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In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case. It is difficult to estimate the upper and lower bounds of the holes due to the coupling of electrons and holes and the degeneracy of electrons at the boundary. Thus, we use the topological degree method to prove the well-posedness of solution. We prove the ill-posedness of subsonic solution under some conditions by direct mathematical analysis and contradiction method. The ill-posedness property shows significant difference to the unipolar model. Another highlight of this paper is the application of specific energy method to obtain the uniqueness of solution in two cases. One case is the relaxation time τ = ∞ , namely, the pure Euler-Poisson case; the other case is j τ ≪ 1 , which means that, when the current flow is sufficiently small and the relaxation time is sufficiently large both can satisfy. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive predator-prey system with fear response delay, where a benefit from the ani-predation response in addition to the cost is taken into account. As a step toward understanding the underlying mechanism of the spatiotemporal pattern formations by rigorous mathematical analysis, we are interested in how the joint effects of time delay, the fear effect strength, the carrying capacity, as well as the diffusion rates could affect the complex spatiotemporal pattern formations of the system. In particular, Turing instability of the Hopf bifurcating periodic solutions is investigated in details. To that end, a general formula is derived to determine Turing instability of the Hopf bifurcating periodic solutions for general 2-component delayed-diffusive system. This extends our earlier general results for non-delayed diffusive system to its delayed-diffusive counterparts. Our general formula tends to be new and can be applied to a variety of 2-component delayed-diffusive systems. [ABSTRACT FROM AUTHOR]

Abstract: In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum forces solutions to have higher integrability as well. In the present paper, we consider -obstacle problems of the nonhomogeneous anisotropic elliptic equations under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe. [Copyright &y& Elsevier]

Abstract: The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244–275]. For instance, it is proved that for a finite non-trivial separable extension , , of Hilbertian fields finitely generated over their prime field, the quotient group , for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent. [Copyright &y& Elsevier]

Abstract: We shall make a slight improvement to a result of p-adic logarithms, which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient . [Copyright &y& Elsevier]

Abstract: We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two. [Copyright &y& Elsevier]

Abstract: The Centre for Kentish Studies in Maidstone, Kent holds the mathematical manuscripts of Philip Stanhope (1714–1786), second Earl of Stanhope. He was an active and capable amateur mathematician. Although the manuscripts cover a wide range of mathematical topics, the current article focuses only on Stanhope''s work in probability, where his interests appear to be on the theoretical rather than the applied side of the subject. His work, mainly derived from De Moivre''s The Doctrine of Chances and Montmort''s Essay d''analyse sur les jeux de hazard, touches on the major probability problems of the day. Among the notes on these two authors there is work that includes an alternate solution to the theory of runs and a simplified solution to a special case of the problem of the duration of play, related to the gambler''s ruin problem. In addition, the manuscript collection contains Stanhope''s transcription of an incorrect solution to the theory of runs by Thomas Bayes. There is also some correspondence with Sir Alexander Cuming that touches on George Berkeley''s criticism of Isaac Newton''s development of the calculus. [Copyright &y& Elsevier]

Abstract: Remark on Polickýʼs paper on circular units of a compositum of quadratic number fields is given. [Copyright &y& Elsevier]

According to the Lighthill scheme of approximation procedures, we establish the Blackstock-Cattaneo model in this paper to describe nonlinear acoustics' propagations with second sound phenomena in perfect gases under the irrotational flow, which is an approximated higher-order equation for Navier-Stokes-Cattaneo equations. This paper is devoted to the mathematical analysis of the linear and nonlinear higher-order evolution equations in R n. Concerning the linearized Cauchy problem for the Blackstock-Cattaneo equation, by means of the Fourier analysis and the WKB analysis, we will derive sharp L 2 estimates of solutions with additional L 1 regularity for initial data. For another, we demonstrate global (in time) well-posedness for the nonlinear Blackstock-Cattaneo equation with small initial data and suitable regularity of Sobolev solutions. [ABSTRACT FROM AUTHOR]

