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We consider the repair problem for Reed-Solomon (RS) codes, evaluated on an $\mathbb{F}_{q}$ -linear subspace $U \subseteq \mathbb{F}_{q^{m}}$ of dimension $d$ , where $q$ is a prime power, $m$ is a positive integer, and $\mathbb{F}_{q}$ is the Galois field of size $q$ . For $q > 2$ , we show the existence of a linear repair scheme for the RS code of length $n=q^{d}$ and codimension $q^{s}, s , evaluated on $U$ , in which each of the $n-1$ surviving nodes transmits only $r$ symbols of $\mathbb{F}_{q}$ , provided that $ms\geq d(m-r)$ . For the case $q=2$ , we prove a similar result, with some restrictions on the evaluation linear subspace $U$ . Our proof is based on a probabilistic argument, however the result is not merely an existence result; the success probability is fairly large (at least 1/3) and there is a simple criterion for checking the validity of the randomly chosen linear repair scheme.

In this paper we are interested in investigating the physical shape-changed propagations to the generalized Equal-Width equation through studying the explicit solutions of Wazwaz-Benjamin-Bona-Mahony model. Both models are of considerable importance in many disciplines of research, including ocean engineering and science, and describe the propagation of equally-width waves. We highlight the effect of the coefficients of both nonlinearity and dispersion terms on changing the physical shape of both models by implementing the new exponential-expansion scheme. 2D and 3D graphical plots are provided to validate the findings of the paper.

In this article, developed the compact implicit difference method based Grunwald Letnikov formula (GLF) to compute the solution of the time-fractional diffusion-wave equation (TFDWE) describing wave propagation phenomenan having order α ( 1 α 2 ) . The fractional derivative is in Caputo sense. The theoretical analysis such as stability, consistency, convergence and solvability of the said scheme are discussed and proved that the scheme is conditionally stable and convergent with the order ( τ 2 + ( Δ x ) 4 ) . The numerical results compared with the recent existed method. The results of the numerical examples show that the GLF and the proposed method is very accurate and efficient for time fractional diffusion-wave equation.

We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proof-term of the formula is produced by some scheme generated by the grammar. As a corollary, we get a similar description of the set of normal proofs of positive formulae for a large class of theories including simple type theory and System F.

In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes DM-CC, a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.

We study the iso-manifold persistence in formulism. We employ a quasilinear iterative scheme to show that the majority of unperturbed tori give rise to invariant tori of the perturbed system starting from the same Riemannian manifold and keep the ratio of certain components of the respective frequencies. We also consider the iso-manifold Melnikov persistence.

This paper investigates the stabilization problem for impulsive Boolean networks (BNs) with stochastic disturbances. Firstly, dynamic evolution of the impulsive BN is converted into an algebraic form by using the semi-tensor product method of matrices. By analyzing the reachable set, the notion of distance between each state of the network and the equilibrium is defined, based on which the algorithm is then proposed for designing the appropriate aperiodic impulsive sequences. After that, based on the state-feedback control method, some criteria are presented to stabilize the Boolean control networks with periodic impulses and stochastic disturbances. Numerical examples are also presented to show feasibility of the theoretical results established.

In this paper k always denotes a finite field of characteristic p > 0 . A variety over k means a reduced scheme separated and of finite type over k. An etale cover is always assumed to be finite (as in [SGA1, I, Def. 4.9] ).

This paper presents a meshless finite point method (FPM) for the numerical analysis of the fractional cable equation. A second-order time discrete scheme is proposed to approximate both integer-order and fractional-order time derivatives. Then, based on the stabilized moving least squares approximation and the meshless smoothed gradients, a new implementation of the FPM is provided to enhance the accuracy and convergence rate in space. Theoretical error of the FPM is analyzed. Numerical results verify the efficiency of the method and show that the method can gain second-order accuracy in time and fourth-order accuracy in space.

The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc

We carry out error estimates for a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme, which is based on the Lagrange Multiplier approach and could also be viewed as a special case of the invariant energy quadratization approach, for the modified phase field crystal equation. At each time step, the proposed scheme only requires solving several linear equations. Rigorous proofs are presented to demonstrate the unique solvability, mass conservation, and unconditional energy stability of the scheme. Various numerical experiments in 2D and 3D are performed to validate the accuracy, unconditional energy stability and mass conservation of the proposed numerical strategy.

