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For a smooth projective scheme X over a ring R on which p is nilpotent that meets some general assumptions we prove that the crystalline cohomology is equipped with the structure of a higher display which is a relative version of Fontaine’s strongly divisible lattices. Frobenius-divisibility is induced by the Nygaard filtration on the relative de Rham–Witt complex. For a nilpotent PD-thickening S/R we also consider the associated relative display and can describe it explicitly by a relative version of the Nygaard filtration on the de Rham–Witt complex associated to a lifting of X over S. We prove that there is a crystal of relative displays if moreover the mod p reduction of X has a smooth and versal deformation space.

One of the key questions in the analysis of numerical approximations to differential equations, and related problems, is in the comparison of two mappings. One mapping would be the exact flow through a specified time step and the other would be a numerical scheme, such as a Runge–Kutta method. The B-series approach to questions like this is to write the Taylor expansions of the two mappings in a special way, in terms of “elementary differentials”, and to then compare the coefficients in corresponding terms. The terms themselves can be indexed in terms of rooted trees and the theory of B-series hinges on this indexing. The chapter includes a discussion of multi-dimensional Taylor series and it is shown how this leads to the formulation of elementary differentials and B-series. Some sample problems, which are both easy and fundamental, are solved, first in a low order introduction and then in full generality. Special attention is given to the flow through a unit step, followed by an implicit variant of the Euler method. It is remarkable that the latter simple example is a direct path to the large family of Runge–Kutta methods. The composition rule for B-series is introduced and some of its wider ramifications are explored.

A multi-party non-interactive key-exchange (\(\mathsf {NIKE}\)) scheme enables N users to securely exchange a secret key \( K \) in a non-interactive manner. It is well-known that \(\mathsf {NIKE}\) schemes can be obtained assuming the existence of indistinguishability obfuscation (\(\mathsf {iO}\)).

This paper provides an introduction to an information theoretic formulation of the embedding problem for Riemannian manifolds developed by the author. The main new construct is a stochastic relaxation scheme for embedding problems and hard constraint systems. This scheme is introduced with examples and context.

In this chapter, we revisit the well-posedness of the model parabolic problem, i.e., we give another proof of Lions’ theorem using the framework of the BNB theorem. In other words, we establish the well-posedness by proving an inf-sup condition. Then we exploit the inf-sup condition to revisit the stability and the error analysis for various approximation techniques investigated in the previous chapters: (1) the space semi-discrete problem; (2) the implicit Euler scheme; (3) the discontinuous Galerkin scheme; (4) the continuous Petrov–Galerkin scheme.

A secret-sharing scheme allows to distribute a secret s among n parties such that only some predefined “authorized” sets of parties can reconstruct the secret, and all other “unauthorized” sets learn nothing about s. The collection of authorized/unauthorized sets can be captured by a monotone function \(f:\{0,1\}^n\rightarrow \{0,1\}\). In this paper, we focus on monotone functions that all their min-terms are sets of size a, and on their duals – monotone functions whose max-terms are of size b. We refer to these classes as (a, n)-upslices and (b, n)-downslices, and note that these natural families correspond to monotone a-regular DNFs and monotone \((n-b)\)-regular CNFs. We derive the following results.

This chapter deals with mathematical models of continuous crystallization described by hyperbolic systems of partial differential equations coupled with ordinary and integro-differential equations. The considered systems admit nonzero steady-state solutions with constant inputs. To stabilize these solutions, we present an approach for constructing control Lyapunov functionals based on quadratic forms in weighted L2-spaces. It is shown that the proposed control design scheme guarantees exponential stability of the closed-loop system.

Uncertain differential equation (UDE) is a type of differential equations driven by canonical Liu’s process.In this paper, we study a new numerical method for solving UDEs the improved predictor corrector. The convergence and stability of the proposed method are presented in details. Furthermore, the Vasicek interest rate model is given for numerical result.