The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In our previous work [38] , we investigate the global existence of strong solutions to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity. However, it is unknown whether the entropy remains its boundedness or not? Thus, the main novelty of this paper is to give a positive response to this problem. In fact, we show that the uniform boundedness of the entropy and the L 2 regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. Modified De Giorgi type iteration techniques and useful singularly weighted estimates are developed to deduce the lower and upper bounds on the entropy. [ABSTRACT FROM AUTHOR]

Particle tracking in a live cell environment is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles, which holds the promise of revealing important new biological insights. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. In contrast to the standard approach, the dynamic reconstruction seeks to simultaneously recover the source locations and velocities from all frames, which enjoys certain advantages. In this paper, we provide a rigorous mathematical analysis for the resolution limit of reconstructing source number, locations, and velocities by general dynamical reconstruction in particle tracking problems, by which we demonstrate the possibility of achieving super-resolution for the dynamic reconstruction. We show that when the location-velocity pairs of the particles are separated beyond certain distances (the resolution limits), the number of particles and the location-velocity pair can be stably recovered. The resolution limits are related to the cut-off frequency of the imaging system, signal-to-noise ratio, and the sparsity of the source. By these estimates we also derive a stability result for a sparsity-promoting dynamic reconstruction. In addition, we further show that the reconstruction of velocities has a better resolution limit which improves constantly as the particles moving. The result is derived by a crucial observation that the inherent cut-off frequency for the velocity recovery can be viewed as the total observation time multiplies the cut-off frequency of the imaging system, which may lead to a better resolution limit as compared to the one for each diffraction-limited frame. This is discovered for the first time and we hope it can inspire many new reconstruction algorithms that significantly improve the resolution of particle tracking in practice. In addition, we propose super-resolution algorithms for recovering the number and values of the velocities in the tracking problem and demonstrate theoretically or numerically their super-resolution capability. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C ∞ [ 0 , 1 ] steady-state solution by the method of upper and lower solution, and the uniqueness in both W 1 , 2 (0 , 1) and L ∞ (0 , 1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C ([ 0 , 1 ] × [ 0 , + ∞)) ∩ C 2 , 1 ([ 0 , 1 ] × (0 , + ∞)) , which, for any continuous initial value, converges to the steady-state solution in C [ 0 , 1 ] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species. [ABSTRACT FROM AUTHOR]

Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

Abstract: W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd–Wong fixed point theorem. Subsequently, I.D. Arand̄elović [I.D. Arand̄elović, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385] established somewhat more general version of Kirk''s result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arand̄elović''s theorem is false. We also explain how to correct this result. [Copyright &y& Elsevier]

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer's confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the mathematical analysis of the Cauchy problem for general cross-diffusion systems without any assumption about its entropic structure. A global existence result of nonnegative solutions is obtained by applying a classical Schauder fixed point theorem. The proof is upgraded for enhancing the regularity of the solution, namely its gradient belongs to the space L r ((0 , T) × Ω) for some r > 2. To this aim, the Schauder's strategy is coupled with an extension of Meyers regularity result for linear parabolic equations. We show how this approach allows to prove the well-posedness of the problem using only assumptions prescribing and admissibility range for the ratios between the diffusion and cross-diffusion coefficients. The results are compared with those that are reachable with an additional regularity assumption on the parabolic operator, namely a small BMO assumption for its coefficients. Finally, the question of the maximal principle is also addressed, especially when source terms are incorporated in the equation in order to ensure the confinement of the solution. [ABSTRACT FROM AUTHOR]