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

We present a new generalization of the extensible bin packing with unequal bin sizes problem. In our generalization the cost of exceeding the bin size depends on the index of the bin and not only on the amount in which the size of the bin is exceeded. This generalization does not satisfy the assumptions on the cost function that were used to present the existing polynomial time approximation scheme (PTAS) for the extensible bin packing with unequal bin sizes problem. In this work, we show the existence of an efficient PTAS (EPTAS) for this new generalization and thus in particular we improve the earlier PTAS for the extensible bin packing with unequal bin sizes problem into an EPTAS. Our new scheme is based on using the shifting technique followed by a solution of polynomial number of $n$-fold programming instances. In addition, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) for the related bin packing type variant of the problem.

In this paper, we present a high-order approach for solving one- and two-dimensional time-space fractional diffusion equations (FDEs) with Caputo-Riesz derivatives. To design the scheme, the Caputo temporal derivative is approximated using a high-order method, and the spatial Riesz derivative is discretized by the second-order weighted and shifted Grunwald difference (WSGD) method. It is proved that the scheme is unconditionally stable and convergent with the order of O ( τ α h 2 + τ 4 ) , where τ and h are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples.

This paper proposes a higher-order numerical approximation scheme to solve singularly perturbed reaction–diffusion boundary value problems. The proposed scheme is a combination of a fourth-order numerical difference method and classical central difference method applied on a non-equidistant grid. The non-equidistant grid is generated by equi-distributing a non-negative monitor function. The theoretical and empirical error analysis demonstrate that the proposed scheme has fourth-order uniform convergence with respect to the perturbation parameter.

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-\Delta)^{s}$ with $s\in(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $\mathcal{O}(h^{k+\frac{1}{2}})$., Comment: 11 pages

Direct Z-scheme g-C3N4/V2O5 photocatalysts were prepared through a sonication-assisted calcination method. The obtained samples were characterised by X-ray diffraction (XRD),Ultraviolet-visible diffuse reflectance spectroscopy (UV-Vis-DRS), Scanning electron microscope (SEM), andPhotoluminescence spectroscopy (PL). Oxidations of tetracycline hydrochloride (TC) were employed to evaluate the photocatalytic activities of the obtained g-C3N4/V2O5materials. Different weight ratios (5, 10, 15, and 20%) of g-C3N4/V2O5 loaded composites were prepared, in which a 15% (CV-15) loaded composite was found to show optimal catalytic performance for the reaction. The degradation conversation of TC has achieved approximately 79.67% in CV-15 after a 2-hour reaction. g-C3N4/V2O5 photocatalystwas more active than the individual g-C3N4 and V2O5 materials, which could be attributed to the efficient separation of photogenerated electron-hole pairs shown in the photocatalytic mechanism of TC degradation.

