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During the Nazi period in Germany, an attempt was made to discern a kind of mathematics that was German as distinct from other ethnic or “racial” types of mathematics: a “Deutsche Mathematik.” While not denying the universal validity of all mathematical truths, such a “German” mathematics stressed ideology in terms of research and pedagogical styles. Because mathematics was nearly independent of anything material, it was—for the “Deutsche Mathematiker”—especially amenable to the Nazi argument that different racial psychological types exhibit different racial characters and modes of thought. This paper is a brief examination of the nature and intellectual content of “Deutsche Mathematik.”

In this paper we obtain the Plancherel formula for the spaces of L 2 -sections of line bundles over the complex projective hyperboloids G=H withGD U.p;qIC/ andHD U.1IC/ U.p 1;qIC/. The Plancherel formula is given in an explicit form by means of spherical distributions associated with a character of the subgroupH . We obtain the Plancherel formula by a special method which is also suitable for other problems, for example, for quantization in the spirit of Berezin. © 2000 Editions scientifiques et medicales Elsevier SAS Keywords: Plancherel formula, Spherical distributions, Complex hyperbolic spaces In this paper we obtain the Plancherel formula for the spaces of L 2 -sections of line bundles over the complex projective hyperboloids G=H with GD U.p;qIC/ and H D U.1IC/ U.p 1;qIC/, i.e. we present the decomposition of L 2 into irreducible represen- tations of the group G of class . In order to leave aside the well-known case of a hyperboloid with compact stabilizer subgroup, see (14), we assumep>1 ;q >0. The Plancherel formula is given in an explicit form by means of spherical distributions associated with a character of the subgroup H. We obtain the Plancherel formula by Molchanov's method, see (9). Namely, we follow the detailed scheme in (1), Sections 4, 7. This method deals with the spectral resolution of the radial part of the Laplace operator. The essential step is setting the boundary conditions at certain special points. Those conditions are prescribed by the behaviour of spherical distributions. Finally, it is necessary to perform various analytic continuations. This method is also suitable for other problems, for example, for quantization in the spirit of Berezin, namely, for the decomposition of the Berezin form. It is therefore why this method has to be preferred to the existing methods, described in (3). We use our results from (13). There we define -spherical distributions, study their asymptotic behaviour and express them by means of hypergeometric functions. We describe the irreducible unitary representations of the group G ,o f class associated with an isotropic cone. We give constructions for the Fourier and Poisson transform, define intertwining operators and diagonalize them. Some of those results are presented in Section 1.

The converse of the Borel–Cantelli Lemma states that if { A i } i = 1 ∞ is a sequence of independent events such that ∑ P ( A i ) = ∞ , then almost surely infinitely many of these events will occur. Erdős and Renyi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them. Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience.

We study approximation of multivariate periodic functions from Besov and Triebel–Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the L q -norm for functions from the Besov spaces B p , θ s ( T d ) and the Triebel–Lizorkin spaces F p , θ s ( T d ) for all s > 0 and admissible 1 ≤ p , θ ≤ ∞ as well as provide analogues of the Littlewood–Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.

Support vector machines regression (SVMR) is an important part of statistical learning theory. The main difference between SVMR and the classical least squares regression (LSR) is that SVMR uses the ϵ-insensitive loss rather than quadratic loss to measure the empirical error. In this paper, we consider SVMR method in the field of functional data analysis under the framework of reproducing kernel Hilbert spaces. The main tool used in our theoretical analysis is the concentration inequalities for suprema of some appropriate empirical processes. As a result, we establish explicit convergence rates of the prediction risk for SVMR, which coincide with the minimax lower bound obtained recently in literature for LSR.

In this paper, weighted extra-weak and weak type inequalities have been characterized for the one-sided Hardy–Littlewood maximal function on the plane. We have addressed conditions on pair of weights for which the dyadic one-sided maximal function on higher dimension is locally integrable. In the process, we characterize weights for which the one-sided Hardy-Littlewood maximal function satisfies restricted weak type inequalities on the plane, thus extending the result of Kerman and Torchinsky to the one-sided Hardy-Littlewood maximal function.

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′ ′ + P ( z ) f = 0 , where P ( z ) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille’s 1969 book Lectures on Ordinary Differential Equations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille’s book.

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 .

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 Calder\'{o}n-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 Fr\'echet--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., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

Fa\`a di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Fa\`a di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions $F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to \mathbb{R}^N$ in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Fa\`a di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series., Comment: 38 pages, 11 figures

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.