Abstract Let n be a non-negative integer and put p n (x) = ∏ i = 0 n (x + i). In the first part of the paper, for given n , we study the existence of integer solutions of the Diophantine equation y m = p n (x) + ∑ i = 1 k p a i (x) , where m ∈ N ≥ 2 and a 1 < a 2 < ... < a k < n. This equation can be considered as a generalization of the Erdős–Selfridge Diophantine equation y m = p n (x). We present some general finiteness results concerning the integer solutions of the above equation. In particular, if n ≥ 2 with a 1 ≥ 2 , then our equation has only finitely many solutions in integers. In the second part of the paper we study the equation y m = ∑ i = 1 k p a i (x i) , for m = 2 , 3 , which can be seen as an additive version of the equation considered by Erdős and Graham. In particular, we prove that if m = 2 , a 1 = 1 or m = 3 , a 2 = 2 , then for each k − 1 tuple of positive integers (a 2 , ... , a k) there are infinitely many solutions in integers. [ABSTRACT FROM AUTHOR]

Abstract Let M be a 3-connected binary matroid; M is internally 4-connected if one side of every 3-separation is a triangle or a triad, and M is (4 , 4 , S) -connected if one side of every 3-separation is a triangle, a triad, or a 4-element fan. Assume M is internally 4-connected and that neither M nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let N be an internally 4-connected proper minor of M. Our aim is to show that M has a proper internally 4-connected minor with an N -minor that can be obtained from M either by removing at most four elements, or by removing elements in an easily described way from a special substructure of M. When this aim cannot be met, the earlier papers in this series showed that, up to duality, M has a good bowtie, that is, a pair, { x 1 , x 2 , x 3 } and { x 4 , x 5 , x 6 } , of disjoint triangles and a cocircuit, { x 2 , x 3 , x 4 , x 5 } , where M \ x 3 has an N -minor and is (4 , 4 , S) -connected. We also showed that, when M has a good bowtie, either M \ x 3 , x 6 has an N -minor and M \ x 6 is (4 , 4 , S) -connected; or M \ x 3 / x 2 has an N -minor and is (4 , 4 , S) -connected. In this paper, we show that, when M \ x 3 , x 6 has no N -minor, M has an internally 4-connected proper minor with an N -minor that can be obtained from M by removing at most three elements, or by removing elements in a well-described way from a special substructure of M. This is a final step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis–Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with 2 u − 1 blocks in the additive group of the finite field of order q e for any prime power q ≡ 2 u + 1 (mod 2 u + 1) with u ⩾ 2 and any positive integer e. In the aforementioned papers of Szekeres, Whiteman, and Wallis–Whiteman, the constructions of skew Hadamard difference families with 2 u − 1 (u = 2 or 3) blocks in (F q e , +) work only for restricted e ;namely e ≡ 1 , 2 , or 3 (mod 4) when u = 2 , and e ≡ 1 (mod 2) when u = 3 , respectively. Our more general construction, in particular, removes the restrictions on e. As a consequence, we obtain new infinite series of skew Hadamard difference families with two or four blocks, and hence skew Hadamard matrices. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the following classical question of extremal set theory: what is the maximum size of a family of subsets of [ n ] such that no s sets from the family are pairwise disjoint? This problem was first posed by Erdős and resolved for n ≡ 0 , − 1 (mod s) by Kleitman in the 60s. Very little progress was made on the problem until recently. The only result was a very lengthy resolution of the case s = 3 , n ≡ 1 (mod 3) by Quinn, which was written in his PhD thesis and never published in a refereed journal. In this paper, we give another, much shorter proof of Quinn's result, as well as resolve the case s = 4 , n ≡ 2 (mod 4). This complements the results in our recent paper, where, in particular, we answered the question in the case n ≡ − 2 (mod s) for s ≥ 5. [ABSTRACT FROM AUTHOR]

Abstract Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B (ℓ p) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space ℓ p of all p -summable sequences. The ℓ p -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓ p -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B (ℓ 2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network. [ABSTRACT FROM AUTHOR]

