and present interesting numerical observations which allow us to state several questions and conjectures. [ABSTRACT FROM AUTHOR]

Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections" of a "network" corresponding to a fractal set. This leads to the definition of the topological Hausdorff dimension of fractals. Finer information might be obtained by considering the Hausdorff dimension of level sets of generic 1-Hölder- α functions, which has a stronger dependence on the geometry of the fractal, as displayed in our previous papers (Buczolich et al., 2022 [9,10]). In this paper, we extend our investigations to the lower and upper box-counting dimensions as well: while the former yields results highly resembling the ones about the Hausdorff dimension of level sets, the latter exhibits a different behavior. Instead of "finding narrow-cross sections", results related to upper box-counting dimension "measure" how much level sets can spread out on the fractal, and how widely the generic function can "oscillate" on it. Key differences are illustrated by giving estimates concerning the Sierpiński triangle. [ABSTRACT FROM AUTHOR]

This paper is a sequel to a previous paper of the authors in which the cohomology of semi-simple Leibniz algebras was computed by using spectral sequences. In the present paper we generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we establish the Leibniz analogue of a non-vanishing theorem of Dixmier for nilpotent Leibniz algebras. In addition, we prove a Fitting lemma for Leibniz bimodules [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of the Eberlein convolution of measures and introduce a reflected version of it. For functions we show that the reflected Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (reflected) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (reflected) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (reflected) Eberlein convolution. [ABSTRACT FROM AUTHOR]

Inspired by the 1995 paper of Baake–Grimm–Pisani, we aim to explain the empirical observation that the distribution of Lee–Yang zeros corresponding to a one-dimensional Ising model appears to follow the gap labelling theorem. This follows by combining two main ingredients: first, the relation between the transfer matrix formalism for the 1D Ising model and an ostensibly unrelated matrix formalism generating the Szegő recursion for orthogonal polynomials on the unit circle, and second, the gap labelling theorem for CMV matrices. [ABSTRACT FROM AUTHOR]

In this paper, we show that all odd-point correlation functions of the balanced Rudin–Shapiro sequence vanish and that all even-point correlation functions depend only on a single number, which holds for any weighted correlation function as well. For the four-point correlation functions, we provide a more detailed exposition which reveals some arithmetic structures and symmetries. In particular, we show that one can obtain the autocorrelation coefficients of its topological factor with maximal pure point spectrum among them. [ABSTRACT FROM AUTHOR]

In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that are counterexamples to the Hasse Principle. [ABSTRACT FROM AUTHOR]

Many interesting combinatorial sequences, such as Apéry numbers and Franel numbers, enjoy the so-called Lucas property modulo almost all primes p. Modulo prime powers p r such sequences have a more complicated behaviour which can be described by matrix versions of the Lucas property called p -linear schemes. They are generalizations of finite p -automata. In this paper we construct such p -linear schemes and give upper bounds for the number of states which, for fixed r , do not depend on p. [ABSTRACT FROM AUTHOR]

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let Γ be a chart, and we denote by Γ m the union of all the edges of label m. A chart Γ is of type (4 , 3) if there exists a label m such that w (Γ) = 7 , w (Γ m ∩ Γ m + 1) = 4 , w (Γ m + 1 ∩ Γ m + 2) = 3 where w (G) is the number of white vertices in G. In this paper, we prove that there is no minimal chart of type (4 , 3). [ABSTRACT FROM AUTHOR]

Given two coprime integers p ≥ 2 and q ≥ 3 , let D p ⊂ [ 0 , 1) consist of all rational numbers which have a finite p -ary expansion, and let K (q , A) = ∑ i = 1 ∞ d i q i : d i ∈ A ∀ i ∈ N , where A ⊂ 0 , 1 , ... , q − 1 with cardinality 1 < # A < q. In 2021 Schleischitz showed that # (D p ∩ K (q , A)) < + ∞. In this paper we show that for any r ∈ Q and for any α ∈ R , # ((r D p + α) ∩ K (q , A)) < + ∞. [ABSTRACT FROM AUTHOR]

