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It is demonstrated that one of the equations from the Lie classification list of second-order ODEs is a first integral of the Schwarz equation. As symmetry-preserving finite-difference schemes have been previously constructed for both equations, the preservation of a similar connection between these schemes is studied. It is shown that the schemes for the Schwarz equation and the second-order ODE (with an arbitrary constant $C$) can be related through a Bäcklund-type difference transformation. In addition, previously unexamined aspects of the difference scheme for the second-order ODE are discussed, including its singular solution and the complete set of difference first integrals for the case $C^2=4$.

High-fidelity numerical simulation serves as a cornerstone for exploring magnetization dynamics in micromagnetics. This work introduces a novel third-order temporally accurate and stable numerical scheme for the Landau-Lifshitz-Gilbert (LLG) equation, aiming to address the limitations in accuracy and efficiency often encountered with conventional approaches. Validation via nanostrip simulations confirms two principal advantages of the proposed method: it attains strict third-order temporal accuracy, surpassing many current techniques, and it offers superior computational efficiency, enabling rapid convergence without sacrificing numerical precision. For Gilbert damping coefficients $α$ ranging from $0.1$ to values below $10$, the scheme preserves strong stability and effectively avoids non-physical magnetization states. The magnetic microstructures predicted by this method are in excellent agreement with those from established benchmark methods, affirming its reliability for quantitative physical analysis. Salient distinctions between the proposed scheme and an existing third-order semi-implicit method include: (1) Solving the linear system associated with the existing scheme deman

The resummed thermodynamics of ${\cal N}=4$ supersymmetric Yang-Mills theory in four space-time dimensions ($\text{SYM}_{4,4}$) has been calculated previously to two loop order within hard thermal loop perturbation theory (HTLpt) using the canonical dimensional regularization (DRG) scheme. Herein, we revisit this calculation using the regularization by dimensional reduction (RDR) scheme. Since the RDR scheme manifestly preserves supersymmetry it is the preferred scheme, however, it is important to assess if and by how much the resummed perturbative results depend on the regularization scheme used. Comparing predictions for the scaled entropy obtained using the DRG and RDR schemes we find that for $λ\lesssim 6$ they are numerically very similar. We then compare the results obtained in both schemes with the strict perturbative result, which is accurate up to order $λ^2$, and a generalized Padé approximant constructed from the known large-$N_c$ weak- and strong-coupling expansions. Comparing the strict perturbative expansion of the two-loop HTLpt result with the perturbative expansion to order $λ^2$, we find that both the DRG and RDR HTLpt calculations result in the same scheme-indepe

We study Hodge loci as leaf schemes of foliations. The main ingredient is the Gauss-Manin connection matrix of families of projective varieties. We also aim to investigate a conjecture on the ring of definition of leaf schemes and its consequences such as the algebraicity of leaf schemes (Cattani-Deligne-Kaplan theorem in the case of Hodge loci). This conjecture is a consequence of a local-global principle for leaf schemes.

Electromagnetic information theory (EIT) is one of the emerging topics for 6G communication due to its potential to reveal the performance limit of wireless communication systems. For EIT, one of the most important research directions is degree of freedom (DoF) analysis. Existing research works on DoF analysis for EIT focus on asymptotic conclusions of DoF, which do not well fit the practical wireless communication systems with finite spatial regions and finite frequency bandwidth. In this paper, we use the theoretical analyzing tools from Slepian concentration problem and extend them to three-dimensional space domain and four-dimensional space-time domain under electromagnetic constraints. Then we provide asymptotic DoF conclusions and non-asymptotic DoF analyzing scheme, which suits practical scenarios better, under different scenarios like three-dimensional antenna array. Moreover, we theoretically prove that the channel DoF is upper bounded by the proposed DoF of electromagnetic fields. Finally, we use numerical analysis to provide some insights about the optimal spatial sampling interval of the antenna array, the DoF of three-dimensional antenna array, the impact of unequal an