Rate distortion theory is concerned with optimally encoding signals from a given signal class $$\mathcal {S}$$ S using a budget of R bits, as $$R \rightarrow \infty $$ R → ∞ . We say that $$\mathcal {S}$$ S can be compressed at rates if we can achieve an error of at most $$\mathcal {O}(R^{-s})$$ O ( R - s ) for encoding the given signal class; the supremal compression rate is denoted by $$s^*(\mathcal {S})$$ s ∗ ( S ) . Given a fixed coding scheme, there usually are some elements of $$\mathcal {S}$$ S that are compressed at a higher rate than $$s^*(\mathcal {S})$$ s ∗ ( S ) by the given coding scheme; in this paper, we study the size of this set of signals. We show that for certain “nice” signal classes $$\mathcal {S}$$ S , a phase transition occurs: We construct a probability measure $$\mathbb {P}$$ P on $$\mathcal {S}$$ S such that for every coding scheme $$\mathcal {C}$$ C and any $$s > s^*(\mathcal {S})$$ s > s ∗ ( S ) , the set of signals encoded with error $$\mathcal {O}(R^{-s})$$ O ( R - s ) by $$\mathcal {C}$$ C forms a $$\mathbb {P}$$ P -null-set. In particular, our results apply to all unit balls in Besov and Sobolev spaces that embed compactly into $$L^2 (\varOmega )$$ L 2 ( Ω ) for a bounded Lipschitz domain $$\varOmega $$ Ω . As an application, we show that several existing sharpness results concerning function approximation using deep neural networks are in fact generically sharp. In addition, we provide quantitative and non-asymptotic bounds on the probability that a random $$f\in \mathcal {S}$$ f ∈ S can be encoded to within accuracy $$\varepsilon $$ ε using R bits. This result is subsequently applied to the problem of approximately representing $$f\in \mathcal {S}$$ f ∈ S to within accuracy $$\varepsilon $$ ε by a (quantized) neural network with at most W nonzero weights. We show that for any $$s > s^*(\mathcal {S})$$ s > s ∗ ( S ) there are constants c, C such that, no matter what kind of “learning” procedure is used to produce such a network, the probability of success is bounded from above by $$\min \big \{1, 2^{C\cdot W \lceil \log _2 (1+W) \rceil ^2 - c\cdot \varepsilon ^{-1/s}} \big \}$$ min { 1 , 2 C · W ⌈ log 2 ( 1 + W ) ⌉ 2 - c · ε - 1 / s } .

Numerous methods have been introduced in the literature for numerical solution of two-dimensional hyperbolic differential equations (telegraph equations). Improved techniques using explicit group methods derived from the standard and skewed (rotated) finite difference operators have been developed over the last few years in solving the linear systems that arise from the discretization of the several types of partial differential equations. In this paper, we present a preliminary study of the formulation of new preconditioned scheme based on explicit group relaxationmethods for the difference solution of the telegraph equations. The efficient and robustness of these new formulations over the existing explicit group schemes demonstrated through numerical experiments.

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split., Comment: The final version to be published in Transactions of the AMS. Results about quadratic forms are strengthened. In the section on Bertini type theorems a correction in the case of a non-perfect residue field is made. Other minor corrections and improvements

The purpose of this paper is to recommend an iterative scheme to approximate a common element of the solution sets of the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem for non-expansive mappings. We prove that the sequences generated by the recommended iterative scheme strongly converge to a common element of solution sets of stated split problems. In the end, we provide a numerical example to support and justify our main result. The result studied in this paper generalizes and extends some widely recognized results in this direction.

In this article, a hybrid Haar wavelet collocation method (HWCM) is proposed for the ill-posed inverse problem with unknown source control parameters. Applying numerical techniques to such problems is a challenging task due to the presence of nonlinear terms, unknown control parameter sources along the solution inside the given region. To find the numerical solution, derivatives are discretized adopting implicit finite-difference scheme and Haar wavelets. The computational stability and theoretical rate of convergence of the proposed HWCM are discussed in detail. Two numerical experiments are incorporated to show the well-condition behavior of the matrix obtained from HWCM and hence not required to supplement some regularization procedures. Moreover, the numerical solutions of the considered experiments illustrate the reliability, suitability, and correctness of HWCM.

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance $\varepsilon$ is at most $O(2^{0.2680n})$ times a polynomial in $1/\varepsilon$. On bipartite graphs, the exponential term in the running time is improved to $O(2^{0.2372n})$. Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, \v{S}tefankovi\v{c} and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless $\mbox{P}=\mbox{NP}$., Comment: 52pp

In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

In this work, an efficient alternating direction implicit (ADI) finite difference scheme is proposed to solve the three-dimensional time-fractional telegraph equation. The fully-discrete scheme is established via the L1 discrete formula in temporal direction and finite difference method in spatial direction. In order to reduce the computational cost of solving three-dimensional problems, an ADI algorithm is designed and applied. Then, via the energy method, we prove the stability and convergence of the scheme in L 2 and H 1 norms, respectively. Eventually, several numerical examples are given to validate the theoretical results.