Although structural maps such as subductions and inductions appear naturally in diffeology, one of the challenges is providing suitable analogues for submersions, immersions, and étale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we consider diffeological or plotwise versions of submersions, immersions, and étale maps as an adaptation of these maps to diffeology by a nonlinear approach. We study their diffeological properties from different aspects in a systematic fashion with respect to the germs of plots. In order to characterize the considered maps from their linear behaviors, we introduce a class of diffeological spaces, so-called diffeological étale manifolds, which not only contains the usual manifolds but also includes irrational tori. We state and prove versions of the rank and implicit function theorems, as well as the fundamental theorem on flows in this class. As an application, we use the results of this work to facilitate the computations of the internal tangent spaces and diffeological dimensions in a few interesting cases. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogeneous spaces of algebraic groups, which we call Frobenius–Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius–Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann–Weil–Thurston principle for Frobenius–Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius–Ehresmann structure are completely determined by their monodromy crystals. [ABSTRACT FROM AUTHOR]

Let X denote a compact set in the complex plane ℂ and z n ≔ { z n , j } j = 0 n be a family of points that lie on X. Then the norm ‖ A z n ‖ of the operator A z n that maps the Banach space C (X) to the space Π n of polynomials of degree ≤ n , defined by A z n f = p n , where p n ∈ Π n is the polynomial that interpolates f ∈ C (X) at z n , is called the Lebesgue constant of the family z n. While the bulk of this paper is on the study of Lagrange polynomial interpolation at the Fejér points z n ∗ ≔ { z n , j ∗ } j = 0 n , and their proper adjustment, that lie on an open arc X = γ in ℂ which does not cross itself, the general spirit of our presentation carries over to other settings of polynomial representations of functions on a simple closed piece-wise smooth curve X = Γ , which is the boundary of a Jordan domain D ⊂ ℂ ; and instead of polynomial interpolation, A z n is replaced by the linear operator A F n , that maps the (generalized) Hardy space H p (D) , for p ≥ 1 , of analytic functions f in D with non-tangential limit f ∗ ∈ L p (Γ) , to the n th partial sum S n (⋅ ; f) of the Faber series representation of f in D. The importance of the Lebesgue constant ‖ A F n ‖ is that if P n ∗ ∈ Π n is the best polynomial approximant of f in H p (D) , with approximation error ϵ n (f ; Γ) ≔ ‖ f − P n ∗ (⋅ ; f) ‖ H p , then S n (⋅ ; f) can be used to replace P n ∗ with error of approximation ‖ f − S n (⋅ ; f) ‖ H p ≤ (1 + ‖ A F n ‖) ϵ n (f ; Γ). This consideration, along with the discussion of related problems and results inspired by the work of Prof. J. Korevaar, will be presented in the final section of the present paper. For Lagrange polynomial interpolation on open arcs X = γ in ℂ , it is well-known that the Lebesgue constant for the family of Chebyshev points x n ≔ { x n , j } j = 0 n on [ − 1 , 1 ] ⊂ R has growth order of O (log (n)). The same growth order was shown in Zhong and Zhu (1995) for the Lebesgue constant of the family z n ∗∗ ≔ { z n , j ∗ ∗ } j = 0 n of some properly adjusted Fejér points on a rectifiable smooth open arc γ ⊂ ℂ. On the other hand, in our recent work Chui and Zhong (2021), it was observed that if the smooth open arc γ is replaced by an L -shape arc γ 0 ⊂ ℂ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz–Zygmund inequalities are no longer valid for the family of Fejér points z n ∗ ≔ { z n , j ∗ } j = 0 n on γ , and that the rate of growth for the corresponding Lebesgue constant L z n ∗ is as fast as c log 2 (n) for some constant c > 0. The main objective of the present paper is 3-fold: firstly, it will be shown that for the special case of the L -shape arc γ 0 consisting of two line segments of the same length that meet at the angle of π / 2 , the growth rate of the Lebesgue constant L z n ∗ is at least as fast as O (log 2 (n)) , with lim sup L z n ∗ log 2 (n) = ∞ ; secondly, the corresponding (modified) Marcinkiewicz–Zygmund inequalities fail to hold; and thirdly, a proper adjustment z n ∗ ∗ ≔ { z n , j ∗ ∗ } j = 0 n of the Fejér points on γ will be described to assure the growth rate of L z n ∗ ∗ to be exactly O (log 2 (n)). [ABSTRACT FROM AUTHOR]