In this paper we provide an efficient construction of a lattice-based polynomial argument and a polynomial batch-protocol, where the latter contains the polynomial argument as a building block. Our contribution is motivated by the discrete log based construction (EUROCRYPT’16), where in our case we employ different techniques to obtain a communication efficient lattice-based scheme. In the zero-knowledge polynomial batch-protocol, we prove the knowledge of an easy relation between two polynomials which also allows batching of several instances of the same relation. Our batch-protocol is applicable to an efficient lattice-based range proof construction which represents a useful application in cryptocurrencies. In contrast to the existing range proof (CRYPTO’19), our proof is more efficient for large number of batched instances.

A distance labeling scheme is an assignment of labels, that is, binary strings, to all nodes of a graph, so that the distance between any two nodes can be computed from their labels without any additional information about the graph. The goal is to minimize the maximum length of a label as a function of the number of nodes. A major open problem in this area is to determine the complexity of distance labeling in unweighted planar (undirected) graphs. It is known that, in such a graph on n nodes, some labels must consist of \(\varOmega (n^{1/3})\) bits, but the best known labeling scheme constructs labels of length \(\mathcal {O}(\sqrt{n}\log n)\) [Gavoille, Peleg, Perennes, and Raz, J. Algorithms, 2004]. For weighted planar graphs with edges of length polynomial in n, we know that labels of length \(\varOmega (\sqrt{n}\log n)\) are necessary [Abboud and Dahlgaard, FOCS 2016]. Surprisingly, we do not know if distance labeling for weighted planar graphs with edges of length polynomial in n is harder than distance labeling for unweighted planar graphs. We prove that this is indeed the case by designing a distance labeling scheme for unweighted planar graphs on n nodes with labels consisting of \(\mathcal {O}(\sqrt{n})\) bits with a simple and (in our opinion) elegant method. We augment the construction with a mechanism that allows us to compute the distance between two nodes in only polylogarithmic time while increasing the length by \(\mathcal {O}(\sqrt{n\log n})\). The previous scheme required \(\varOmega (\sqrt{n})\) time to answer a query in this model.

In this chapter, we continue the study started in the previous chapter on higher-order time approximation schemes using a space-time functional framework. The trial functions are now continuous in time and piecewise polynomials with a polynomial degree that is one order higher than that of the test functions. The resulting technique is called continuous Petrov–Galerkin method, and its lowest-order version is the Crank–Nicolson scheme. Like the discontinuous Galerkin schemes, the continuous Petrov–Galerkin schemes are implicit one-step methods. They can also be interpreted as implicit Runge–Kutta methods.

We construct a two-round Schnorr-based signature scheme (DWMS) by delinearizing two pre-commitments supplied by each signer. DWMS is a secure signature scheme in the algebraic group model (AGM) and the random oracle model (ROM) under the assumption of the hardness of the one-more discrete logarithm problem and the 2-entwined sum problem that we introduce in this paper. Our new m-entwined sum problem tweaks the k-sum problem in a scalar field using the associated group. We prove the hardness of our new problem in the AGM assuming the hardness of the discrete logarithm problem in the associated group. We believe that our new problem simplifies the security proofs of multi-signature schemes that use the delinearization of commitments.

In this lecture we look at the convergence of the implicit Euler scheme with a more variational perspective, providing convergence results valid in general metric spaces (E, d). A basic reference for this part is [12].

We study from a numerical point of view the solution of the system of partial differential equations arising in the theory of isotropic dipolar thermoelasticity with double porosity. We write the variational formulation and introduce fully discrete approximations by using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the first-order time derivatives. By using these algorithms, we perform numerical simulations in order to show the behaviour of the solution. To this end, we use the finite element software FreeFem++.

An implementation of Discontinuous Galerkin method designed for unsteady computations is presented. Explicit 4-th order 5-stage strong stability-preserving Runge-Kutta scheme is used together with global time stepping technique. Shape functions are taken to be orthonormal polynomials in physical space. Polynomial orders K of up to 5 are used, formally giving a K + 1 accuracy order. Bassi and Rebay 2 approximation of viscous fluxes is adopted. Test cases presented are DNS of Taylor-Green Vortex at Re = 1600 and ILES/DDES computations of ERCOFTAC Periodic Hills test case at Re = 10 595. The results include accuracy versus computational cost comparisons.

