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We propose an efficient version of ensemble Hartree–Fock/density functional theory to calculate a set of two charge-transfer states for systems with odd-numbers of electrons. The approach does require definitions of donor/acceptor fragments; however, the approach is not very sensitive to such definitions—even in the limit of very strong electronic coupling. The key ansatz is that, by mandating that the vector space spanned by the active orbitals projects equally onto the donor and acceptor fragments, such a constraint eliminates all intra-molecular local excitations and makes it far easier to generate potential energy surfaces that are smooth over a wide region of configuration space. The method is fast, working with only two electron configurations, and should be useful for ab initio non-adiabatic dynamics in the near future. [ABSTRACT FROM AUTHOR]

We demonstrate that, for systems with spin–orbit coupling and an odd number of electrons, the standard fewest switches surface hopping algorithm does not conserve the total linear or angular momentum. This lack of conservation arises not so much from the hopping direction (which is easily adjusted) but more generally from propagating adiabatic dynamics along surfaces that are not time reversible. We show that one solution to this problem is to run along eigenvalues of phase-space electronic Hamiltonians H(R, P) (i.e., electronic Hamiltonians that depend on both nuclear position and momentum) with an electronic–nuclear coupling Γ · P [see Eq. (25)], and we delineate the conditions that must be satisfied by the operator Γ. The present results should be extremely useful as far as developing new semiclassical approaches that can treat systems where the nuclear, electronic orbital, and electronic spin degrees of freedom altogether are all coupled together, hopefully including systems displaying the chiral-induced spin selectivity effect. [ABSTRACT FROM AUTHOR]

Numerical difficulties associated with computing matrix elements of operators between Hartree–Fock–Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree–Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan–Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the skew polynomial ring R = ( p + u p) [ x ; ] and its quotient ring R n = R / 〈 x n − 1 〉 , where p is an odd prime number, u 2 = 0 , and (u) = − u. We give an explicit structure of the ideals in R and R n and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian. [ABSTRACT FROM AUTHOR]

Given an odd prime number p and a p-stabilized Artin representation \rho over \mathbb {Q}, we introduce a family of p-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a p-adic Stark conjecture which can be seen as an explicit strengthening of conjectures by Perrin-Riou and Benois in the context of Artin motives. We show that these conjectures imply the p-part of the Tamagawa number conjecture for Artin motives at s=0 and we obtain unconditional results on the torsionness of Selmer groups. We also relate our new conjectures with various main conjectures and variants of p-adic Stark conjectures that appear in the literature. In the case of monomial representations, we prove that our conjectures are essentially equivalent to some newly introduced Iwasawa-theoretic conjectures for Rubin-Stark elements. We derive from this a p-adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which p is inert. Along the way, we prove that the Gross-Kuz'min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

In this paper we consider Gaussian type quadrature rules for trigonometric polynomials where an even number of nodes is fixed in advance. For an integrable and nonnegative weight function w on the interval E = [ a , a + 2 π) , a ∈ R , these quadrature rules have the following form ∫ E t (x) w (x) d x = ∑ i = 1 2 k a i t (y i) + ∑ i = 1 2 (n + γ) A i t (x i) , t ∈ T 2 (n + γ) + k − 1 , where the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance, γ ∈ { 0 , 1 / 2 } and T n = { cos ⁡ k x , sin ⁡ k x | k = 0 , 1 , ... , n } , n ∈ N. Also, for γ = 1 / 2 , i.e., for the case of quadrature rules for trigonometric polynomials with odd number of nodes, we consider the optimal sets of quadrature rules in the sense of Borges (see [1,13]) for trigonometric polynomials with even number of fixed nodes. Let n = (n 1 , n 2 , ... , n r) , r ∈ N , be a multi-index and let W = (w 1 , w 2 , ... , w r) be a system of weight functions on the interval E = [ a , a + 2 π) , a ∈ R. The optimal set of quadrature rules with respect to (W , n) , with even number of fixed nodes, have the form ∫ E f (x) w m (x) d x ≈ ∑ i = 1 2 k a m , i f (y i) + ∑ i = 1 2 | n | + 1 A m , i f (x i) , m = 1 , 2 , ... r , where | n | = n 1 + n 2 + ⋯ + n r and the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance. For r = 1 the optimal set of quadrature rules reduces to Gaussian quadrature rule for trigonometric polynomials with odd number of nodes. For all mentioned quadrature rules, in addition to the theoretical results, we will present the method for construction and give appropriate numerical examples. [ABSTRACT FROM AUTHOR]

The abundance of data about social relationships allows the human behavior to be analyzed as any other natural phenomenon. Here we focus on balance theory, stating that social actors tend to avoid establishing cycles with an odd number of negative links. This statement, however, can be supported only after a comparison with a benchmark. Since the existing ones disregard actors' heterogeneity, we extend Exponential Random Graphs to signed networks with both global and local constraints and employ them to assess the significance of empirical unbalanced patterns. We find that the nature of balance crucially depends on the null model: while homogeneous benchmarks favor the weak balance theory, according to which only triangles with one negative link should be under-represented, heterogeneous benchmarks favor the strong balance theory, according to which also triangles with all negative links should be under-represented. Biological networks, instead, display strong frustration under any benchmark, confirming that structural balance inherently characterizes social networks. According to balance theory, social actors avoid establishing cycles with an odd number of negative links. This statement can be supported only after a comparison with a benchmark. The authors find that the level of balance depends on the null-model employed: homogeneous ones favor the weak balance theory; heterogeneous ones favor the strong balance theory. [ABSTRACT FROM AUTHOR]