The monotonicity and stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability and monotonicity of a non-linear scheme in terms of its corresponding scheme in variations. Such an approach leads to application of the stability theory for linear equation systems to establish stability of the corresponding non-linear scheme. The main methodological innovation is the theorems establishing the notion that a non-linear scheme is stable (and monotone) if the corresponding scheme in variations is stable (and, respectively, monotone). Criteria are developed for monotonicity and stability of difference schemes associated with the numerical analysis of systems of partial differential equations. The theorem of Friedrichs (1954) is generalized to be applicable to variational schemes with non-symmetric matrices. A new modification of the central Lax-Friedrichs (LxF) scheme is developed to be of the second order accuracy. A monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question. The stability and monotonicity of

An exponential time-integrator scheme of second-order accuracy based on the predictor-corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge-Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computationa

Calysto Scheme is written in Scheme in Continuation-Passing Style, and converted through a series of correctness-preserving program transformations into Python. It has support for standard Scheme functionality, including call/cc, as well as syntactic extensions, a nondeterministic operator for automatic backtracking, and many extensions to allow Python interoperation. Because of its Python foundation, it can take advantage of modern Python libraries, including those for machine learning and other pedagogical contexts. Although Calysto Scheme was developed with educational purposes in mind, it has proven to be generally useful due to its simplicity and ease of installation. It has been integrated into the Jupyter Notebook ecosystem and used in the classroom to teach introductory Programming Languages with some interesting and unique twists.

On the basis of the recent group classification of the one-dimensional magnetohydrodynamics (MHD) equations in cylindrical geometry, the construction of symmetry-preserving finite-difference schemes with conservation laws is carried out. New schemes are constructed starting from the classical completely conservative Samarsky-Popov schemes. In the case of finite conductivity, schemes are derived that admit all the symmetries and possess all the conservation laws of the original differential model, including previously unknown conservation laws. In the case of a frozen-in magnetic field (when the conductivity is infinite), various schemes are constructed that possess conservation laws, including those preserving entropy along trajectories of motion. The peculiarities of constructing schemes with an extended set of conservation laws for specific forms of entropy and magnetic fluxes are discussed.

Unconditional security in quantum key distribution (QKD) relies on authenticating the identities of users involved in key distribution. While classical identity authentication schemes were initially utilized in QKD implementations, concerns regarding their vulnerability have prompted the exploration of quantum identity authentication (QIA) protocols. In this study, we introduce a new protocol for QIA, derived from the concept of controlled secure direct quantum communication. Our proposed scheme facilitates simultaneous authentication between two users, Alice and Bob, leveraging Bell states with the assistance of a third party, Charlie. Through rigorous security analysis, we demonstrate that the proposed protocol withstands various known attacks, including impersonation, intercept and resend and impersonated fraudulent attacks. Additionally, we establish the relevance of the proposed protocol by comparing it with the existing protocols of similar type.

We advance the thesis that the simulation of quantum circuits is fundamentally about the efficient management of a large (potentially exponential) number of delimited continuations. The family of Scheme languages, with its efficient implementations of first-class continuations and with its imperative constructs, provides an elegant host for modeling and simulating quantum circuits.

We provide some conditions for the image of a morphism of abelian schemes to again be an abelian scheme. For context, in characteristic 0, the image is always an abelian scheme; in mixed and positive characteristic the image can fail to be an abelian scheme, and so it is in this setting that the conditions we provide are pertinent.

This paper presents a construction of a proper and stable labelled sample compression scheme of size $O(\VCD^2)$ for any finite concept class, where $\VCD$ denotes the Vapnik-Chervonenkis Dimension. The construction is based on a well-known model of machine teaching, referred to as recursive teaching dimension. This substantially improves on the currently best known bound on the size of sample compression schemes (due to Moran and Yehudayoff), which is exponential in $\VCD$. The long-standing open question whether the smallest size of a sample compression scheme is in $O(\VCD)$ remains unresolved, but our results show that research on machine teaching is a promising avenue for the study of this open problem. As further evidence of the strong connections between machine teaching and sample compression, we prove that the model of no-clash teaching, introduced by Kirkpatrick et al., can be used to define a non-trivial lower bound on the size of stable sample compression schemes.

