Pick’s Theorem in Two-Dimensional Subspace of R3 by Lin Si

“…In the Euclidean space R3, denote the set of all points with integer coordinate by Z3. …”
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Adjoint curves of special Smarandache curves with respect to Bishop frame by Esra Damar

“…In this paper, adjoint curves generated by means of integral curves of special Smarandache curves with respect to the Bishop frame in three-dimensional Euclidean space were introduced. Relations between the main curve and the Bishop apparatus of these adjoint curves were obtained. …”
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Boundedness of solutions of matrix nonlinear volterra difference equations by Michael I. Gil', Rigoberto Medina

“…Nonlinear discrete-time Volterra equations in a Euclidean space are considered. Conditions for the boundedness of solutions are established by virtue of recent estimates for the norm of the resolvent of Volterra operators. …”
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On Linear Difference Equations for Which the Global Periodicity Implies the Existence of an Equilibrium by István Győri, László Horváth

“…This result is extended for higher order linear inhomogeneous system in a real or complex Euclidean space. The work was highly motivated by the early works of Smith (1934, 1941) and the papers of Kister (1961) and Bas (2011).…”
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Estimation of the Value at Risk Using the Stochastic Approach of Taylor Formula by Vini Yves Bernadin Loyara, Remi Guillaume Bagré, Diakarya Barro

“…A higher dimensional extension of the Taylor–Young formula is used for this estimation in a Euclidean space. Moreover, a time-varying and conditional copula is used for the modeling of the VaR.…”
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On CR-Lightlike Product of an Indefinite Kaehler Manifold by Rakesh Kumar, Jasleen Kaur, R. K. Nagaich

“…Mixed foliate CR-lightlike submanifold of indefinite complex space form has also been discussed and showed that the indefinite Kaehler manifold becomes the complex semi-Euclidean space.…”
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Study of Differential Equations on Warped Product Semi-Invariant Submanifolds of the Generalized Sasakian Space Forms by Ibrahim Al-Dayel

“…More precisely, by analyzing Bochner’s formula on these inequalities, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to Euclidean space. We also look at the effects of certain differential equations on warped product semi-invariant product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.…”
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Linear algebra and differential geometry on abstract Hilbert space by Alexey A. Kryukov

“…However, while n-dimensional spaces in applications are always realized as the Euclidean space Rn, Hilbert spaces admit various useful realizations as spaces of functions. …”
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Weyl-Euler-Lagrange Equations of Motion on Flat Manifold by Zeki Kasap

“…Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. …”
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Small sets in convex geometry and formal independence over ZFC by Menachem Kojman

“…To each closed subset S of a finite-dimensional Euclidean space corresponds a σ-ideal of sets 𝒥 (S) which is σ-generated over S by the convex subsets of S. …”
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Some inequalities for eigenvalues of an elliptic differential operator by Shahroud Azami, Mosayeb Zohrehvand, Ghodratallah Fasihi-Ramandi

“…Applying this inequality, we give universal estimates  for eigenvalues on compact domains of  complete submanifolds in an Euclidean space, and of complete manifolds admitting special functions. …”
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A ranking hashing algorithm based on listwise supervision by Anbang YANG, Jiangbo QIAN, Yihong DONG, Huahui CHEN

“…Recently,learning to hash technology has been used for the similarity search of large-scale data.It can simultaneous increase the search speed and reduce the storage cost through transforming the data into binary codes.At present,most ranking hashing algorithms compare the consistency of data in the Euclidean space and the Hamming space to construct the loss function.However,because the Hamming distance is a discrete integer value,there may be many data points sharing the same Hamming distance result in the exact ranking cannot be performed.To address this challenging issue,the encoded data was divided into several subspaces with the same length.Each subspace was set with different weights.The Hamming distance was calculated according to different subspace weights.The experimental results show that this algorithm can effectively sort the data in the Hamming space and improve the accuracy of the query compared with other learning to hash algorithms.…”
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Noether Symmetries of the Area-Minimizing Lagrangian by Adnan Aslam, Asghar Qadir

“…It is shown that the Lie algebra of Noether symmetries for the Lagrangian minimizing an (n-1)-area enclosing a constant n-volume in a Euclidean space is so(n)⊕sℝn and in a space of constant curvature the Lie algebra is so(n). …”
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The Euclidean model of space and time, and the wave nature of matter by Radovan Machotka

“…The EMST offers a unified description of all particles of matter as waves moving through four-dimensional Euclidean space at the speed of light. Unlike the usual description in three dimensions, where the group and phase velocities of a particle differ, in four-dimensional space the wave and the associated particle can be treated as a single object. …”
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Optimal scalable redeployment algorithm of relay nodes in wireless sensor networks by Bin ZENG, Jun WEI, Lu YAO

“…A relay node redeployment algorithm was proposed to find the optimal location of redeployment relay node when an existed relay node was overloaded considering the multi-dimensional localization and bandwidth constraint.The algorithm translated the redeployment problem into the optimal location planning in a multi-dimensional Euclidean space and then gave the suitable redeployment plan through searching the intersections of transmission areas heuristically.The correctness and completeness of the algorithm were proved.Furthermore,an optimization method was proposed to reduce the complexity of the algorithm to the linear function of the number of sensor nodes.The simulation results show that the algorithm can balance the loads of the overloaded relay nodes by joining in the candidate nodes and then prolong the network lifetime.…”
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Lax Triples for Integrable Surfaces in Three-Dimensional Space by Jan L. Cieśliński, Artur Kobus

“…., Lax representations consisting of three linear equations) associated with families of surfaces immersed in three-dimensional Euclidean space E3. We begin with a natural integrable deformation of the principal chiral model. …”
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A note on surfaces with prescribed oriented Euclidean Gauss map by Ricardo Sa Earp, Eric Toubiana

“…We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. …”
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An Adaptive Subspace Similarity Search Approach by Jianxin Ren, Huahui Chen

“…In recent years，such database fields as multimedia information retrieval，similarity join and time series matching，where similarity search has attracted much attention.Existing researches mostly compute nearest neighbor to solve problems about search target set，such as kNN and kNNJ，by metric distance functions in the Euclidean space.But some studies showed that high dissimilarity dimensions had got great effect on the accuracy of answer and flexibility and robustness still were lacked in corresponding solutions.Thus centralized dynamic subspace or partial dimensions similarity search problem and algorithms were proposed at first.Furthermore，with the emerge of very large dataset，centralized algorithms can，t extend very well.Finally，the distributed ones under hadoop framework were proposed.Experiments prove that distributed algorithms outperform centralized ones in the performance without accuracy loss.…”
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Global well-posedness of 2D incompressible Navier–Stokes–Darcy flow in a type of generalized time-dependent porosity media by Linlin Tan, Bianru Cheng

“…This study investigates the global well-posedness of a coupled Navier–Stokes–Darcy model incorporating the Beavers–Joseph–Saffman–Jones interface boundary condition in two-dimensional Euclidean space. We establish the existence of global strong solutions for the system in both linear and nonlinear cases where porosity depends on pressure. …”
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SOME ITERATIVE METHODS FOR QUASI-EQUILIBRIUM PROBLEMS by Jean Jacques Strodiot, Nguyễn Thị Thu Vân, Nguyễn Văn Hiền

“…In this paper we present two new extensions of Korpelevich's extragraclient method for solving the scalar Quasi-Equilibrium Problem (QEP) in Euclidean space. In both extensions we replace the second orthogonal projection in the correction step of the extragradient method by an Armijo linesearch adapted to (QEP). …”
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