Highlights • Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. • M. Herzog et al. in [J. Algebra, 2018] give two new criteria for solvability of finite groups. • They proved that if G is a group of order n and ψ (G) ≥ ψ (C n) / 6.68 , then G is solvable. • They conjectured that: Conjecture. If G is a group of order n and ψ (G) > 211 1617 ψ (C n) , then G is solvable. • As the main result of this paper we prove the validity of this conjecture. Abstract Let G be a finite group and ψ (G) = ∑ g ∈ G o (g) , where o (g) denotes the order of g ∈ G. In [M. Herzog, et al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: Conjecture. If G is a group of order n and ψ (G) > 211 ψ (C n) / 1617 , where C n is the cyclic group of order n, then G is solvable. In this paper we prove the validity of this conjecture. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with blow-up solutions to the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type { u t = ∇ ⋅ (∇ u m − u q − 1 ∇ v) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R N (N ≥ 3), m ≥ 1 , q ≥ 2. As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q > m + 2 N ([5]). In the parabolic–elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic–parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic–parabolic case. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the regularity of the weak solutions to a quasilinear parabolic systems which is a generalization of p -Laplacian of the type u t i − (A α i (∇ u)) x α = f i (x , t , u , ∇ u) , i = 1 , … , N where the main part satisfies some ellipticity and f i satisfies certain growth conditions. We prove boundedness of the solutions and the gradients of solutions to the systems by the means of the energy estimates and a nonlinear iteration procedure of the Moser type in this paper. [ABSTRACT FROM AUTHOR]

Abstract There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the q -Airy equation is studied. [ABSTRACT FROM AUTHOR]

When studying non-symmetric nonlocal operators on R d : L f ( x ) = ∫ R d ( f ( x + z ) − f ( x ) − ∇ f ( x ) ⋅ z 1 { | z | ⩽ 1 } ) κ ( x , z ) | z | d + α d z , where 0 < α < 2 , d ⩾ 1 , and κ ( x , z ) is a function on R d × R d that is bounded between two positive constants, it is customary to assume that κ ( x , z ) is symmetric in z . In this paper, we study heat kernel of L and derive its two-sided sharp bounds without the symmetric assumption κ ( x , z ) = κ ( x , − z ) . In fact, we allow the kernel κ to be time-dependent and x → κ ( t , x , z ) to be only locally β -Hölder continuous with Hölder constant possibly growing at a polynomial rate in | z | . We also derive gradient estimate when β ∈ ( 0 ∨ ( 1 − α ) , 1 ) as well as fractional derivative estimate of order θ ∈ ( 0 , ( α + β ) ∧ 2 ) for the heat kernel. Moreover, when α ∈ ( 1 , 2 ) , drift perturbation of the time-dependent non-local operator L t with drift in Kato's class is also studied in this paper. As an application, when κ ( x , z ) = κ ( z ) does not depend on x , we show the boundedness of nonlocal Riesz's transformation: for any p > 2 d / ( d + α ) , ‖ L 1 / 2 f ‖ p ≍ ‖ Γ ( f ) 1 / 2 ‖ p , where Γ ( f ) : = 1 2 L ( f 2 ) − f L f is the carré du champ operator associated with L , and L 1 / 2 is the square root operator of L defined by using Bochner's subordination. Here ≍ means that both sides are comparable up to a constant multiple. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [ABSTRACT FROM AUTHOR]

This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II superstring amplitudes. These modular graph forms are multiple sums associated with decorated Feynman graphs on the world-sheet torus. The action of standard differential operators on these modular graph forms admits an algebraic representation on the decorations. First order differential operators are used to map general non-holomorphic modular graph functions to holomorphic modular forms. This map is used to provide proofs of the identities between modular graph functions for weight less than six conjectured in earlier work, by mapping these identities to relations between holomorphic modular forms which are proven by holomorphic methods. The map is further used to exhibit the structure of identities at arbitrary weight. [ABSTRACT FROM AUTHOR]