In this work, we focus on a binary stellar system consisting of two elliptical galaxies. Specifically, our investigation centers around prolate and oblate elliptical galaxies, which we model using the circular restricted three-body problem. Employing various standard numerical methods, we determine the positions of equilibrium points and analyze their linear stability and dynamical characteristics. To comprehensively explore the influence of mass and shape on the equilibrium points and their linear stability, we systematically investigate and discretize the parameter space of the system. Our study reveals a consistent pattern of an odd number of equilibrium points, which varies from 5 to 11. Notably, the system consistently displays linearly stable points, with the number of collinear points varying between 3 and 5. [ABSTRACT FROM AUTHOR]

The document titled "Problems and Solutions" is an article from the American Mathematical Monthly that provides a list of proposed problems and solutions in mathematics. The problems cover various mathematical concepts and are contributed by mathematicians from different countries. The article includes instructions for submitting proposed problems and solutions, as well as guidelines for submitting classics. The solutions to the problems are provided by individuals from different countries. The text also discusses mathematical concepts related to matrices, limits, inequalities, functional equations, recurrence relations, graphs, and integrals. It presents equations, solutions, and proofs for these concepts. The text is written in a technical manner and may be of interest to researchers or individuals studying mathematics. [Extracted from the article]

In 1971, Wilson (1971) proved that "almost all" finite games have an odd number of mixed Nash equilibria. Since then, several other proofs have been given, but always for mixed extensions of finite games. In this paper, we present a new oddness theorem for large classes of polynomial payoff functions and semi-algebraic sets of strategies. Additionally, we provide some applications to recent models of games on networks such that Patacchini-Zenou's model about juvenile delinquency and conformism (Patacchini and Zenou, 2012), Calvó-Armengol-Patacchini-Zenou's model about social networks in education (Calvó-Armengol et al., 2009), Konig-Liu-Zenou's model about R&D networks (König et al., 2019), Helsley-Zenou's model about social networks and interactions in cities (Helsley and Zenou, 2014). [ABSTRACT FROM AUTHOR]

Polarization properties of chiral fiber gratings with the core enclosed by a non-circular equiwidth curve have been analyzed and examined numerically. Since the curve has multiple odd number degree of symmetries, the specific analysis is mainly based on the simplest triple-helix chiral fiber gratings and difference of properties caused by small geometric deformation. Results show that, only right-handed circularly polarized core modes couples with cladding modes. Besides, if the shape of the fiber core was closer to a circle, rather than a triangle with curved edges, both the shortest total power transfer length and the distance between the relevant resonant dips of the transmission spectrum would increase. Therefore, as a multiple-helix chiral fiber grating, it might have some potential applications, such as filters, polarizers, and sensors. [ABSTRACT FROM AUTHOR]

Let p be an odd prime number, q = p m for a positive integer m , let q be the finite field with q elements and ω be a primitive element of q . We first give an orthogonal decomposition of the ring R = q + ν q , where ν 2 = a 3 , and a = ω 2 l for a fixed integer l. In addition, Galois dual of a linear code over R is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring R are investigated as well. Remarkably, we obtain that if linear codes and are a complementary pair, then the code and the dual code ⊥ E of are equivalent to each other. [ABSTRACT FROM AUTHOR]

We study the problem of perfect tiling in the plane and explore the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not. Previously, it has been proved that tiling the plane is not feasible using a set of odd numbers or an infinite sequence of natural numbers including exactly two odd numbers. The problem is open for different situations in which the number of odd numbers is arbitrary. In addition to providing a solution to this special case, we discuss some open problems to tile the plane and rectangles in this paper. [ABSTRACT FROM AUTHOR]

Let p be a prime such that p ≡ 3 (mod 4). Then, we show that there is no imaginary cyclic quartic extension K of ℚ whose class number is p. Suppose L / ℚ is a cyclic extension of number fields with an odd degree. Then, we show that 2 does not divide the class number of L if the class group of L is cyclic. We also construct some families of number fields whose class number is not divisible by a fixed prime. [ABSTRACT FROM AUTHOR]

Oscillations arise in many real-world systems and are associated with both functional and dysfunctional states. Whether a network can oscillate can be estimated if we know the strength of interaction between nodes. But in real-world networks (in particular in biological networks) it is usually not possible to know the exact connection weights. Therefore, it is important to determine the structural properties of a network necessary to generate oscillations. Here, we provide a proof that uses dynamical system theory to prove that an odd number of inhibitory nodes and strong enough connections are necessary to generate oscillations in a single cycle threshold-linear network. We illustrate these analytical results in a biologically plausible network with either firing-rate based or spiking neurons. Our work provides structural properties necessary to generate oscillations in a network. We use this knowledge to reconcile recent experimental findings about oscillations in basal ganglia with classical findings. [ABSTRACT FROM AUTHOR]

