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We extend Morley’s trisector theorem in the plane to an isosceles tetrahedron in three-dimensional space. We will show that the Morley tetrahedron of an isosceles tetrahedron is also isosceles tetrahedron. Furthermore, by the formula for distance in barycentric coordinate, we introduce and prove a general theorem on an isosceles tetrahedron.

We consider small perturbations of analytic non-twist area preserving mappings, and prove the existence of invariant curves with prescribed frequency by KAM iteration. Generally speaking, the frequency of invariant curve may undergo some drift, if the twist condition is not satisfied. But in this paper, we deal with a degenerate situation where the unperturbed rotation angle function r → w + r2n+1 is odd order degenerate at r = 0, and prove the existence of invariant curve without any drift in its frequency. Furthermore, we give a more general theorem on the existence of invariant curves with prescribed frequency for non-twist area preserving mappings and discuss the case of degeneracy with various orders.

In M. Ru and P. Vojta, A birational Nevanlinna constant and its consequences, Amer. J. Math. 142(3) (2020) 957–991, among other things, proved the so-called general theorem (arithmetic part) which can be viewed as an extension of Schmidt’s subspace theorem. In this paper, we extend their result by replacing the divisors by closed subschemes.

© 2021 Taylor & Francis Group, LLC.This paper presents a general theorem dealing with absolute Cesàro summability of the series (Formula presented.) under weaker conditions.

Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer $d$-invariants for pairs of L-spaces that are related by distance one Dehn fillings and satisfy a certain condition in first homology. These results imply that $T(2, 5)$ is the only torus knot $T(2, m)$ with $m$ square-free that admits a chirally cosmetic banding, i.e. a band surgery operation to its mirror image. We conclude with a discussion on the scarcity of chirally cosmetic bandings., Comment: The main theorem has been strengthened. This version accepted for publication in the Bulletin of the London Mathematical Society

This paper proves a precisification of Hume’s Law—the thesis that one cannot get an ought from an is—as an instance of a more general theorem which establishes several other philosophically interesting, though less controversial, barriers to logical consequence.

We establish an algorithm for producing formulas for p(n, m,N), the function enumerating partitions of n into at most m parts with no part larger than N. Recent combinatorial results of H. Hahn et al. on a collection of partition identities for p(n, m,N) are considered. We offer direct proofs of these identities and then place them in a larger context of the unimodality of Gaussian polynomials \( [^{N+m}_{m}] \) whose coefficients are precisely p(n, m,N). We give complete characterizations of the maximal coefficients of \( [^{M}_{3}] \) and \( [^{M}_{4}] \) . Furthermore, we prove a general theorem on the period of quasipolynomials for central/maximal coefficients of Gaussian polynomials. We place some of Hahn’s identities into the context of some known results on differences of partitions into at most m parts, p(n,m), which we then extend to p(n, m,N).

A proof of a general theorem for the calculation of conditional mean duration of a finite absorbing discrete time Markov chain is presented. In the simplest case, this result is equivalent to one suggested in the book of Kemeny and Snell (1976). In addition, we prove that the mean duration and mean conditional duration of a finite absorbing continuous time Markov chain can be calculated via the fundamental matrix of the embedded discrete time chain. These results are also extended to certain non-absorbing Markov chains. Applications are presented to illustrate the utility of these results.

We prove that in any Hausdorff space, the Rothberger game is equivalent to the k-Rothberger game, i.e. the game in which player II chooses k open sets in each move. This result follows from a more general theorem in which we show these games are equivalent to a game we call the restricted Menger game. In this game I knows immediately in advance of playing each open cover how many open sets II will choose from that open cover. This result illuminates the relationship between the Rothberger and Menger games in Hausdorff spaces. The equivalence of these games answers a question posed by Aurichi, Bella, and Dias [1] , at least in the context of Hausdorff spaces.

We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures., Comment: 17 pages; to appear in GAFA

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi to B is constrained in a specified way., 26 pages, no figures. Updated presentation

Many rigorous works have examined the behavior of Kaldor business cycle models. In this paper, utilizing the sustainability over sets (SOS) concept, a general theorem is proven establishing necessary and sufficient conditions for the existence and identification of rectangular sustainable sets for a general Kaldor model. These conditions can be readily verified, either in the two-dimensional space of lower and upper bounds on income, or in the one-dimensional space of upper bound on income, thus enabling identification of the region of economic sustainability for the considered Kaldor model. The Theorem’s power is illustrated with a case study of a particular Kaldor business cycle model from the literature that can exhibit rich dynamic behavior, and can give rise to economic sustainability regions that encompass multiple stable equilibrium points. Furthermore, a methodology is also presented that identifies non-rectangular sustainable sets, when the imposition of tight lower and upper bounds on one of the economic system variables (e.g., income) prevents the existence of a sustainable rectangular set.