Neural networks (NN) have generated extensive interest in the field of machine condition monitoring (MCM). Many applications are however adapting structures and approaches from fields where NNs are highly established (e.g., image classification). This is done despite many neural network layers presenting strong similarities to signal processing steps that are well established in MCM. This paper aims at providing a mathematical analysis of typical NN architectures and layers, with specific attention to similarities to common signal processing steps such as filtering, downsampling, and enveloping. The set of strong analogies between NN and traditional MCM tools enables this work to provide MCM-specific guidelines for the design and tuning of NN architectures for MCM-specific applications. Most MCM applications are often characterised by periodic and cyclostationary signals, a situation not frequently encountered in many other NN applications. Hence, the Fourier-based explanation of the typical NN layers will be specific to the solution of typical MCM problems (e.g., diagnostics of gears and bearings). The study is aligned with the current trends of explainable AI and physics informed neural networks (PINN), in the sense that it aims at producing mathematical rules for a rational design of the network, drawing on the expertise accumulated in decades of signal processing for MCM. This is however done without upsetting traditional NN architectures, but rather trying to provide clear reasons for the setting of network parameters. The analytical work will be verified with numerical tests, which will also enable practical considerations, and by application of the suggested guidelines to the popular Case Western Reserve University (CWRU) experimental dataset. [ABSTRACT FROM AUTHOR]

Machine learning methods are widely used in the natural sciences to model and predict physical systems from observation data. Yet, they are often used as poorly understood "black boxes," disregarding existing mathematical structure and invariants of the problem. Recently, the proposal of Hamiltonian Neural Networks (HNNs) took a first step towards a unified "gray box" approach, using physical insight to improve performance for Hamiltonian systems. In this paper, we explore a significantly improved training method for HNNs, exploiting the symplectic structure of Hamiltonian systems with a different loss function. This frees the loss from an artificial lower bound. We mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn. This allows us to prove and numerically analyze the errors made by HNNs which, in turn, renders them fully explainable. Finally, we present a novel post-training correction to obtain the true Hamiltonian only from discretized observation data, up to an arbitrary order. • The loss of Hamiltonian Neural Networks suffers from an artificial lower bound. • Using a symplectic integrator during training significantly improves performance. • Rigorous mathematical analysis yields an exact handle on the discretization error. • The learnt Hamiltonian can be corrected post-training, to any arbitrary order. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlinear embeddings between Banach spaces. More specifically, the goal of this paper is to study weaker versions of coarse and uniform embeddability, and to provide suggestive evidences that those weaker embeddings may be stronger than one would think. We do such by proving that many known results regarding coarse and uniform embeddability remain valid for those weaker notions of embeddability. [ABSTRACT FROM AUTHOR]

Text A set A of nonnegative integers is said to be complete if every sufficiently large natural number is the sum of distinct terms taken from A . In 1959, Birch confirmed a conjecture of Erdős by proving that the set { p n q m : n , m = 0 , 1 , … } is complete, where p and q are two coprime integers greater than 1. In this paper, we study extensively the set P ( S p 1 , … , p r ) of all vectors ( m 1 , … , m r ) with integer coordinates which can be represented as ∑ ( p 1 α 1 , … , p r α r ) , where p 1 , … , p r are integers greater than 1 and α 1 , … , α r are nonnegative integers. We find many regular parts in P ( S p 1 , … , p r ) , for example, discrete rectangles. To study the problem, we find a new matching principle. Video For a video summary of this paper, please visit https://youtu.be/qXUe1E4jlLQ . [ABSTRACT FROM AUTHOR]

Text Let k , l , a , b , c be positive integers such that gcd ⁡ ( k a , l b ) = 1 , min ⁡ { a , b , c } > 1 , a ≠ 3 , b ≠ 3 and 2 ∤ c . In this paper, we prove that there are at most four solutions in positive integers ( x , y , z ) to the equation k a x + l b y = c z and at most two solutions when 2 ∤ ( u ( l / k ) ) , where u ( m ) is the least positive integer t with m t ≡ 1 ( mod c ) . Video For a video summary of this paper, please visit https://youtu.be/Dt3Y7TDMxlg . [ABSTRACT FROM AUTHOR]