The groups mentioned in the title are certain matrix groups of infinite size over a finite field $\mathbb F_q$. They are built from finite classical groups and at the same time they are similar to reductive $p$-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. We examine first the group $\mathbb{GLB}$, a topological completion of the inductive limit group $\varinjlim GL(n, \mathbb F_q)$. As was shown by Gorin, Kerov, and Vershik [arXiv:1209.4945], the traceable factor representations of $\mathbb{GLB}$ admit a complete classification, achieved in terms of harmonic functions on the Young graph $\mathbb Y$. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with a deformed version of harmonic functions on $\mathbb Y$. Here the deformation means that the edges of $\mathbb Y$ are endowed with certain formal multiplicities coming from the simplest version of Pieri rule (multiplication by the first power sum $p_1$) for the Hall-Littlewood (HL) symmetric functions with parameter $t:=q^{-1}$. This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. The latter are still related to the HL functions, but the novelty is that now the formal edge multiplicities come from the multiplication by $p_2$ (not $p_1$) and the HL parameter $t$ turns out to be negative (as in Ennola's duality)., 46 pages; v2: minor modifications, shortened abstract, journal version, to appear in Adv. Math

In this paper we study the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. This is achieved by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of uncertainty principle for measures on the complex sphere which subsumes certain classical results about pluriharmonic measures., Submitted

Assume E is an imaginary quadratic field and O is its ring of integers. For each positive integer m, let I m be the free Hermitian lattice of rank m over O having an orthonormal basis. For each positive integer n, let S O ( n ) be the set of all Hermitian lattices of rank n over O that can be represented by some I m . Denote by g O ( n ) the smallest positive integer g such that each Hermitian lattice in S O ( n ) can be represented by I g . In this paper, we shall provide an explicit upper bound for g O ( n ) for all imaginary quadratic fields E and all positive integers n.

In this paper, we are mainly concerned with the following integral equations: (0.1) u ( x ) = C n ∫ R n ln ⁡ ( 1 | x − y | ) f ( y , u ( y ) ) d y + γ , x ∈ R n , where n ≥ 2 , γ ∈ R , u ∈ C ( R n ) and f ( x , u ) may change signs and satisfies some assumptions. By using the method of scaling spheres developed by Dai and Qin in [14] , we first derive nonexistence of positive solutions to the above IEs under some assumptions. Then, based on the equivalence between the above IEs and the following 2D PDEs: (0.2) − Δ u ( x ) = f ( x , u ) , x ∈ R 2 , we also obtain nonexistence of positive solutions to the 2D PDEs under some assumptions. One should note that there are no growth conditions on u and hence f ( x , u ) can grow exponentially (or even faster) on u.

This paper considers backward stochastic differential equations (BSDEs for short) on the infinite horizon under consistent nonlinear expectations which are dominated by consistent sublinear expectations. We derive the existence and uniqueness result by discretization and approximation method. Besides, we show the existence of the solution of Markovian ergodic BSDEs under G ˜ -expectations and state some applications.

This paper is devoted to the study of existence of solutions to some integro-differential problems. We establish the existence of solutions by employing a singular perturbations method as a natural tool. The perturbed problems are classical, nonlocal, non-linear, hyperbolic and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. The existence of a solution to the nonlocal hyperbolic problem can be ensured using the Schauder fixed point theorem.

In this paper we study pseudo-differential operators with operator valued symbols on the Similitude group, SIM ( 2 ) . We characterize the unique infinite-dimensional, unitary, irreducible representations of SIM ( 2 ) and using that the Fourier inversion formula is given. The operator valued Wigner transform and the Weyl transform are also studied here. Furthermore we also obtain a necessary and sufficient condition on the symbols for which the pseudo-differential operators SIM ( 2 ) are Hilbert-Schmidt and trace class operators. Finally, we also give the trace formula for these operators.

The goal of this paper is to establish the relation between the notion of sets of finite perimeter and the theory of the Ornstein-Uhlenbeck semigroup in the setting of the Gauss space. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the Ornstein-Uhlenbeck semigroup and we also give a new characterization of BV functions in terms of a near-diagonal energy in this space.

In this paper, we study the long-time behavior of the solution for the linearized ideal MHD around sheared velocity and magnetic field under Stern stability condition. We prove that the velocity and magnetic field will converge to sheared velocity and magnetic field as time approaches infinity. Moreover a new depletion phenomenon is proved: the horizontal velocity and magnetic field at the critical points will decay to 0 as time approaches infinity.

The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable – reversibility depending – Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.

We show in this paper that the VC-dimension of the family of d-dimensional axis-parallel boxes and cubes on the d-dimensional torus are both asymptotically d log 2 ⁡ d . This is especially surprising as in most other examples the VC-dimension usually grows linearly with d in similar settings.