Let \(N=pq\) be an RSA modulus with balanced prime factors. In 2018, Murru and Saettone presented a variant of the RSA cryptosystem based on a cubic Pell equation in which the public key (N, e) and the private key (N, d) satisfy \(ed\equiv 1\pmod {\left( p^2+p+1\right) \left( q^2+q+1\right) }\). They claimed that the classical small private attacks on RSA such as Wiener’s continued fraction attack do not apply to their scheme. In this paper, we show that, on the contrary, Wiener’s method as well as the small inverse problem technique of Boneh and Durfee can be applied to attack their scheme. More precisely, we show that the proposed variant of RSA can be broken if \(d

In this lecture, we will prove the Brezis-Komura Theorem 11.7. We start with a brief presentation of the classical proof, in the more general context of the theory of maximal monotone operators, based on the so-called Yosida regularization. Then, we describe in detail another strategy to prove the result, based on the so-called implicit Euler scheme.

Recently, a new code based signature scheme, called LESS, was proposed with three concrete instantiations, each aiming to provide 128 bits of classical security [3]. Two instantiations (LESS-I and LESS-II) are based on the conjectured hardness of the linear code equivalence problem, while a third instantiation, LESS-III, is based on the conjectured hardness of the permutation code equivalence problem for weakly self-dual codes. We give an improved algorithm for solving both these problems over sufficiently large finite fields. Our implementation breaks LESS-I and LESS-III in approximately 25 s and 2 s respectively on an Intel i5-8400H CPU. Since the field size for LESS-II is relatively small \((\mathbb {F}_7)\) our algorithm does not improve on existing methods. Nonetheless, we estimate that LESS-II can be broken with approximately \(2^{44}\) row operations.

The problem of fitting multidimensional reduced data \(\mathcal{M}_n\) is discussed here. The unknown interpolation knots \(\mathcal{T}\) are replaced by optimal knots which minimize a highly non-linear multivariable function \(\mathcal{J}_0\). The numerical scheme called Leap-Frog Algorithm is used to compute such optimal knots for \(\mathcal{J}_0\)via the iterative procedure based in each step on single variable optimization of \(\mathcal{J}_0^{(k,i)}\). The discussion on conditions enforcing unimodality of each \(\mathcal{J}_0^{(k,i)}\) is also supplemented by illustrative examples both referring to the generic case of Leap-Frog. The latter forms a new insight on fitting reduced data and modelling interpolants of \(\mathcal{M}_n\).

The object of study in this chapter is a scheme over the spectrum of a local ring. A separately standing Section 10.1 is devoted to the concepts of a split variety and of a split fibre of a morphism of varieties; for arithmetic applications and for the calculation of the Brauer group, split fibres should be considered as ‘good’ or ‘non-degenerate’.

These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to see how motives may be used to enhance cohomological methods, giving natural ways to prove independence of $\ell$ results for traces and zeta-functions, and constructions of characteristic classes (as $0$-cycles). This leads to the Grothendieck-Lefschetz formula, of which we give a new motivic proof. There are also a few additions to what have been told in the lectures: a proof of Grothendieck-Verdier duality of etale motives on schemes of finite type over a regular quasi-excellent scheme (which slightly improves the level of generality in the existing literature); a proof that $\mathbf{Q}$-linear motivic sheaves are virtually integral; a proof of the motivic generic base change formula.