The N th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large N th linear complexity. However, in this paper we show that for q -automatic sequences over F q the converse is not true. We prove that any (not ultimately periodic) q -automatic sequence over F q has N th linear complexity of order of magnitude N . For some famous sequences including the Thue–Morse and Rudin–Shapiro sequence we determine the exact values of their N th linear complexities. These are non-trivial examples of predictable sequences with N th linear complexity of largest possible order of magnitude. [ABSTRACT FROM AUTHOR]

This paper presents an efficient nonparametric time domain nonlinear system identification method. It is shown how truncated Volterra series models can be efficiently estimated without the need of long, transient-free measurements. The method is a novel extension of the regularization methods that have been developed for impulse response estimates of linear time invariant systems. To avoid the excessive memory needs in case of long measurements or large number of estimated parameters, a practical gradient-based estimation method is also provided, leading to the same numerical results as the proposed Volterra estimation method. Moreover, the transient effects in the simulated output are removed by a special regularization method based on the novel ideas of transient removal for Linear Time-Varying (LTV) systems. Combining the proposed methodologies, the nonparametric Volterra models of the cascaded water tanks benchmark are presented in this paper. The results for different scenarios varying from a simple Finite Impulse Response (FIR) model to a 3rd degree Volterra series with and without transient removal are compared and studied. It is clear that the obtained models capture the system dynamics when tested on a validation dataset, and their performance is comparable with the white-box (physical) models. [ABSTRACT FROM AUTHOR]

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points. [ABSTRACT FROM AUTHOR]

This paper examines the Euler–Rodrigues formula in dual 3-space D 3 by analyzing its variations such as vectorial form, exponential map, point–line theory and quaternions which have some intrinsic relations. Contrary to the Euclidean case, dual rotation in dual 3-space corresponds to a screw motion in Euclidean 3-space. This paper begins by explaining dual motion in terms of the given dual axis and angle. It will then go on to express dual Euler–Rodrigues formula with algebraic methods. Furthermore, an application of dual Euler–Rodrigues formula to point–line geometry is accomplished and point–line displacement operator is obtained by dual Euler–Rodrigues formula. Finally, dual Euler–Rodrigues formula is presented with the help of dual Euler–Rodrigues parameters that is expressed as a dual quaternion. [ABSTRACT FROM AUTHOR]

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L 2 -spaces. However, in subsequent papers, we intend to extend the study to L p -spaces ( 1 < p < ∞ ) . [ABSTRACT FROM AUTHOR]

The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that 4 n r − 4 r 2 generic measurements are both necessary and sufficient for the recovery of rank- r matrices in C n × n . Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound 4 n r − 4 r 2 is tight provided n = 2 k + r , k ∈ Z + . Motivated by Vinzant's work [19] , we construct 11 matrices in R 4 × 4 by computer random search and prove they define injective measurements on rank-1 matrices in R 4 × 4 . This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than 2 n − 1 orthogonal projections are possible for the recovery of x ∈ R n from the norm of them, which gives a negative answer for a question raised in [1] . [ABSTRACT FROM AUTHOR]

This paper is a complement of our recent works on the semilinear Tricomi equations in [9] and [10] . For the semilinear Tricomi equation ∂ t 2 u − t Δ u = | u | p with initial data ( u ( 0 , ⋅ ) , ∂ t u ( 0 , ⋅ ) ) = ( u 0 , u 1 ) , where t ≥ 0 , x ∈ R n ( n ≥ 3 ), p > 1 , and u i ∈ C 0 ∞ ( R n ) ( i = 0 , 1 ), we have shown in [9] and [10] that there exists a critical exponent p crit ( n ) > 1 such that the solution u , in general, blows up in finite time when 1 < p < p crit ( n ) , and there is a global small solution for p > p crit ( n ) . In the present paper, firstly, we prove that the solution of ∂ t 2 u − t Δ u = | u | p will generally blow up for the critical exponent p = p crit ( n ) and n ≥ 2 , secondly, we establish the global existence of small data solution to ∂ t 2 u − t Δ u = | u | p for p > p c r i t ( n ) and n = 2 . Thus, we have given a systematic study on the blowup or global existence of small data solution u to the equation ∂ t 2 u − t Δ u = | u | p for n ≥ 2 . [ABSTRACT FROM AUTHOR]