The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L : = ∇ X ⋅ ( A ( X , Y , t ) ∇ X ) + X ⋅ ∇ Y − ∂ t , in unbounded domains of the form Ω = { ( X , Y , t ) = ( x , x m , y , y m , t ) ∈ R m − 1 × R × R m − 1 × R × R | x m > ψ ( x , y , y m , t ) } , where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L . Concerning A = A ( X , Y , t ) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in R m ). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

By means of the regularity of the differential operators generated by the Szasz-Mirakjan operators, this paper further investigates the relation between the approximation rate of the linear combinations of the Szasz-Mirakjan operators and the smoothness of the approximated function. We precisely give both the upper and lower bounds, which are of the same order, of the approximation of the linear combinations of these operators. Our results under some conditions include the direct theorem, the inverse theorem and the saturation of the uniform approximation of the linear combinations of these operators, and hence the approximation order of the uniform approximation of the linear combinations of the Szasz-Mirakjan operators are further characterized.

In this paper, for new defined Bernstein type operators, we show that taking derivatives and taking the limit are sequence independent when the Voronovskaja-type expansion is differentiated.

This paper presents various functional-geometric aspects of the logarithmic Sobolev capacity Cap log ⁡ , γ , p - a capacity generated by the logarithmic Sobolev space W p log ⁡ , γ , where ( γ , p ) ∈ ( 0 , ∞ ) × [ 1 , ∞ ) . This new nonlinear-inhomogeneous-nontrivial capacity is closely related to the logarithmic Hausdorff capacity. Not only continuously sharp embedding property of W p log ⁡ , γ into certain logarithmic weighted Lebesgue space and the tracing inequalities are established, but also the logarithmic perimeter and the Lebesgue measure are utilized to: reformulate Cap log ⁡ , γ , p ; find the first variation of the logarithmic perimeter via the corresponding logarithmic mean curvature; characterize the hypersurface of a constant logarithmic mean curvature.

In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $\infty$-categories., Comment: 33 pages

Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie algebra $\mathfrak{g} = \operatorname{Lie}(G)$ admits a non-degenerate Ad$(G)$-invariant symmetric bilinear form. Given a linear function $\chi$ on $\mathfrak{g}$ we denote by $U_\chi(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $\chi$. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible $U_\chi(\mathfrak{g})$-module has dimension divisible by $p^{d(\chi)}$ where $2d(\chi)$ is the dimension of the coadjoint $G$-orbit containing $\chi$. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra $U_\chi(\mathfrak{g})$ admits a module of dimension $p^{d(\chi)}$., Comment: The present version is accepted for publication

This paper is devoted to interior estimates for eigenfunctions of the restricted fractional Laplacian on a bounded domain in R d . We prove that the eigenfunctions satisfy the expected L p bounds analogous to the classical results by Sogge [24] . As the fractional Laplacian is nonlocal, the standard method for Laplacian eigenfunction estimates can no longer work here. In the proof, we mainly reduce L p bounds to a kind of L 2 commutator estimates, which can be handled by the explicit integral expression of the restricted fractional Laplacian and its heat kernel estimates.

Let G be a connected reductive complex algebraic group with a maximal torus T. We denote by Λ the coweight lattice of T. Let Λ + ⊂ Λ be the submonoid of dominant coweights. For λ ∈ Λ + , μ ∈ Λ , μ ⩽ λ , in [7] , authors defined a generalized transversal slice W ‾ μ λ . This is an algebraic variety of the dimension 〈 2 ρ ∨ , λ − μ 〉 , where 2 ρ ∨ is the sum of positive roots of G. In this paper, we construct an isomorphism W ‾ μ λ ≃ W ‾ μ + λ × A 〈 2 ρ ∨ , μ + − μ 〉 for μ ∈ Λ such that 〈 α ∨ , μ 〉 ⩾ − 1 for any positive root α ∨ , here μ + ∈ W μ is the dominant representative in the Weyl group orbit of μ. We consider the example when λ is minuscule, μ ∈ W λ and describe natural coordinates, Poisson structure on W ‾ μ λ ≃ A 〈 2 ρ ∨ , λ − μ 〉 and its T × C × -character. We apply these results to compute T × C × -characters of tangent spaces at fixed points of convolution diagrams W ˜ μ λ _ with minuscule λ i . We also apply our results to construct open coverings by affine spaces of convolution diagrams W ˜ μ λ _ over slices with μ such that 〈 α ∨ , μ 〉 ⩾ − 1 for any positive root α ∨ and minuscule λ i and to compute Poincare polynomials of such convolution diagrams W ˜ μ λ _ .