We consider the graph k-partitioning problem under the min-max objective, termed as \(\textsc {Minmax}\;k\text {-}\textsc {cut}\). The input here is a graph \(G=(V,E)\) with non-negative edge weights \(w:E\rightarrow \mathbb {R}_+\) and an integer \(k\ge 2\) and the goal is to partition the vertices into k non-empty parts \(V_1, \ldots , V_k\) so as to minimize \(\max _{i=1}^k w(\delta (V_i))\). Although minimizing the sum objective \(\sum _{i=1}^k w(\delta (V_i))\), termed as \(\textsc {Minsum}\;k\text {-}\textsc {cut}\), has been studied extensively in the literature, very little is known about minimizing the max objective. We initiate the study of \(\textsc {Minmax}\;k\text {-}\textsc {cut}\) by showing that it is NP-hard and W[1]-hard when parameterized by k, and design a parameterized approximation scheme when parameterized by k. The main ingredient of our parameterized approximation scheme is an exact algorithm for \(\textsc {Minmax}\;k\text {-}\textsc {cut}\) that runs in time \((\lambda k)^{O(k^2)}n^{O(1)}\), where \(\lambda \) is the value of the optimum and n is the number of vertices. Our algorithmic technique builds on the technique of Lokshtanov, Saurabh, and Surianarayanan (FOCS, 2020) who showed a similar result for \(\textsc {Minsum}\;k\text {-}\textsc {cut}\). Our algorithmic techniques are more general and can be used to obtain parameterized approximation schemes for minimizing \(\ell _p\)-norm measures of k-partitioning for every \(p\ge 1\).

We consider the multi-broadcast problem in arbitrary connected radio networks consisting of n nodes. There are k designated source nodes for some fixed \(k \in \{1,\ldots ,n\}\), and each source node has a piece of information that it wants to share with all nodes in the network. We set out to determine the shortest possible labels so that multi-broadcast can be solved deterministically in the labeled radio network by some deterministic distributed algorithm. First, we show that every radio network G with maximum degree \(\varDelta \) can be labeled using \(O(\min \{\log k,\log \varDelta \})\)-bit labels in such a way that multi-broadcast with k sources can be accomplished. This bound is tight for certain network topologies (e.g., complete graphs), but there are networks where significantly shorter labels are sufficient, e.g., we show how to construct a tree with maximum degree \(\varTheta (\sqrt{n})\) in which gossiping (i.e., multi-broadcast with n sources) can be solved after labeling each node with O(1) bits. For all trees, we provide a labeling scheme and algorithm that will solve gossiping, and, we prove an impossibility result that demonstrates that our labeling scheme is optimal for gossiping in each tree. In particular, we prove that \(\varTheta (\log D(G))\)-bit labels are necessary and sufficient in each tree G, where D(G) denotes the distinguishing number of G.

We construct the first decentralized multi-authority attribute-based encryption (\(\mathsf {MA}\text {-}\mathsf {ABE}\)) scheme for a non-trivial class of access policies whose security is based (in the random oracle model) solely on the Learning With Errors (LWE) assumption. The supported access policies are ones described by \(\mathsf {DNF}\) formulas. All previous constructions of \(\mathsf {MA}\text {-}\mathsf {ABE}\) schemes supporting any non-trivial class of access policies were proven secure (in the random oracle model) assuming various assumptions on bilinear maps.

In this paper, we present an Asymptotic Preserving scheme for a stochastic linear kinetic equation. Its construction is based on a micro-macro decomposition. We start by explaining how we build it and then perform the formal numerical limit. After stating some stability results proved in [1], some numerical tests confirming the good performances of our scheme are finally presented.

We prove that for \(n\ge 2\), the size \(b(t_n)\) of the smallest bidirectional scheme for the nth Thue–Morse word \(t_n\) is \(n+2\). Since Kutsukake et al. [SPIRE 2020] show that the size \(\gamma (t_n)\) of the smallest string attractor for \(t_n\) is 4 for \(n \ge 4\), this shows for the first time that there is a separation between the size of the smallest string attractor \(\gamma \) and the size of the smallest bidirectional scheme b, i.e., there exist string families such that \(\gamma = o(b)\).