Text In a series of papers C. Dartyge and A. Sárközy (partly with other coauthors) studied pseudorandom measures of subsets. In this paper we extend the theory of C. Dartyge and A. Sárközy to several dimensions. We introduce measure for multi-dimensional pseudorandom subsets, and study the connection between measures of different orders. Large families of multi-dimensional pseudorandom subsets are given by using the squares in F q . Video For a video summary of this paper, please visit https://youtu.be/7iH9x7nyyfQ . [ABSTRACT FROM AUTHOR]

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSpace -complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSpace -complete. Additionally, we provide simpler constructions for the uniform word problems of these classes. For the uniform word problem for automaton groups (without rational constraints), we show NL -hardness. Finally, we investigate a question asked by Cain about a better upper bound for the length of a word on which two distinct elements of an automaton semigroup must act differently. A detailed listing of the contributions of this paper can be found at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G , whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has an abelian hyperfocal subgroup of rank 3. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following Keller–Segel–Navier–Stokes system (⋆) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − c + n , x ∈ Ω , t > 0 , u t + κ ( u ⋅ ∇ ) u = Δ u + ∇ P + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , where Ω ⊂ R 3 is a bounded domain with smooth boundary ∂ Ω , κ ∈ R and S denotes a given tensor-valued function fulfilling | S ( x , n , c ) | ≤ C S ( 1 + n ) α with some C S > 0 and α > 0 . As the case κ = 0 has been considered in [25] , it is shown in the present paper that the corresponding initial–boundary problem with κ ≠ 0 admits at least one global weak solution if α ≥ 3 7 . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions to a class of p ( x ) -curl systems arising in electromagnetism. The results obtained in this paper extend several contributions concerning the p -curl operator and we focus on new existence results which are due to the presence of variable exponent. To our best knowledge, our results are new even in the semilinear case. [ABSTRACT FROM AUTHOR]

This note is in the nature of an appendix to the recent paper by Paul Sobaje, [5] . In that paper, Section 2.2 the question of the uniqueness of G structures on (or liftings of) principal indecomposable G 1 -module in the case G = SL 3 is raised. We here answer this question positively as well as making some more general observations. [ABSTRACT FROM AUTHOR]

We consider the algorithm for verified integration of piecewise analytic functions presented in Petras’ paper (Petras, 2002). The analysis of the algorithm contained in that paper is limited to a narrow class of functions and gives upper bounds only. We present an estimate of the cost (measured by the number of evaluations of the integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. We show examples of functions with cost Θ ( ∣ ln ε ∣ / ε p − 1 ) , for any p > 1 , where ε is the desired accuracy of the computed integral. [ABSTRACT FROM AUTHOR]

In this note, we study properties of certain parabolas which were recently defined by Hutchinson and Zaharescu in their paper on parabolas infiltrating the Ford circles. We examine this connection between Ford circles and these parabolas. In particular, we characterize the intersection points of these parabolas in terms of tangent Ford circles. Further, we provide a finer analysis of area under the arcs of these parabolas with end points joining the centers of two neighboring Ford circles, than discussed in their paper. [ABSTRACT FROM AUTHOR]

In this paper we characterize Birkhoff–James orthogonality of linear operators defined on a finite dimensional real Banach space X . We also explore the left symmetry of Birkhoff–James orthogonality of linear operators defined on X . Using some of the related results proved in this paper, we finally prove that T ∈ L ( l p 2 ) ( p ≥ 2 , p ≠ ∞ ) is left symmetric with respect to Birkhoff–James orthogonality if and only if T is the zero operator. [ABSTRACT FROM AUTHOR]