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon., The exposition has been improved, we now present the case of the number of irreducible factors both counting and not counting multiplicities. We also add some results on the possible values of the bias

In this paper, we show that one can naturally associate a limiting dynamical system $F: T\longrightarrow T$ on an $\R$-tree to any degenerating sequence of rational maps $f_n: \hat\C \longrightarrow \hat\C$ of fixed degree. The construction of $F$ is in $2$ steps: first we use barycentric extension to get $\E f_n : \Hyp^3 \longrightarrow \Hyp^3$; second, we take appropriate limit on rescalings of hyperbolic space. An important ingredient we prove here is that the Lipschitz constant depends only on the degree of the rational map. We show that the dynamics of $F$ records the limiting length spectra of the sequence $f_n$., Comment: 41 pages, 5 figure. Final version

We examine an unstudied manuscript of N.~S.~Koshliakov over $150$ pages long and containing the theory of two interesting generalizations $\zeta_p(s)$ and $\eta_p(s)$ of the Riemann zeta function $\zeta(s)$, which we call \emph{Koshliakov zeta functions}. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each $p\in\mathbb{R^{+}}$. The first one is a generalization of Ramanujan's famous formula for $\zeta(2m+1)$ and the second is an elegant extension of a modular relation on page $220$ of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for $\zeta(4m+3)$., Comment: 34 pages, submitted for publication; comments are welcome

In this paper, the question of when the k-subdirect sum of two Dashnic-Zusmanovich matrices is also a Dashnic-Zusmanovich matrix is studied, and some sufficient conditions are given. Furthermore, those sufficient conditions only depend on the elements of the given matrices. Moreover, examples are presented to illustrate the corresponding results.

In this paper, we consider the following question: how many degree d curves are there in P 3 (passing through the right number of generic lines and points), whose image lies inside a P 2 , having δ nodes and one singularity of codimension k? We obtain an explicit formula for this number when δ + k ≤ 4 (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in the paper by A. Zinger [30] and which is further pursued by S. Basu and the second author in [1] , [2] and [3] . Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by S. Kleiman and R. Piene (in [17] ) and by T. Laarakker (in [19] ). We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author (in [22] ), where they compute the characteristic number of rational planar curves in P 3 with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande in [23] , regarding the enumerativity of BPS numbers for P 3 .

This paper deals with the long-time dynamics for a class of highly nonlinear fractional nonclassical diffusion equations with nonlinear colored noise and time delay defined on the whole space R n . The existence and uniqueness of tempered pullback random attractors of the equations are established in C ( [ − ρ , 0 ] , H α ( R n ) ) ( ρ > 0 and α ∈ ( 0 , 1 ) ) for polynomial growth drift and diffusion terms as well as Lipschitz time-delay terms. In a special case where the nonlinear diffusion term depends only on the space variable, the approximation of those random attractors is also investigated in C ( [ − ρ , 0 ] , H α ( R n ) ) when the correlation time of the colored approaches zero. The pullback asymptotical compactness of the solutions in C ( [ − ρ , 0 ] , H α ( R n ) ) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates developed by Wang (1999) [59] in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains as well as the weakly dissipative distinguishing structures of the equations.

The stability and hopf bifurcation of the multi-lingual rumor spreading model are mainly studied in this paper. A nonlinear inhibition mechanism is introduced to formulate an IS2R2 model with concerning the cross propagation in the multi-lingual environment. Next, the existence and stability of equilibrium points are analyzed systematically here. Meanwhile, choosing the key parameter and time delay of inhibition functions as bifurcation parameters, the characteristics of Hopf bifurcation are further displayed through certain critical values. Besides, an optimal control strategy involving inhibition functions is designed, and simulated via forward-backward sweep algorithm. Finally, the comparative analysis of various parameters in the numerical simulation is presented to justify our theoretical results.

This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequence is self-similar and (3) Every Pascal 3-triangle has a free group presentation.

This paper provides a derivation of Zipf-Pareto laws directly from the principle of least effort. A probabilistic functional of efficiency is introduced as the consequence of an extension of the nonadditivity of the efficiency of thermodynamic engine to a large number of living agents assimilated to engines, all randomly distributed over their output. Application of the maximum calculus to this efficiency yields the Zipf-Pareto laws., Comment: 26 pages, 1 figure

A new discrete fractional-order Peano-Baker series is established in this paper. Based on this series, the explicit solutions of Caputo linear discrete fractional-order equations (DFOEs) with variable matrix coefficients involving the homogeneous and inhomogeneous cases are obtained. On the basis of the comparison theorem and the explicit solutions of Caputo linear DFOEs with variable coefficients, the asymptotic behaviors of the solutions of the homogeneous and inhomogeneous Caputo linear DFOEs with variable coefficients are presented. Finally, the applicability and effectiveness of the proposed results are illustrated by four examples.