This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the q -Askey scheme and uniform ways of parametrizing the families. For a system of polynomials p n (x) in the q -Askey scheme satisfying L p n = h n p n with L a second order q -difference operator the q -Zhedanov algebra is the algebra generated by operators L and X (multiplication by x). It has two relations in which essentially five coefficients occur. Vanishing of one or more of the coefficients corresponds to a subfamily or limit family of the Askey–Wilson polynomials. An arrow from one family to another means that in the latter family one more coefficient vanishes. This yields the q -Zhedanov scheme given in this paper. The q -hypergeometric expression of p n (x) can be interpreted as an expansion of p n (x) in terms of certain Newton polynomials. In our previous paper (Contemporary Math. 780) we used Verde-Star's clean parametrization of such expansions and we obtained a q -Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators L and X on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the q -Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the q -Verde-Star scheme and the q -Zhedanov scheme, which are discussed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the large deviation principle (LDP) for two types (Type I and Type II) of multiplicative Ising models. For Types I and II, the explicit formulas for the free energy functions and the associated rate functions are derived. Furthermore, we prove that those free energy functions are differentiable, which indicates that both systems are characterized by a lack of phase transition phenomena. [ABSTRACT FROM AUTHOR]

In this paper we prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values. [ABSTRACT FROM AUTHOR]

In this paper, among other things, we explicit a G δ -dense set of Liouville numbers, for which the triple power tower of any of its elements is a transcendental number. [ABSTRACT FROM AUTHOR]

The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces H [ X (w) ] built upon translation-invariant Banach function spaces X with weights w such that w ∈ X and w − 1 ∈ X ′ , where X ′ is the associate space of X. We prove that if X is separable, then H [ X (w) ] has the BCAP with the approximation constant M (H [ X (w) ]) ≤ 2. Moreover, if X is reflexive, then H [ X (w) ] has the BCAP and the DCAP with the approximation constants M (H [ X (w) ]) ≤ 2 and M ∗ (H [ X (w) ]) ≤ 2 , respectively. In the case of classical weighted Hardy space H p (w) = H [ L p (w) ] with 1 < p < ∞ , one has a sharper result: M (H p (w)) ≤ 2 | 1 − 2 / p | and M ∗ (H p (w)) ≤ 2 | 1 − 2 / p | . [ABSTRACT FROM AUTHOR]

Let (u j) be a decreasing sequence of psh functions in the domain of definition D of the Monge–Ampère operator on a domain Ω of ℂ n such that u = inf j u j is plurisubharmonic on Ω. In this paper we are interested in the problem of finding conditions insuring that lim j → + ∞ ∫ φ (d d c u j) n = ∫ φ NP (d d c u) n for any continuous function on Ω with compact support, where NP (d d c u) n is the nonpolar part of (d d c u) n , and conditions implying that u ∈ D. For u j = max (u , − j) these conditions imply also that lim j → + ∞ ∫ K (d d c u j) n = ∫ K NP (d d c u) n for any compact set K ⊂ { u > − ∞ }. [ABSTRACT FROM AUTHOR]

In a recent paper (Temme, 2021) new asymptotic expansions are given for the Kummer functions M (a , b , z) and U (a , b + 1 , z) for large positive values of a and b , with z fixed and special attention for the case a ∼ b. In this paper we extend the approach and also accept large values of z. The new expansions are valid when at least one of the parameters a , b , or z is large. We provide numerical tables to show the performance of the expansions. [ABSTRACT FROM AUTHOR]

We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Sections 2 and 3, using the technique together with Mahler's method, we can reduce the algebraic independence of the infinite set mentioned above to the linear independence of certain rational functions modulo the rational function field of many variables. The latter one is solved by the discussions involving a certain valuation and a generic point in Sections 3 and 4. [ABSTRACT FROM AUTHOR]

In Schmidt (1960), Schmidt studied a quantitative type of Khintchine–Groshev theorem for general (higher) dimensions. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition (Alam et al., 2021). In this paper, we generalize this new approach to S -arithmetic spaces and obtain a quantitative version of an S -arithmetic Khintchine–Groshev theorem. During the process, we consider a new, but still natural S -arithmetic analog of Diophantine approximation, which is different from the one formerly established (see Kleinbock and Tomanov, 2007). Hence for the sake of completeness, we also deal with the convergent case of the Khintchine–Groshev theorem, based on this new generalization. [ABSTRACT FROM AUTHOR]

For 0 < s < 1 , let { z n } be a sequence in the open unit disk such that ∑ n (1 − | z n | 2) s δ z n is an s -Carleson measure. In this paper, we consider the connections between this s -Carleson measure and the theory of Möbius invariant F(p, p-2, s) spaces by the Volterra type operator, the reciprocal of a Blaschke product, and second order complex differential equations having a prescribed zero sequence. [ABSTRACT FROM AUTHOR]

In this paper, we provide several characterisations for uniform amenability concerning a family of finitely generated groups. More precisely, we show that the Hulanicki–Reiter condition for uniform amenability can be weakened in several directions, including cardinalities of supports and certain operator norms. [ABSTRACT FROM AUTHOR]