Prime period sequences can serve as the fundamental tool to construct arbitrary composite period sequences. This is a review study of the prime period perfect Gaussian integer sequence (PGIS). When cyclic group { 1 , 2 , ... , N − 1 } can be partitioned into k cosets, where N = k f + 1 is an odd prime number, the construction of a degree-(k + 1) PGIS can be derived from either matching the flat magnitude spectrum criterion or making the sequence with ideal periodic autocorrelation function (PACF). This is a systematic approach of prime period N = k f + 1 PGIS construction, and is applied to construct PGISs with degrees 1, 2, 3 and 5. However, for degrees larger than 3, matching either the flat magnitude spectrum or achieving the ideal PACF encounters a great challenge of solving a system of nonlinear constraint equations. To deal with this problem, the correlation and convolution operations can be applied upon PGISs of lower degrees to generate new PGISs with a degree of 4 and other higher degrees, e.g., 6, 7, 10, 11, 12, 14, 20 and 21 in this paper. In this convolution-based scheme, both degree and pattern of a PGIS vary and can be indeterminate, which is rather nonsystematic compared with the systematic approach. The combination of systematic and nonsystematic schemes contributes great efficiency for constructing abundant PGISs with various degrees and patterns for the associated applications. [ABSTRACT FROM AUTHOR]

We intend to continue our previous papers on dense ball packing hyperbolic space H³ by equal balls, but here with centres belonging to different orbits of the fundamental group Cw(2z, 3 ≤ z ∈ N, odd number), of our new series of tube or cobweb manifolds Cw = H³/Cw with z-rotational symmetry. As we know, Cw is a fixed-point-free isometry group, acting on H³ discontinuously with appropriate tricky fundamental domain Cw, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every Cw(2z) is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group W(u; v;w = u) that is a complete Coxeter orthoscheme reflection group, extended by the half-turn h (0 ↔ 3, 1 ↔ 2) of the complete orthoscheme A0A1A2A3 ∼ b0b1b2b3 (see Figure 1). The vertices A0 and A3 are outer points of the (Beltrami-Cayley-Klein) B-C-K model of H³, as 1/u+1/v ≤ 1/2 is required, 3 ≤ u = w, v for the above orthoscheme parameters. For the above simple manifold-construction we specify u = v = w = 2z. Then the polar planes a0 and a3 of A0 and A3, respectively, make complete with reflections a0 and a3 the Coxeter reflection group, where the other reflections are denoted by b0, b¹, b², b³ in the sides of the orthoscheme b0b1b2b3. The situation is described first in Figure 1 of the half trunc-orthoscheme W and its usual extended Coxeter diagram, moreover, by the scalar product matrix (bij) = (⟨bi, bj⟩) in formula (1) and its inverse (Ajk) = (⟨Aj, Ak⟩) in (3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme W, then its ball packings, densities, then those of the manifolds Cw(2z). As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molecules by equal balls, for general W(u; v;w = u) as well, summarized at the end of our paper. [ABSTRACT FROM AUTHOR]

An odd c - coloring of a graph is a proper c -coloring such that each non-isolated vertex has a color appearing an odd number of times within its open neighborhood. A proper conflict-free c - coloring of a graph is a proper c -coloring such that each non-isolated vertex has a color appearing exactly once within its neighborhood. Clearly, every proper conflict-free c -coloring is also an odd c -coloring. Cranston conjectured that every graph G with maximum average degree mad (G) < 4 c c + 2 (where c ≥ 4) has an odd c -coloring, and he proved this conjecture for c ∈ { 5 , 6 }. Note that the bound 4 c c + 2 is best possible. Cho et al. solved Cranston's conjecture for c ≥ 5 , strengthening the result by transitioning from odd c -coloring to proper conflict-free c -coloring. However, they did not provide all the extremal non-colorable graphs G with mad (G) = 4 c c + 2 , which remains an open question of interest. In this paper, we tackle this intriguing extremal problem. We aim to characterize all non-proper conflict-free c -colorable graphs G with mad (G) = 4 c c + 2 . For the case of c = 4 , Cranston's conjecture is not true, as evidenced by the existence of a counterexample: a graph whose every block is a 5-cycle. Cho et al. proved that a graph G with mad (G) < 22 9 and no induced 5-cycles has an odd 4-coloring. We improve this result by proving that a graph G with mad (G) ≤ 22 9 (with equality allowed) is not odd 4-colorable if and only if G belongs to a specific class of graphs. On the other hand, Cho et al. established that a planar graph with girth at least 5 has an odd 6-coloring; we improve it by proving that a planar graph without 4 − -cycles adjacent to 7 − -cycles also has an odd 6-coloring. [ABSTRACT FROM AUTHOR]