In this paper, a problem of the analysis of the randomly forced patterns in spatially distributed systems with diffusion is considered. For the approximation of mean-square deviations of random solutions from the unforced deterministic pattern-attractors, we suggest a constructive method based on the stochastic sensitivity technique. To demonstrate an efficiency of this method, we consider the Levin-Segel model with formation of non-homogeneous structures of the phytoplankton and herbivore populations. The spatial peculiarities of probabilistic distributions near patterns are investigated. The dependence of the stochastic sensitivity on the variation of system parameters is studied. An application of the stochastic sensitivity technique to the study of noise-induced transitions between coexisting spatial structures is demonstrated.

This paper evaluated the correlation among the activations of eyes and brain in visual stimulation. Because of the complex structure of eye fluctuations and brain signals, we employed fractal theory and sample entropy to decode their correlation. Ten subjects looked at four different images with different complexities while we collected their eye fluctuations and EEG signals simultaneously. The results indicated that the variations of the complexity of eye fluctuations and EEG signals are strongly correlated (r = -0.8043 in case of fractal dimension and r = -0.9740 in case of sample entropy), which reflects the correlation among the activations of the eyes and brain. This analysis can be conducted to examine how the activations of other organs are linked to brain activation.

In this paper, we derive risk-neutral density functions of multi-asset to model the price of European options by incorporating a simple constrained minimization of the Kullback measure of relative information. Based on the theoretical analysis, when the underlying financial asset price follows a geometric Brownian motion, we obtain a two-dimensional quanto-option Black-Scholes equation. In addition, to evaluate the explicit solution of this multi-asset option pricing model, we design a Liouville-Caputo time-fractional derivative and use the Laplace homotopy perturbation method to obtain the explicit scheme in the form of convergent series under suitable regularity conditions.

In this paper, a Green's function satisfying free surface condition, which is more suitable for solving potential flow problems, is improved, proposed and applied to FMM-BEM to solve the problems of large amount of calculation and long calculation time in the traditional boundary element method. The results show that the FMM-BEM method is more efficient than the BEM method in hydrodynamic analysis of large-scale structures.

The risk of propagation of infectious diseases such as avian influenza and COVID-19 is generally controlled or reduced by quarantine measures. Considering this situation, a network-based SIQS (susceptible-infected-quarantined-susceptible) infectious disease model with nonmonotone incidence rate is established and analyzed in this paper. The psychological impact of the transmission of certain diseases in heterogeneous networks at high levels of infection may be characterized by the related nonmonotone incidence rate. The expressions of the basic reproduction number and equilibria of the model are determined analytically. We demonstrate in detail the uniform persistence of system and the global asymptotic stability of the disease-free equilibrium. The global attractivity of the unique endemic equilibrium is discussed by using monotone iteration technique. We obtain that the endemic equilibrium is globally asymptotically stable under certain conditions by constructing appropriate Lyapunov function. In addition, numerical simulations are performed to indicate the theoretical results.

Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m − n . Our RME depends on three parameters: the network size n , the size of the smaller set m , and the connectivity between the two sets α , where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r , the Shannon entropy of the eigenvectors S , and the Randic index R . First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ ≡ ξ ( n , m , α ) that scales the average normalized measures X ¯ > (with X representing r , S and R ). Specifically, we show that (i) ξ ∝ α n with a weak dependence on m , and (ii) for ξ 1 / 10 most vertices in the mutualistic network are isolated, while for ξ > 10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1 / 10 ξ 10 . Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves X ¯ > vs. ξ show good correspondence with the properties of real-world networks.

The Leslie-Gower type predator-prey system with the ratio-dependent Holling III functional response and Neumann boundary conditions is investigated in this paper. First, the boundedness results of both parabolic and elliptic equations are presented. Hereafter, the existence of the codimension-two Turing-Hopf point (C2THP) is identified, where the Turing and the Hopf modes intersect. To further explore the spatiotemporal dynamics near the C2THP, it is necessary to derive the amplitude equations, however, there are few results about that in the two-dimensional domain. Here the method of weakly nonlinear analysis is adopted to derive the amplitude equations. The temporal patterns, hexagonal patterns, and plane wave patterns, as well as the sufficient conditions of their existence and stability, can be presented through amplitude equations.