The Commutative Supersingular Isogeny Diffie-Hellman key exchange (CSIDH) uses supersingular elliptic curves of Montgomery form over \(\mathbb {F}_p\) with \(p\equiv 3\) \((\mathrm{mod}\) 8), while CSURF considered those of Montgomery\(^-\) form with \(p\equiv 7\) \((\mathrm{mod}\) 8). The two protocols both have ratio (1:1) between the coefficients and the \(\mathbb {F}_p\)-isomorphism classes. Castryck and Decru showed that the ratio became (2:1) when Montgomery supersingular curves with endomorphism ring \(\mathbb {Z}[(1+\sqrt{-p})/2]\) and \(p\equiv 7\) \((\mathrm{mod}\) 8) were considered. The fact that the coefficients are not unique for each \(\mathbb {F}_p\)-isomorphism class is the major obstacle to use this case in the scheme.

We investigate semi-discrete maximal surfaces with singularities in Minkowski 3-space. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space admit a Weierstrass-type representation and they generally have singularities. In this paper, we first describe semi-discrete isothermic maximal surfaces in Minkowski 3-space and give a Weierstrass-type representation for them determined from integrable system principles. Furthermore, we show that semi-discrete isothermic maximal surfaces admit associated one-parameter families of deformations whose mean curvature remains identically 0. Finally we give a criterion that naturally describes the unified scheme of the “singular set” for these semi-discrete maximal surfaces, including the associated family.

It is immediately apparent from the definition, and the basic examples we studied, that the concept of a scheme is far more general than the concept of a variety as introduced in Chap. 1, just as a topological space is much more general than a subset of \(\mathbb {R}^n\). What are the properties of schemes we should study?

An indistinguishability obfuscator is a polynomial-time probabilistic algorithm that takes a circuit as input and outputs a new circuit that has the same functionality as the input circuit, such that for any two circuits of the same size that compute the same function, the outputs of the indistinguishability obfuscator are indistinguishable. Here, we study schemes for indistinguishability obfuscation for quantum circuits. We present two definitions for indistinguishability obfuscation: in our first definition (\(qi\mathcal {O}\)) the outputs of the obfuscator are required to be indistinguishable if the input circuits are perfectly equivalent, while in our second definition (\(qi\mathcal {O}_\mathbf{D}\)), the outputs are required to be indistinguishable as long as the input circuits are approximately equivalent with respect to a pseudo-distance D. Our main results provide (1) a computationally-secure scheme for \(qi\mathcal {O}\) where the size of the output of the obfuscator is exponential in the number of non-Clifford (\(\mathsf{T}\) gates), which means that the construction is efficient as long as the number of \(\mathsf{T}\) gates is logarithmic in the circuit size and (2) a statistically-secure \(qi\mathcal {O}_\mathbf{D},\) for circuits that are close to the \(k\)th level of the Gottesman-Chuang hierarchy (with respect to D); this construction is efficient as long as \(k\) is small and fixed.

We propose a symplectic finite difference scheme for the Boussinesq equations with sixth order dispersion terms. This scheme conserves exactly the discrete mass and approximately with error \(O(h^2+\tau ^2)\) the discrete Hamiltonian. The numerical experiments for quadratic and cubic nonlinearity confirm the theoretical results.

The size b of the smallest bidirectional macro scheme, which is arguably the most general copy-paste scheme to generate a given sequence, is considered to be the strictest reachable measure of repetitiveness. It is strictly lower-bounded by measures like \(\gamma \) and \(\delta \), which are known or believed to be unreachable and to capture the entropy of repetitiveness. In this paper we study another sequence generation mechanism, namely compositions of a morphism. We show that these form another plausible mechanism to characterize repetitive sequences and define NU-systems, which combine such a mechanism with macro schemes. We show that the size \(\nu \le b\) of the smallest NU-system is reachable and can be \(o(\delta )\) for some string families, thereby implying that the limit of compressibility of repetitive sequences can be even smaller than previously thought. We also derive several other results characterizing \(\nu \).

In this chapter, we consider the same space semi-discrete problem as in the previous chapter, but we now discretize it in time using an explicit scheme. We first discuss generic properties of explicit Runge–Kutta schemes (ERK). Then we analyze the explicit Euler scheme, second-order two-stage ERK schemes, and third-order three-stage ERK schemes. The key advantage of explicit schemes over implicit schemes is that the linear algebra at each time step is greatly simplified since one has to invert only the mass matrix. However, the stability of ERK schemes requires that the time step be limited by a CFL-like condition.