In this article, a prescribed-time state-feedback stabilization design strategy is proposed for a class of p-norm stochastic nonlinear strict feedback systems. In previous work on prescribed-time stabilization of stochastic systems, only stochastic nonlinear systems with fractional power less than or equal to one are considered. To overcome this problem, we improve the existing method and discuss the issue of prescribed-time stabilization of stochastic nonlinear systems with fractional power is arbitrary positive odd rational number. First, a prescribed-time controller is designed by combining the Lyapunov function with adding a power integrator technique. It should be pointed out that the homogeneous domination approach is adopted when dealing with the nonlinear terms of the system. Then, according to the stochastic prescribed-time stability theorem, it is proved that the designed controller can ensure the closed-loop system is prescribed-time mean-square stable. Finally, three simulation examples are given to investigate the validity of the presented method, in which the last one is an electromechanical system example. [ABSTRACT FROM AUTHOR]

In this paper, we study the invariants of twisted knots derived from the invariants of virtual knots such as the arc shift number and the odd writhe. We provide a class of twisted knots such that the arc shift number of every member is 1. [ABSTRACT FROM AUTHOR]

Node-containing straws exhibit superior mechanical properties compared to node-free straw plants, particularly in terms of shear resistance and compression resistance. We explore the relationship between the structure and mechanical properties of straw materials, providing deeper insights for the field of biomechanics. In this study, we focused on two node-containing straw plants, namely sorghum and reed. The main characteristics of sorghum and reed stalks were compared using macroscopic observation, stereomicroscopy, scanning electron microscopy, infrared spectroscopy, and EDS analysis. This study revealed numerous similarities and differences in the macro- and microstructures as well as the elemental composition of sorghum and reed stalks. The functional groups in sorghum and reed stalks were largely similar, with the primary elements being C and O. Distinguishing features included a higher tapering and a slightly larger reduction in wall thickness in sorghum stalks compared to reed stalks. The cross-section of sorghum stalks was filled with pith structures, while reed stalks exhibited a hollow structure. The vascular bundles in sorghum typically showed a paired arrangement, whereas those in reeds were arranged in odd numbers. Furthermore, sorghum straws contained more Cl and no Br, while the parenchyma of reed straws contained higher Br. The C and O proportions of sorghum straws and reed straws are 50–53% (50–51%) and 45–46% (48–49%), respectively. These variations in elemental composition are believed to be correlated with the mechanical properties of the materials. By conducting a detailed study of the micro/macrostructures and material composition of sorghum and reed straw, this paper provides valuable insights for the field of biomechanics. [ABSTRACT FROM AUTHOR]

Background: Rotator cuff tendinopathy (RCT) is a main source of disability work inefficiency and overall inefficiency. Platelet-rich plasma (PRP) has been postulated to be of great advantage in the management of RCT. Steroidal formulations are base of all joint morbidities since long for inflammatory and degenerative conditions in orthopedics. Aim and Objectives: The aim of the study was to compare the effect of PRP injections versus steroid Injection (triamcinolone) in subacromial space on pain control and improved shoulder functions in patients having chronic RCT. Materials and Methods: The study was conducted on 40 patients (aged more than 18 years) who presented in emergency and Outpatient Department with symptoms of shoulder pain and decreased mobility at shoulder. The patients were divided into two groups. Every odd number of patient presenting to us was given PRP injection (Study group) and every even number patient was given injection triamcinolone (control group) along with physical therapy in both study and control group. Patient was followed up subsequently after 4-week and 12-week time for resolution of symptoms and improved pain-free activities. Outcome assessment criterion used included VAS system and Oxford Shoulder Scoring System. Results: Comparison of the patients in the two groups revealed significant difference between the groups in VAS and OSS at 4-week and 12-week follow-ups. Long-term effect was more in case of PRP group as compared to steroid formulation which was almost similarly effective acutely. Conclusion: Subacromial PRP injection was found to be more effective in long-term in improving overall quality of life, disability, pain, improved work efficiency, and improved shoulder movements in patients with chronic RCT than those treated by subacromial steroidal injection along with exercise program. [ABSTRACT FROM AUTHOR]

Optic disc drusen (ODD) are calcium‐containing deposits in the optic nerve head, capable of causing visual field defects and sudden visual loss. The underlying pathophysiology remains inadequately understood and treatment options are missing. In this paper, we systematically reviewed prevalence studies of ODD in non‐selected populations to provide an overview of its prevalence, conducted meta‐analyses to determine modality‐specific prevalence estimates and performed a forecasting study to estimate current and future global population number of individuals with ODD. We searched 11 literature databases on 25 October 2022 for prevalence studies of ODD in non‐selected populations. Eight eligible studies provided data from a total of 27 463 individuals. Prevalence estimates were stratified according to diagnostic modalities: ophthalmoscopy 0.37% (95% CI: 0.10–0.95%), fundus photography 0.12% (95% CI: 0.03–0.24%), spectral domain optical coherence tomography with enhanced depth imaging 2.21% (95% CI: 1.25–3.42%) and histopathology 1.82% (95% CI: 1.32–2.38%). Using histopathology‐based summary prevalence estimate, we forecast 145 million individuals with ODD currently, a number expected to increase further due to world population growth. These numbers underscore the importance of including ODD in health education and highlight the necessity of continuing research in ODD. [ABSTRACT FROM AUTHOR]