This article investigates the output regulation problem of probabilistic $k$ -valued logical systems with delays by an intermittent control scheme. Two types of event-triggered control are designed via semi-tensor product (STP) of matrices. According to the algebraic state-space representation of probabilistic $k$ -valued systems with delays, the problem is transformed into the existence of solutions of algebraic equations. Then we obtain the sufficient and necessary condition for the output regulation problem of probabilistic $k$ -valued logical systems with delays. Two types of approaches are given to design the event-triggered control laws. Eventually, two exemplifications are provided to explain the effectiveness and utility of the acquired results.

The aim of this paper is to prove the existence and uniqueness of common fixed points for a pair of $$(\psi -\varphi )$$ ( ψ - φ ) -weak contractive self-maps in the setting of b-metric spaces satisfying the minimal requirement of weakly compatibility, and other weak commuting properties as compatibility, R-weakly commuting and R-weakly commuting of types ($$A_T$$ A T ), ($$A_S$$ A S ) and ($$A_P$$ A P ). Also, we will analyze the convergence and stability of the Jungck-Noor iterative scheme for this class of pairs of mappings on b-metric spaces endowed with a convex structure.

We develop a randomized algorithm (that succeeds with high probability) for generating an $$\epsilon $$ -net in a sphere of dimension n. The basic scheme is to pick an alphabet consisting of $$O (n\ln (1/\epsilon )+\ln (1/\delta ))$$ random rotations, form all possible words of length $$O (n\ln (1/\epsilon ))$$ from this alphabet, and require these words act on a fixed point. We show the set of points so generated is equidistributed at a scale of $$\epsilon $$ .

We investigate the total number of edge crossings (i.e., the crossing number) of the Euclidean minimum weight Laman graph \(\mathsf {MLG}(P)\) on a planar point set P. Bereg et al. (2016) showed that the upper and lower bounds for the crossing number of \(\mathsf {MLG}(P)\) are \(6|P|-9\) and \(|P|-3\), respectively. In this paper, we improve these upper and lower bounds given by Bereg et al. (2016) to \(2.5|P|-5\) and \((1.25-\varepsilon )|P|\) for any \(\varepsilon > 0\), respectively. Especially, for improving the upper bound, we introduce a novel counting scheme based on some geometric observations.

This paper considers the numerical treatment of the time-dependent Gross–Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank–Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions and allows to capture the time invariants with an accuracy of order O ( H 6 ) \mathcal {O}(H^6) with respect to the chosen mesh size H H . This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the L ∞ ( L 2 ) L^{\infty }(L^2) -norm with order O ( τ 2 + H 4 ) \mathcal {O}(\tau ^2 + H^4) , where τ \tau denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.

In this paper, we develop a higher order numerical method for the fractional Bagley-Torvik equation. The main tools used include a new fourth-order approximation for the fractional derivative based on the weighted shifted Grünwald-Letnikov difference operator and a discrete cubic spline approach. We show that the theoretical convergence order improves those of previous work. Five examples are further presented to illustrate the efficiency of our method and to compare with other methods in the literature.

We obtain asymptotic accuracy of the poissonisation in the infinite occupancy scheme. All results are obtained for integer-valued random variables having a regularly varying distribution.

In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.

In this paper, we give an approach to the zeta values of a (proper regular) arithmetic scheme X at the integers r>=d:=dim(X), using \'etale cohomology of X with Q_p(r) and Z_p(r)-coefficients., Comment: 62 pages. Remark 2.2 has been added. Proposition 7.3 has been modified. Some details of the proof of Lemma 7.9 have been improved, and many typos have been corrected. To appear in Journal of Number Theory

Concentration properties of functionals of general Poisson processes are studied. Using a modified Phi-Sobolev inequality a recursion scheme for moments is established, which is of independent interest. This is applied to derive moment and concentration inequalities for functionals on abstract Poisson spaces. Applications of the general results in stochastic geometry, namely Poisson cylinder models and Poisson random polytopes, are presented as well. (C) 2021 Elsevier B.V. All rights reserved.

We demonstrate that in the mass-independent renormalization scheme, the renormalization group equations associated with the unphysical parameters that characterize the renormalization scheme and the mass scale leads to summation that results in a cancellation between the implicit and explicit dependence on these parameters. The resulting perturbative expansion is consequently independent of these arbitrary parameters. We illustrate this by considering R, the cross section for e+e– → hadrons.