Let α > 1 be irrational and of finite type, β ∈ R. In this paper, it is proved that for R ⩾ 13 and any fixed c ∈ (1 , c R) , there exist infinitely many primes in the intersection of Beatty sequence B α , β and ⌊ n c ⌋ , where c R is an explicit constant depending on R herein, n is a natural number with at most R prime factors, counted with multiplicity. [ABSTRACT FROM AUTHOR]

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calderón–Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fréchet–Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method. [ABSTRACT FROM AUTHOR]

In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L (V ∗) = L (V) ∗ , via representations of the form L (V) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , ... , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , ... , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L. [ABSTRACT FROM AUTHOR]

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a , generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations. The recipe involves folding and contracting a -invariant and θ -twisted symmetric classical dynamical r -matrices along an involutive automorphism θ. In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann's classical dynamical r -matrices which are a -invariant and θ -twisted. The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories. [ABSTRACT FROM AUTHOR]

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal I meets a unique largest nilpotent orbit O I in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals I such that O I = O for a fixed nilpotent orbit O. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan–Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed. [ABSTRACT FROM AUTHOR]

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions (i , j) with i + j ≥ 0 and small negative jumps, i.e., i , j ≥ − 1. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow us to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory. [ABSTRACT FROM AUTHOR]

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers in single queues and open Kelly–Whittle networks of queues. For positive recurrent birth–death processes we show that the sequence of sample maxima is stochastically compact. For transient birth–death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact. We show that the Markov chain recording the total number of customers in a Kelly–Whittle network is a birth–death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly–Whittle networks of queues. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the stationary workload of queues with heavy-tailed (regularly varying) characteristics. We adopt a transform perspective to illuminate a close connection between the tail asymptotics and heavy-traffic limit in infinite-variance scenarios. This serves as a tribute to some of the pioneering results of J.W. Cohen in this domain. We specifically demonstrate that reduced-load equivalence properties established for the tail asymptotics of the workload naturally extend to the heavy-traffic limit. [ABSTRACT FROM AUTHOR]

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider. [ABSTRACT FROM AUTHOR]

In 1987, J.W. Cohen analyzed the so-called Serve the Longest Queue (SLQ) queueing system, where a single server attends two non-symmetric M / G / 1 -type queues, exercising a non-preemptive priority switching policy. Cohen further analyzed in 1998 a non-symmetric 2-queue Markovian system, where newly arriving customers follow the Join the Shortest Queue (JSQ) discipline. The current paper generalizes and extends Cohen's works by studying a combined JSQ–SLQ model, and by broadening the scope of analysis to a non-symmetric 3-queue system, where arriving customers follow the JSQ strategy and a single server exercises the preemptive priority SLQ discipline. The system states' multi-dimensional probability distribution function is derived while applying a non-conventional representation of the underlying process's state-space. The analysis combines both Probability Generating Functions and Matrix Geometric methodologies. It is shown that the joint JSQ–SLQ operating policy achieves extremely well the goal of balancing between queue sizes. This is emphasized when calculating the Gini Index associated with the differences between mean queue sizes: the value of the coefficient is close to zero. Extensive numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper we consider the notions of binomial thinning, binomial mixing, their generalizations, certain interplay between them, associated limit theorems and provide various examples. [ABSTRACT FROM AUTHOR]

Building on the works of S. Bochner on equivalence of modular relation with functional equation associated to the Dirichlet series, K. Chandrasekharan and R. Narasimhan obtained new equivalences between the functional equation and some arithmetical identities. Sister Ann M. Heath considered the functional equation in the Hawkins and Knopp context and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we use techniques of Chandrasekharan and Narasimhan to prove results analogous to those of Sister Ann M. Heath. Specifically, we establish equivalence of two arithmetical identities with a functional equation associated with automorphic integrals involving log-polynomial-period functions on the discrete Hecke groups. [ABSTRACT FROM AUTHOR]

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function. [ABSTRACT FROM AUTHOR]

In an author's previous work, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is shown under the same assumptions that were used in the author's previous work to prove analogous decomposition results for analytic immersive curves. Together with the results obtained there, this completely classifies 1-dimensional analytic objects (analytic curves and analytic 1-submanifolds) w.r.t. their symmetry under a given regular and separately analytic Lie group action. [ABSTRACT FROM AUTHOR]

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m (q) × GU n (q) on M m × n (q 2) (which is the analog of the singular value decomposition). The proof involves Kronecker's theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields. [ABSTRACT FROM AUTHOR]