In discrete mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Chemical graph theory is concerned with non-trivial applications of graph theory to the solution of molecular problems. Its main goal is to use numerical invariants to reduce the topological structure of a molecule to a single number that characterizes its properties. Topological indices are numerical invariants associated with the chemical constitution, for the purpose of the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. They have found important application in predicting the behavior of chemical substances. The Graovac–Ghorbani ( A B C G G ) index is a topological descriptor that has improved predictive potential compared to analogous descriptors. It is used to model both the boiling point and melting point of molecules and is applied in the pharmaceutical industry. In the recent years, the number of publications on its mathematical properties has increased. The aim of this work is to partially solve an open problem, namely to find the structure of unicyclic graphs that minimize the A B C G G index. We characterize unicyclic graphs with even girth that minimize the A B C G G index, while we also present partial results for odd girths. As an auxiliary result, we compare the A B C G G indices of paths and cycles with an odd number of vertices. [ABSTRACT FROM AUTHOR]

In this work, we have studied, within density functional theory, the interaction of NO with pure and oxidized silver clusters, both anionic and cationic, composed from 11 to 13 Ag atoms. In that size interval, shell closing effects are not expected, and structural and electronic odd–even effects will determine the strength of interaction. First, we obtained that species Ag n ± and AgnO± with odd number of electrons (n = 12) adsorb NO with higher energy than their neighbors (n = 11 and 13). This result is in agreement with the facts observed in recent mass spectroscopy measurements, which were performed, however, at finite temperature. The adsorption energy is about twice for oxidized clusters compared to pure ones and higher for anions than for cations. Second, the adsorption of another NO molecule on AgnNO± forms Ag n (NO) 2 ± , with the dimer (NO)2 in cis configuration, and binding the two N atoms with two neighbor Ag atoms. The n = 12 species show the higher adsorption energy again. Third, in the absence of reaction barriers, all complexes Ag n (NO) 2 ± dissociate spontaneously into AgnO± and N2O, except the n = 12 anion. The maximum high barrier along the dissociation path of Ag 13 (NO) 2 − is about 0.7 eV. Further analysis of projected density of states for Ag 11 − 13 (NO) x ± (x = 0, 1, 2) molecules shows that bonding between NO and Ag clusters mainly occurs in the energy range between −3.0 and 3.0 eV. The overlap between 4d of Ag and 2p of N and O is larger for Ag 12 (NO) 2 ± than for neighbor sizes. For n = 12, the d bands are close to the (NO)2 2π orbital, leading to extra back-donation charge from the 4d of Ag to the closer 2π orbital of (NO)2. [ABSTRACT FROM AUTHOR]

This paper presents a proof of Goldbach's conjectures, both strong and weak, which state that: "Every even number greater than 2 can be written as the sum of two prime numbers" and that "Every odd number greater than 5 can be written as the sum of three prime numbers. To carry out these demonstrations, all the combinations were initially analyzed to obtain the even and odd numbers generated by adding two or three prime numbers, respectively. In the end, two very revealing mathematical relationships were obtained, demonstrated by the method of mathematical induction, which allows verifying the validity of these conjectures. [ABSTRACT FROM AUTHOR]

Non-reciprocal interactions of discrete or continuous systems may induce surprising responses such as flutter instabilities. It is shown in this paper that a finite one-dimensional lattice under non-symmetrical elastic interactions may flutter for sufficiently strong unsymmetrical interactions. An exact solution is presented for the vibration of such one-dimensional lattices with direct and non-symmetrical elastic interactions. An internal force controlling the interactions is included in the model as an additional force for each mass, which acts proportionally to the elongation of a spring at its position. This non-conservative problem due to this circulatory interaction is solved from the resolution of a linear difference equation for this unsymmetrical repetitive lattice. It is possible to derive the exact eigenfrequency dependence with respect to the unsymmetrical interaction parameter, which plays the role of a bifurcation parameter. Divergence and flutter instabilities of this fixed–fixed non-conservative axial lattice under non-Hermitian interactions are theoretically predicted, from a direct approach or by solving the difference equation whatever the number of masses of the lattice. It is shown that the system may flutter for sufficiently strong unsymmetrical interactions, whatever the size of the system, for even or odd number of masses. However, divergence instability may arise in such a system only for even number of masses. The drastic change of response of the present system for odd or even number of particles is specific of the discrete nature of the dynamic system. [ABSTRACT FROM AUTHOR]

An odd hole is an induced odd cycle of length at least five. Let l ≥ 2 be an integer, and let G l denote the family of graphs which have girth 2 l + 1 and have no holes of odd length at least 2 l + 3. Chudnovsky and Seymour proved that every graph in G 2 is three-colorable. Following the idea of Chudnovsky and Seymour, Wu, Xu and Xu proved that every graph in G 3 is three-colorable. In 2022, Wu, Xu and Xu conjectured that every graph G ∈ ⋃ l ≥ 2 G l is three-colorable. In this paper, we prove that every graph G ∈ G l with radius at most l + 3 is three-colorable. [ABSTRACT FROM AUTHOR]