We extend the notion of an enumeration scheme developed by Zeilberger and Vatter to the case of vincular patterns (also called "generalized patterns" or "dashed patterns"). In particular we provide an algorithm which takes in as input a set $B$ of vincular patterns and search parameters and returns a recurrence (called a "scheme") to compute the number of permutations of length $n$ avoiding $B$ or confirmation that no such scheme exists within the search parameters. We also prove that if $B$ contains only consecutive patterns and patterns of the form $σ_1σ_2 ... σ_{t-1}-σ_t$, then such a scheme must exist and provide the relevant search parameters. The algorithms are implemented in Maple and we provide empirical data on the number of small pattern sets admitting schemes. We make several conjectures on Wilf-classification based on this data. We also outline how to refine schemes to compute the number of $B$-avoiding permutations of length $n$ with $k$ inversions.

Voting forms the most important tool for arriving at a decision in any institution. The changing needs of the civilization currently demands a practical yet secure electronic voting system, but any flaw related to the applied voting technology can lead to tampering of the results with the malicious outcomes. Currently, blockchain technology due to its transparent structure forms an emerging area of investigation for the development of voting systems with a far greater security. However, various apprehensions are yet to be conclusively resolved before using blockchain in high stakes elections. Other than this, the blockchain based voting systems are vulnerable to possible attacks by upcoming noisy intermediate scale quantum (NISQ) computer. To circumvent, most of these limitations, in this work, we propose an anonymous voting scheme based on quantum assisted blockchain by enhancing the advantages offered by blockchain with the quantum resources such as quantum random number generators and quantum key distribution. The purposed scheme is shown to satisfy the requirements of a good voting scheme. Further, the voting scheme is auditable and can be implemented using the currently availa

The following three types of objects are considered in a dual functorial formalism: (i) ind-scheme of mappings between two schemes, (ii) for a quantum group G, ind-scheme of G-mappings between two G-schemes, and (iii) ind-scheme of group homomorphisms between two quantum group. By schemes and quantum groups here we mean objects which are respectively dual to unital associative algebras and Hopf algebras.

This article studies an extended Nori and local fundamental group schemes of Abelian varieties. We also discuss the birational invariance of these group schemes and study their behaviour under the Albanese and étale morphisms.

Despite continued efforts to improve classification accuracy, it has been reported that offline accuracy is a poor indicator of the usability of pattern recognition-based myoelectric control. One potential source of this disparity is the existence of transitions between contraction classes that happen during regular use and are reported to be problematic for pattern recognition systems. Nevertheless, these transitions are often ignored or undefined during both the training and testing processes. In this work, we propose a set of metrics for analyzing the transitions that occur during the voluntary changes between contraction classes during continuous control. These metrics quantify the common types of errors that occur during transitions and compare them to existing metrics that apply only to the steady-state portions of the data. We then use these metrics to analyze transition characteristics of 6 commonly used classifiers on a novel dataset that includes continuous transitions between all combinations of seven different contraction classes. Results show that a linear discriminant classifier consistently outperforms other conventional classifiers during both transitions and steady

Directed signature scheme is applicable when the signed message contains information sensitive to the receiver, because only receiver can directly verify the signature and that he/she can prove its validity to any third party, whenever necessary. This paper presents two applications of directed signature scheme. (i) Directed –Delegated Signature Scheme. This scheme combines the idea of proxy signatures with directed signature scheme. (ii) Allocation of registration number. This scheme proposes a registration scheme in which the registration number cannot be forged and misused.