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications., Comment: 32 Pages, 9 Figures

In a mathematics workshop with $mn$ mathematicians from $n$ different areas, each area consisting of $m$ mathematicians, we want to create a collaboration network. For this purpose, we would like to schedule daily meetings between groups of size three, so that (i) two people of the same area meet one person of another area, (ii) each person has exactly $r$ meeting(s) each day, and (iii) each pair of people of the same area have exactly $\lambda$ meeting(s) with each person of another area by the end of the workshop. Using hypergraph amalgamation-detachment, we prove a more general theorem. In particular we show that above meetings can be scheduled if: $3 \ | \ rm$, $2 \ | \ rnm$ and $r \ | \ 3\lambda(n-1)\binom{m}{2}$. This result can be viewed as an analogue of Baranyai's theorem on factorizations of complete multipartite hypergraphs., Comment: 9 Pages. arXiv admin note: text overlap with arXiv:1909.09643

Our work presents several mechanisms to calculate indicative prices for forex and cryptocurrency markets in terms of a numeraire. One of the mechanisms is tailored for the practitioner and is thus accompanied by analytic estimates that maximize its computational efficiency. Additionally, we discuss how to leverage the prices provided by the different mechanisms in terms of a numeraire to deduce pairwise prices between all currencies. Finally, we prove a general theorem that guarantees the inability to induce an arbitrage for all the pricing mechanisms presented.

In this paper, firstly we proved a general theorem dealing with absolute Riesz summability factors of infinite series under weaker conditions. And secondly we applied it to the trigonometric Fourie...

We first prove the existence, uniqueness and continuous dependence of mild solutions to stochastic delay evolution equations with a Caputo fractional derivative: D t α C y ( t ) = A y ( t ) + f ( t , y t ) + g ( t , y t ) d W ( t ) d t , 1 2 α 1 . Then, we investigate the asymptotic behavior of mild solutions to fractional stochastic delay evolution equations of the form D t α C y ( t ) = A y ( t ) + I t 1 − α f ( t , y t ) + [ I t 1 − α g ( t , y t ) ] d W ( t ) d t , 0 α 1 . In particular, the existence of a global forward attracting set in the mean-square topology is established. A general theorem on the existence of mild solutions is obtained by using α-order fractional resolvent operator theory and the Schauder fixed point theorem.

The coexistence of subharmonic periodic solutions of various orders is investigated to the first-order vector system of impulsive (upper-) Caratheodory differential equations and inclusions on tori. As the main tool, our recent Sharkovsky-type results for multivalued maps on tori are applied via the associated Poincare translation operators along the trajectories of given systems. The solvability criteria are formulated, under natural bi-periodicity assumptions imposed on the right-hand sides, in terms of the Lefschetz numbers of admissible impulsive maps. Since the criteria become effective on the circle, the main general theorem can be improved and reformulated there in a more transparent way. The obtained results can be regarded in a certain sense as a nontrivial extension of those due to Poincare [28] , Denjoy [17] and van Kampen [24] .

In this paper, we proved a general theorem dealing with the absolute Cesaro summability factors by using an almost increasing sequence. This new theorem also contains as particular cases several known and new results on the absolute Cesaro summability factors of infinite series.

In this article, we prove a general theorem dealing with an application of quasi-f-power increasing sequences and δ-quasi monotone sequences. This theorem also includes some known and new results.

For almost any compact connected Lie group$G$and any field$\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra$H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if$p$is odd or$p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology$HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over$\mathbb{F}_{2}$, such an isomorphism of Batalin–Vilkovisky algebras does not hold when$G=\text{SO}(3)$or$G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

We consider a special discrete time Leslie-Gower competition models for three species: \begin{document}$ x_i(t+1) = \frac{a_ix_i(t)}{1+x_i(t) +c \sum_{j\not = i} x_j(t)} $\end{document} for \begin{document}$ 1\leq i \leq 3 $\end{document} and \begin{document}$ t \geq 0 $\end{document} . Here \begin{document}$ c $\end{document} is the interspecific coefficient among different species. Assume \begin{document}$ a_1>a_2>a_3>1 $\end{document} . It is shown that when \begin{document}$ 0 , a unique interior equilibrium \begin{document}$ E^* $\end{document} exists and is locally stable. Then from a general theorem in Balreira, Elaydi and Luis (2017), it follows that \begin{document}$ E^* $\end{document} is globally asymptotically stable. Using a result of Ruiz-Herrera [ 11 ], it is shown that the unique positive equilibrium in the \begin{document}$ x_1x_2 $\end{document} -plane is globally asymptotically stable for \begin{document}$ c_0 . Then it is shown that \begin{document}$ (a_1-1, 0, 0) $\end{document} is globally asymptotically stable for \begin{document}$ \beta_{21} . This partially generalizes a result in Chow and Hsieh (2013) and Ackleh, Sacker and Salceanu (2014). For \begin{document}$ c>\beta_{12} $\end{document} , it is shown that there are multiple asymptotically stable equilibria.