We construct the first efficient, unconditionally secure MPC protocol that only requires black-box access to a non-commutative ring R. Previous results in the same setting were efficient only either for a constant number of corruptions or when computing branching programs and formulas. Our techniques are based on a generalization of Shamir’s secret sharing to non-commutative rings, which we derive from the work on Reed Solomon codes by Quintin, Barbier and Chabot (IEEE Transactions on Information Theory, 2013). When the center of the ring contains a set \(A = \{\alpha _0, \ldots , \alpha _n\}\) such that \(\forall i \ne j, \alpha _i \,-\, \alpha _j \in R^*\), the resulting secret sharing scheme is strongly multiplicative and we can generalize existing constructions over finite fields without much trouble.

In this paper, we extend the notion of server-aided revocable identity-based encryption (SR-IBE) to the hierarchical IBE (HIBE) setting to obtain the definition of server-aided revocable hierarchical IBE (SR-HIBE), and consider a stronger security called SSR-a/sID-CPA security. Specifically, the security required that the challenge identity \(\mathsf {ID}^*\) is revoked when both the private key for \(\mathsf {ID}^*\) or some ancestors of \(\mathsf {ID}^*\) and the public key for \(\mathsf {ID}^*\) are revealed, or both the private key and the secret key for \(\mathsf {ID}^*\) or some (may be different) ancestors of \(\mathsf {ID}^*\) are revealed. And the adversary can have access to the transformation keys oracle. Then, we propose a generic construction of SR-HIBE schemes from any L-level revocable HIBE scheme and (L + 1)-level HIBE scheme. The security of our generic SR-HIBE scheme inherits those of the underlying building blocks. Furthermore, when the maximum hierarchical depth is one, we obtain a generic construction of SR-IBE schemes from any IBE scheme and two-level HIBE scheme.

In this paper, a space source identification problem for parabolic equation with involution and Neumann condition is studied. The well-posedness theorem on the differential equation of the source identification parabolic problem is established. For the approximate solution of the problem, a stable difference scheme and its stability estimates are presented. The theoretical results are supported by the numerical results of a test problem.

Let \(E_a/ \mathbb {F}_{2}: y^2+xy=x^3+ax^2+1\) be a Koblitz curve. The window \(\tau \)-adic non-adjacent form (window \(\tau \)NAF) is currently the standard representation system to perform scalar multiplications on \(E_a/ \mathbb {F}_{2^m}\) utilizing the Frobenius map \(\tau \). This work focuses on the pre-computation part of scalar multiplication. We first introduce \(\mu \bar{\tau }\)-operations where \(\mu =(-1)^{1-a}\) and \(\bar{\tau }\) is the complex conjugate of \(\tau \). Efficient formulas of \(\mu \bar{\tau }\)-operations are then derived and used in a novel pre-computation scheme. Our pre-computation scheme requires 6M\(\,+\,6\)S, 18M\(\,+\,17\)S, 44M\(\,+\,32\)S, and 88M\(\,+\,62\)S (\(a=0\)) and 6M\(\,+\,6\)S, 19M\(\,+\,17\)S, 46M\(\,+\,32\)S, and 90M\(\,+\,62\)S (\(a=1\)) for window \(\tau \)NAF with widths from 4 to 7 respectively. It is about two times faster, compared to the state-of-the-art technique of pre-computation in the literature. The impact of our new efficient pre-computation is also reflected by the significant improvement of scalar multiplication. Traditionally, window \(\tau \)NAF with width at most 6 is used to achieve the best scalar multiplication. Because of the dramatic cost reduction of the proposed pre-computation, we are able to increase the width for window \(\tau \)NAF to 7 for a better scalar multiplication. This indicates that the pre-computation part becomes more important in performing scalar multiplication. With our efficient pre-computation and the new window width, our scalar multiplication runs in at least 85.2% the time of Kohel’s work (Eurocrypt’2017) combining the best previous pre-computation. Our results push the scalar multiplication of Koblitz curves, a very well-studied and long-standing research area, to a significant new stage.