In this article, the $H_{\infty }$ problem of nonlinear descriptor systems (NDSs) is investigated. For regular or nondescriptor systems, this problem can be solved via the event-triggered control law. But this control law may be ineffective for the same problem of descriptor systems. The reason is that the discontinuity of the event-triggered controller will cause sudden failure of the algegbraic part of descriptor systems at each triggering instant. From the perspective of mathematics, the algebraic equation will have no solution at each triggering instant. To address this problem, a novel hybrid control scheme is introduced. The main characteristic of this control scheme is that it consists of an event-triggered controller and an impulsive controller. It is shown that the impulsive controller plays an extremely important role in adjusting system states to satisfy the algebraic equation. Based on this hybrid control scheme, some improved results are proposed compared with the existing ones. Finally, an example about circuit systems is given to illustrate the effectiveness of the results.

We show that the one- and two-loop $\beta$-functions of the closed, bosonic string can be written in a manifestly O($D$,$D$)-covariant form. Based on this result, we prove that 1) Poisson-Lie symmetric $\sigma$-models are two-loop renormalisable and 2) their $\beta$-functions are invariant under Poisson-Lie T-duality. Moreover, we identify a distinguished scheme in which Poisson-Lie symmetry is manifest. It simplifies the calculation of two-loop $\beta$-functions significantly and thereby provides a powerful new tool to advance into the quantum regime of integrable $\sigma$-models and generalised T-dualities. As an illustrating example, we present the two-loop $\beta$-functions of the integrable $\lambda$- and $\eta$-deformation., Comment: 44 pages, 2 tables

In this paper, we provide a semi-implicit scheme based on the Euler–Maruyama method to study second-order stochastic systems driven by multiplicative independent additive noises. Using the Milstein’s fundamental convergence theorem, mean-square convergence analysis of the proposed approach is investigated. We examine the convergence of the stochastic system by considering the Lipschitz and linear growth bound conditions, Holder’s and Gronwall’s inequalities and also the Ito isometry. Finally, the numerical simulations demonstrate the potential of the introduced scheme to investigate the behavior of the second-order stochastic systems.

In this article, Modified Backward Euler Scheme is developed to solve the diffusion equation subject to nonlinear nonlocal boundary conditions. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differ- entiation for the time derivative term. Nonlinear terms in boundary conditions are linearized by Taylor expansion. Numerical examples are provided to verify the accuracy and efficiency of our proposed method.

В данной работе рассматривается первая краевая задача для уравнения Аллера дробного по времени порядка с обобщенными функциями памяти. Для численного решения поставленной задачи построены две разностные схемы повышенного порядка аппроксимации. В случае переменных коэффициентов предложена разностная схема второго порядка аппроксимации, как по времени, так и по пространству. А для обобщенного уравнения Аллера с постоянными коэффициентами предложена компактная разностная схема четвертого порядка аппроксимации по пространственной переменной и второго порядка по времени. Методом энергетических неравенств получены априорные оценки для решений предложенных разностных схем. Доказана их безусловная устойчивость и сходимость. Показано, что скорость сходимости совпадает с порядком погрешности аппроксимации в случае достаточно гладкого решения исходной задачи. На базе предложенных алгоритмов проведены численные расчеты тестовых задач, подтверждающие полученные теоретические результаты. Все вычисления выполнялись с помощью языка программирования Julia v1.5.1.

A CNF formula is harder than another CNF formula with the same number of clauses if it requires a longer resolution proof. In this paper we introduce resolution hardness numbers; they give for m=1,2,... the length of a shortest proof of a hardest formula on m clauses. We compute the first ten resolution hardness numbers, along with the corresponding hardest formulas. To achieve this, we devise a candidate filtering and symmetry breaking search scheme for limiting the number of potential candidates for hardest for- mulas, and an efficient SAT encoding for computing a shortest resolution proof of a given candidate formula.

The recent growing interest in urban air mobility (UAM) worldwide has led to the demand for physical analyses of the aerodynamic performance and aeroacoustic characteristics of electric vertical ta...

Richardson's leapfrog scheme is notoriously unconditionally unstable in well-posed, forward, linear dissipative evolution equations. Remarkably, that scheme can be stabilized, marched backward in t...