A signed graph is a graph G together with a signature σ : E (G) → { 1 , − 1 }. For a real number r ≥ 1 , C r is a circle of circumference r. For two points a , b ∈ C r , the distance d (mod r) (a , b) between a and b is the length of the shorter arc of C r connecting a and b. For x ∈ C r , the antipodal x ̄ of x is the unique point in C r of distance r / 2 from x. A circular r -colouring of (G , σ) is a mapping f : V (G) → C r such that for each positive edge e = u v , d (mod r) (f (u) , f (v)) ≥ 1 , and for each negative edge e = u v , d (mod r) (f (u) , f (v) ¯) ≥ 1. The circular chromatic number of a signed graph (G , σ) is the minimum r such that (G , σ) is circular r –colourable. Let g ∗ (G , σ) be the length of the shortest cycle in (G , σ) with an odd number of positive edges. For a positive integer g , let SP g be the family of signed series–parallel graphs with g ∗ (G , σ) ≥ g. This paper determines, for any positive integer g , the supremum value of χ c (G , σ) of signed graphs in SP g. [ABSTRACT FROM AUTHOR]

The understanding of magnetic properties of hybrid compounds is important for the design of magnetic devices. In this contribution a prominent odd–even effect in both the structural characteristics and magnetic properties of four new hybrid compounds comprised of n-carboxyalkylammonium cations and perchloridomanganate anions is reported. When the n-carboxyalkylammonium cations contain an even number of carbon atoms, compounds of the formula (NH3(CH2)nCOOH)2[MnCl4], with n = 3 and 5, are formed, which display the two-dimensional (2D) hybrid halide perovskite structure in which bridging chlorido ligands link Mn2+ ions. Compounds containing n-carboxyalkylammonium cations with an odd number of carbon atoms have the formula (NH3(CH2)nCOOH)2[MnCl4(H2O)2], with n = 2 and 4, and display a zero-dimensional (0D) structure with hydrogen bonding interactions linking neighbouring [MnCl4(H2O)2]2− anions. The odd–even effect is also evident in the magnetic properties of the compounds, which are linked to the structural differences observed in these compounds. Compounds containing an even number of carbon atoms show antiferromagnetic (AFM) interactions and spin canting at temperature TN, with 2JK = −8.28(5) K and TN = 45.0(5) K when n = 3 and 2JK = −7.72(4) K and TN = 43(1) K when n = 5. Much weaker AFM interactions and no spin canting is observed in compounds containing an odd number of carbon atoms, with 2JK = −0.14(2) K when n = 2 and 2JK = −0.14(2) K when n = 4. [ABSTRACT FROM AUTHOR]

Mixture experimental designs have been in widespread use in agricultural, pharmaceutical, and other industrial research for many years. Much of the previous work mainly focuses on optimal design for mixture experiments when observations are uncorrelated, in large part because of the intractability of the optimal mixture experimental design on correlated case. When observations have certain correlation structures within each block, the order of the observations in each block matters and this order impacts the optimality of the design. Thus, there is need of research into construction of these useful designs when correlation structures might exist within blocks. In this paper, we propose D-optimal minimum support designs for Scheffe's quadratic mixture model with odd number of components when observations in blocks are circulant correlated and hub correlated respectively, and Scheffe's mixture model with any components when observations are in block-structured correlation. [ABSTRACT FROM AUTHOR]

The scientific literature contains many accounts of application of polymethine dyes, including cyanine dyes, as imaging agents, i.e., "biological stains," for microscopic investigation of biological materials. Currently, many such dyes are used as probes for living cells, i.e., "fluorescent probes." Polymethine dyes are defined here by two criteria. First, they possess a conjugated chain of (2n + 1) sp2-hybridized carbon atoms that connect a terminal π-electron-accepting (π-electron withdrawing) group with a terminal π-electron-donating group. Second, they have an odd number (2n + 3) of π-centers and an even number (2n + 4) of π-electrons in this chain, where n equals the number of –CR2=CR3– groups, usually vinylene groups –CH=CH–. Commercialization of diverse chemical types of many polymethine dyes has been attempted. The dyes that have achieved wide application, however, are limited in number and it is these dyes that are emphasized here. Because these polymethine dyes sometimes have been described by confusing, and sometimes confused, names, we clarify here the chemical categories and names of such dyes for the nonchemist, biomedical end user of such imaging agents. Nevertheless, the nomenclature presented here is not intended to replace the traditional "chromophore" categories of dyestuff chemistry, because the latter are held in place both by wide usage and by venerable authorities, such as the Colour Index. [ABSTRACT FROM AUTHOR]

We show that for each positive integer a there exist only finitely many prime numbers p such that a appears an odd number of times in the period of continued fraction of p or 2 p. We also prove that if p is a prime number and D = p or 2 p is such that the length of the period of continued fraction expansion of D is divisible by 4, then 1 appears as a partial quotient in the continued fraction of D. Furthermore, we give an upper bound for the period length of continued fraction expansion of D , where D is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas [MU]. [ABSTRACT FROM AUTHOR]