We study the K-item knapsack problem (i.e., 1.5-dimensional knapsack problem), a generalization of the famous 0–1 knapsack problem (i.e., 1-dimensional knapsack problem) in which an upper bound K is imposed on the number of items selected. This problem is of fundamental importance and is known to have a broad range of applications in various fields. It is well known that, there is no fully polynomial time approximation scheme (FPTAS) for the d-dimensional knapsack problem when \(d\ge 2\), unless P \(=\) NP. While the K-item knapsack problem is known to admit an FPTAS, the complexity of all existing FPTASs have a high dependency on the cardinality bound K and approximation error \(\varepsilon \), which could result in inefficiencies especially when K and \(\varepsilon ^{-1}\) increase. The current best results are due to [Mastrolilli and Hutter, 2006], in which two schemes are presented exhibiting a space-time tradeoff–one scheme with time complexity \(O(n+Kz^{2}/\varepsilon ^{2})\) and space complexity \(O(n+z^{3}/\varepsilon )\), and another scheme that requires a run-time of \(O(n+(Kz^{2}+z^{4})/\varepsilon ^{2})\) but only needs \(O(n+z^{2}/\varepsilon )\) space, where \(z=\min \{K,1/\varepsilon \}\).

This research examines and compares the construction of protein-protein interaction (PPI) networks of CD4\(^{+}\) and CD8\(^{+}\)T cells and investigates why studying these cells is critical after HIV infection. This study also examines a mathematical model of fractional HIV infection of CD4\(^{+}\)T cells and proposes a new numerical procedure for this model that focuses on a recent kind of orthogonal polynomials called discrete Chebyshev polynomials. The proposed scheme consists of reducing the problem by extending the approximated solutions and by using unknown coefficients to nonlinear algebraic equations. For calculating unknown coefficients, fractional operational matrices for orthogonal polynomials are obtained. Finally, there is an example to show the effectiveness of the recommended method. All calculations were performed using the Maple 17 computer code.

The classical Adams operations are defined on the Grothendieck group of algebraic vector bundles on a scheme, and are related to Grothendieck’s Riemann-Roch Theorem. In these notes, I review the theory of Adams operations for Grothendieck groups with supports, following the work of Gillet and Soule, and give two applications in local algebra: A proof of the Serre Vanishing Conjecture for arbitrary regular local rings (due to Gillet and Soule) and a proof of the Total Rank Conjecture for many local rings.

We consider a family of singular Laplace-Beltrami operators, focussing our attention on the problem of so-called quantum confinement in the half-plane equipped with Riemannian metrics of Grushin type degenerate at the boundary. By introducing a constant-fiber direct integral scheme we are able to rigorously characterize the presence or absence of self-adjointness of these operators. We also compare our technique and result with the already studied problem of quantum confinement in the half-cylinder.

This paper is devoted to algebraically stabilized finite element methods for the numerical solution of convection–diffusion–reaction equations. First, the algebraic flux correction scheme with the popular Kuzmin limiter is presented. This limiter has several favourable properties but does not guarantee the validity of the discrete maximum principle for non-Delaunay meshes. Therefore, a generalization of the algebraic flux correction scheme and a modification of the limiter are proposed which lead to the discrete maximum principle for arbitrary meshes. Numerical results demonstrate the advantages of the new method.

This work is an extension and improvement of [1] which dealt with a convergence analysis of a FVS (Finite Volume Scheme) using the Characteristic method for non-stationary LINEAR advection-diffusion equations. In this note, we address the case of non-stationary SEMILINEAR advection-diffusion equations. We establish two FVSs, one is linear and the other is nonlinear, which uses the discrete gradient developed in [5] and an approximation of the equation using the Characteristic method. For the sake of simplicity of the present note, we only focus on the linear scheme and we prove its convergence. The convergence analysis relies mainly on a well developed new discrete a prior estimate.