Density functional theory methods are applied to crystal structures and stabilities of phases from the aleksite homologous series, PbnBi4Te4Sn+2 (n = homologue number). The seven phases investigated correspond to n = 0 (tetradymite), 2 (aleksite‐21R and ‐42R), 4 (saddlebackite‐9H and ‐18H), 6 (unnamed Pb6Bi4Te4S8), 8 (unnamed Pb8Bi4Te4S10), 10 (hitachiite) and 12 (unnamed Pb12Bi4Te4S14). These seven phases correspond to nine single‐ or double‐module structures, each comprising an odd number of atom layers, 5, 7, (5.9), 9, (7.11), 11, 13, 15 and 17, expressed by the formula: S(MpXp+1)·L(Mp+1Xp+2), where M = Pb,Bi and X = Te,S, p ≥ 2, and S and L = number of short and long modules, respectively. Relaxed structures show a and c values within 1.5% of experimental data; a and the interlayer distance dsub decrease with increasing PbS content. Variable Pb—S bond lengths contrast with constant Pb—S bond lengths in galena. All phases are n‐fold superstructures of a rhombohedral subcell with c/3 = dsub*. Electron diffraction patterns show two brightest reflections at the centre of dsub*, described by the modulation vector qF = (i/N) · dsub*, i = S+L. A second modulation vector, q = γ · csub*, shows a decrease in γ, from 1.8 to 1.588, across the n = 0 to n = 12 interval. The linear relationship between γ and dsub allows the prediction of any theoretical phases beyond the studied compositional range. The upper PbS‐rich limit of the series is postulated as n = 398 (Pb398Bi4Te4S400), a phase with dsub (1.726 Å) identical to that of trigonal PbS within experimental error. The aleksite series is a prime example of mixed layer compounds built with accretional homology principles. [ABSTRACT FROM AUTHOR]

The transmission of a vertex v of a (chemical) graph G is the sum of distances from v to other vertices in G. If any two vertices of G have different transmissions, then G is a transmission irregular graph. It is shown that for any odd number n ≥ 7 there exists a transmission irregular chemical tree of order n. A construction is provided which generates new transmission irregular (chemical) trees. Two additional families of chemical graphs are characterized by property of transmission irregularity and two sufficient condition provided which guarantee that the transmission irregularity is preserved upon adding a new edge. [ABSTRACT FROM AUTHOR]

There is a need for a universal model to describe higher harmonic generation (HHG) in different states of matter. Based on an electromagnetic model (EM), the generation of odd higher harmonic (HHG) and supercontinuum (SC) from intense fs and ps pulses for visible, NIR, and MIR lasers is simulated based on the parameters from experimental observation. HHG and SC depend critically on the different Kerr material response times τ from the ultrafast on the order of 100 as for electronic cloud distortion to fast ~ 10 fs from plasma and molecular redistribution and to the slower picoseconds rotational and vibrational molecular processes. The number of odd HHG generated is shown to depend critically on the fastest Kerr response time on the order of ~ 1 fs from electronic self-phase modulation (ESPM). In this study, different states of matter from noble gas Argon to condensed matter ZnO and LBG are simulated showing the dependence on the Kerr response time to produce HHG for various applications in Physics, Chemistry, Biology, and Engineering. The EM model is universal to produce HHG and SC in different states of matter. [ABSTRACT FROM AUTHOR]

Andrews and Newman introduced the mex-function mex A , a (λ) for an integer partition λ of a positive integer n as the smallest positive integer congruent to a modulo A that is not a part of λ . They then defined p A , a (n) to be the number of partitions λ of n satisfying mex A , a (λ) ≡ a (mod 2 A) . They found the generating function for p t , t (n) and p 2 t , t (n) for any positive integer t, and studied their arithmetic properties for some small values of t. In this article, we study the partition function p m t , t (n) for all positive integers m and t. We show that for sufficiently large X, the number of all positive integers n ≤ X such that p m t , t (n) is an even number is at least O (X / 3) for all positive integers m and t. We also prove that for sufficiently large X, the number of all positive integers n ≤ X such that p m p , p (n) is an odd number is at least O (log log X) for all m ≢ 0 (mod 3) and all primes p ≡ 1 (mod 3) . Finally, we establish identities connecting the ordinary partition function to p m t , t (n) . [ABSTRACT FROM AUTHOR]

The structurally related odd and even numbered wheels [FeIII11ZnII4(tea)10(teaH)1(OMe)Cl8] (1) and [FeIII12ZnII4(tea)12Cl8] (2) can be synthesized under ambient conditions by reacting FeIII and ZnII salts with triethanolamine (teaH3), the change in nuclearity being dictated by the solvents employed. An antiferromagnetic exchange between nearest neighbors, J = ‐10.0 cm−1 for 1 and J = −12.0 cm−1 for 2, leads to a frustrated S = 1/2 ground state in the former and an S = 0 ground state in the latter. [ABSTRACT FROM AUTHOR]

Over binary-input memoryless symmetric (BMS) channels, the performance of polar codes under successive cancellation list (SCL) decoding can approach maximum likelihood (ML) algorithm when the list size L is greater than or equal to 2 M F , where MF, known as mixing factor of code, represents the number of information bits before the last frozen bit. Recently, Yao et al. showed the upper bound of the mixing factor of decreasing monomial codes with length n = 2 m and rate R ≤ 1 2 when m is an odd number; moreover, this bound is reachable. Herein, we obtain an achievable upper bound in the case of an even number. Further, we propose a new decoding hard-decision rule beyond the last frozen bit of polar codes under BMS channels. [ABSTRACT FROM AUTHOR]

Every odd prime number p has exactly (p + 1) / 2 different expressions as a sum ab + cd of two ordered products ab and cd such that min (a , b) > max (c , d) . An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4 N as sums of two squares. [ABSTRACT FROM AUTHOR]

We study mutually unbiased maximally entangled bases (MUMEBs) in bipartite system C d ⊗ C d with d ≥ 3 a power of an odd prime number. By using the theory of finite fields, we provide a new and intuitive method to construct MUMEBs in C d ⊗ C d . And we construct d (d - 1) MUMEBs in bipartite system C d ⊗ C d explicitly. [ABSTRACT FROM AUTHOR]

Background: Cataract surgery is aimed at providing early visual rehabilitation and good vision. MSICS has added advantage over phacoemulsification of being done in every cataract type along with being cost-effective, stable, consistent, successful in visual recovery, safe, less technology-dependent, short-duration, and sutureless. Aim: The present study was conducted to comparatively evaluate astigmatism induced surgically following Blumenthal versus Frown corneoscleral tunnel incision in MSICS. Methods: The study included 88 subjects from both genders divided into 2 groups of 44 subjects each via odd and even numbers where Group A subjects underwent MSICS with Blumenthal Corneoscleral incision and Group B subjects with Frown Corneoscleral incision. Follow-up of all subjects was done at 1 day, 1, 4, and 6 weeks postoperatively where the assessment was done with Keratometry, fundoscopy, slit-lamp biomicroscopy and vision. Results: Mean best-corrected visual acuity was higher in Group B (Frown incision) compared to Group A with Blumenthal incision with respective values of 0.630±0.210 and 0.691±0.229. However, this difference was statistically non-significant with p=0.199. In the group, The mean astigmatism was continuously increasing till the 3rd week and then it decreased. But there were no statistically significant differences were found between different time intervals (P=0.453). In group B it was gradually increasing till 3rd week and after 6th week it slightly reduced. A statistically significant difference was found between time intervals (P=0.026). Mean visual acuity, preoperatively, was statistically non-significant between the two groups with respective values of 0.191±0.408 and 0.123±0.136 respectively for Group A and B (p=0.297). On postoperative day1, 1st week, 3rd week, and 6th week the visual acuity between the two groups was statistically non-significant with respective pvalues of 0.402, 0.387, 0.227, and 0.087 Conclusion: The present study concludes that in MSICS, both Frown and Blumenthal corneoscleral incisions have minimal astigmatism and best-corrected visual acuity postoperatively. However, surgically induced astigmatism has a non-significant difference in the two study groups. Hence, both the incisions can be used for cataract management;however, the Blumenthal incision is better for big nuclei and hard cataracts compared to the Frown incision. [ABSTRACT FROM AUTHOR]

Nickel(II) complexes with pyrazole-based ligands are widely employed in catalysis of ethylene oligomerization and subsequent Friedel-Crafts alkylation of toluene. We have prepared ten new nickel(II) dibromide complexes with various substituted bis(azolyl)methanes. They have been characterized using ¹H NMR, IR, high resolution mass spectrometry and elemental analysis. The structures of three complexes have been unambiguously established using X-ray diffraction. It was found that these complexes in the presence of Et2AlCl or Et3Al2Cl3 are active both in ethylene oligomerization and Friedel-Crafts alkylation processes (activity up to 3720 kgoligomer·mol[Ni] -1·h-1). The use of Et3Al2Cl3 results in a higher share of alkylated products (up to 60%). Moreover, catalytic systems activated with Et3Al2Cl3 produced small amounts of odd carbon number olefins (up to 0.8%). The Friedel-Crafts alkylation was used as a trap for previously undetected short-chain odd carbon number olefins (C3 and C5). [ABSTRACT FROM AUTHOR]

In this study, we employ the Sachs graph theory to formulate the conduction properties of a single-molecular junction consisting of a molecule in which one carbon atom of an alternant hydrocarbon is replaced with a heteroatom. The derived formula includes odd and even powers of the adjacency matrix, unlike the graph of the parental structure. These powers correspond to odd- and even-length walks. Furthermore, because the heteroatom is represented as a self-loop of unit length in the graph, an odd number of passes of the self-loop will change the parity of the length of the walk. To confirm the aforementioned effects of heteroatoms on conduction in an actual sample, the conduction behavior of meta-connected molecular junctions consisting of a heterocyclic six-membered ring, whose conductive properties have already been experimentally determined, was analyzed based on the enumerated number of walks. [ABSTRACT FROM AUTHOR]