In this work, post-buckling problems of structures like shell are investigated using an implicit conservative time integration dynamic scheme. We have proposed the use of a four nodded quadrilateral flat shell element, with the drilling rotation degree of freedom (D. Boutagouga., A new enhanced assumed strain quadrilateral membrane element with drilling degree of freedom and modified shape functions, Int. J. Numer. Meth. Eng. 2017; 110:573–600) and based upon the updated Lagrangian formulation. An implicit conservative scheme (K.J, Bathe., G, Noh., Insight into an implicit time integration scheme for structural dynamics. Comput. Struct. 2012; 98–99: 1–6) is chosen to obtain the solution to the nonlinear dynamic behaviour and geometric nonlinear post-buckling response.

We are interested in the numerical approximation of the shallow water equations in two space dimensions. We propose a well-balanced, all-regime, and positive scheme. Our approach is based on a Lagrange-projection decomposition which allows to naturally decouple the acoustic and transport terms.

We show new partial and full instantiation results under chosen-ciphertext security for the widely implemented and standardized RSA-OAEP encryption scheme of Bellare and Rogaway (EUROCRYPT 1994) and two variants. Prior work on such instantiations either showed negative results or settled for “passive” security notions like IND-CPA. More precisely, recall that RSA-OAEP adds redundancy and randomness to a message before composing two rounds of an underlying Feistel transform, whose round functions are modeled as random oracles (ROs), with RSA. Our main results are: Either of the two oracles (while still modeling the other as a RO) can be instantiated in RSA-OAEP under IND-CCA2 using mild standard-model assumptions on the round functions and generalizations of algebraic properties of RSA shown by Barthe, Pointcheval, and Baguelin (CCS 2012). The algebraic properties are only shown to hold at practical parameters for small encryption exponent (\(e=3\)), but we argue they have value for larger e as well. Both oracles can be instantiated simultaneously for two variants of RSA-OAEP, called “t-clear” and “s-clear” RSA-OAEP. For this we use extractability-style assumptions in the sense of Canetti and Dakdouk (TCC 2010) on the round functions, as well as novel yet plausible “XOR-type” assumptions on RSA. While admittedly strong, such assumptions may nevertheless be necessary at this point to make positive progress.

We suggest a general framework to study dependency schemes for dependency quantified Boolean formulas (DQBF). As our main contribution, we exhibit a new tautology-free DQBF dependency scheme that generalises the reflexive resolution path dependency scheme. We establish soundness of the tautology-free scheme, implying that it can be used in any DQBF proof system. We further explore the power of DQBF resolution systems parameterised by dependency schemes and show that our new scheme results in exponentially shorter proofs in comparison to the reflexive resolution path dependency scheme when used in the expansion DQBF system \(\mathsf {\forall {Exp{\text{+ }}Res}}\).

Multi-agent path finding with continuous movements and time (denoted MAPF\(^\mathcal {R}\)) is addressed. The task is to navigate agents that move smoothly between predefined positions to their individual goals so that they do not collide. Recently a novel solving approach for obtaining makespan optimal solutions called SMT-CBS\(^\mathcal {R}\) based on satisfiability modulo theories (SMT) has been introduced. We extend the approach further towards the sum-of-costs objective which is a more challenging case in the yes/no SMT environment due to more complex calculation of the objective. The new algorithm combines collision resolution known from conflict-based search (CBS) with previous generation of incomplete propositional encodings on top of a novel scheme for selecting decision variables in a potentially uncountable search space. We experimentally compare SMT-CBS\(^\mathcal {R}\) and previous CCBS algorithm for MAPF\(^\mathcal {R}\).

We study Godunov’s method for diffusion and advection-diffusion problems. The numerical fluxes for the finite volume scheme are based on an approximation of the generalized Riemann Problem. Hereby, approximate Riemann solvers are constructed, which approximate the solutions by a space-time discontinuous Galerkin approach. The implementation to a Godunov-type finite volume method and numerical results are discussed.