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L2(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ1 and μ2 are close to each other in a sense quantified relative to a proximate order. Consider the sections of corresponding chains of de Branges spaces C1 and C2 which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces. Among other situations, we apply—and refine—this general theorem in two important particular situations In the proof of the main result, we employ a method used by P. Yuditskii in the context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

Simple cycles, also known as self-avoiding polygons, are cycles on graphs which are not allowed to visit any vertex more than once. We present an exact formula for enumerating the simple cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from an Hopf algebra, which we construct explicitly, and which provides further means of counting simple cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate simple cycles.

A theorem related to the theory of zero-sum games is proved. Rather general assumptions on the payoff function are found that are sufficient for an optimal strategy of one of the players to be chosen in the class of mixed strategies concentrated in at most m + 1 points if the opponent chooses a pure strategy in a finite-dimensional convex compact set and m is its dimension. This theorem generalizes results of several authors, starting from Bohnenblust, Karlin, and Shapley (1950).

A theorem of Sierpinski says that every infinite set Q of reals contains an infinite number of disjoint subsets whose outer Lebesgue measure is the same as that of Q. He also has a similar theorem involving Baire property. We give a general theorem of this type and its corollaries, strengthening classical results.

Moessner's theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1 n , 2 n , 3 n , … . Several generalizations of Moessner's theorem exist. Recently, Kozen and Silva gave an algebraic proof of a general theorem that subsumes Moessner's original theorem and its known generalizations. In this note, we describe the formalization of this theorem that the first author did in Nuprl . On the one hand, the formalization remains remarkably close to the original proof. On the other hand, it leads to new insights in the proof, pointing to small gaps and ambiguities that would never raise any objections in pen and pencil proofs, but which must be resolved in machine formalization.

Causal sets (or causets) are a particular class of partially ordered sets, which are proposed as basic models of discrete space-time, specially in the field of quantum gravity. In this context, we show the existence of temporal foliations for any causal set, or more generally, for a causal space. Moreover, we show that (order-preserving) automorphisms of a large class of infinite causal sets fall into two classes 1) Automorphisms of spacelike hypersurfaces in some given foliation (i.e. spacelike automorphisms), or 2) Translations in time. More generally, we show that for any automorphism $$\Phi $$ of a generic causal set $${\mathcal {C}}$$ , there exists a partition of $${\mathcal {C}}$$ into finitely many subcausets, on each of which (1) or (2) above hold. These subcausets can be assumed connected if, in addition, there are enough distinct orbits under $$\Phi $$ .

We prove convergence in probability for particular sequences defined in terms of the digits appearing in Oppenheim Series expansions and Oppenheim Continued Fractions expansions of real numbers. Our results are obtained by first proving a general theorem (Theorem 2.2) having both kinds of expansion as particular cases.

We prove a general theorem that gives a linear recurrence for tuples of paths in every cylindrical network. This can be seen as a cylindrical analog of the Lindstr\"om-Gessel-Viennot theorem. We illustrate the result by applying it to Schur functions, plane partitions, and domino tilings., Comment: 32 pages, 9 figures; v3: references updated and added a conjecture on total positivity

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R n → R n is a vector-function belonging to the Caratheodory class corresponding to the matrix-function A: [a, b] → R n×n with bounded total variation components, and h: BVs([a, b],R n ) → R n is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t1(x)) = B(x) · x(t 2(x))+c 0, where t i: BVs([a, b],R n ) → [a, b] (i = 1, 2) and B: BVs([a, b], R n ) → R n are continuous operators, and c 0 ∈ R n .

We prove that if a smoothly bounded strongly pseudoconvex domain $$D \subset {\mathbb {C}}^n$$ , $$n \ge 2$$ , admits at least one Monge-Ampere exhaustion smooth up to the boundary (i.e., a plurisubharmonic exhaustion $$\tau : {\overline{D}} \rightarrow [0,1]$$ , which is $${\mathscr {C}}^\infty $$ at all points except possibly at the unique minimum point x and with $$u {:}{=} \log \tau $$ satisfying the homogeneous complex Monge-Ampere equation), then there exists a bounded open neighborhood $${\mathscr {U}}\subset D$$ of the minimum point x, such that for each $$y \in {\mathscr {U}}$$ there exists a Monge-Ampere exhaustion with minimum at y. This yields that for each such domain D, the restriction to the subdomain $${\mathscr {U}}\subset D$$ of the Kobayashi pseudo-metric $$\kappa _D$$ is a smooth Finsler metric for $${\mathscr {U}}$$ and each pluricomplex Green function of D with pole at a point $$y \in {\mathscr {U}}$$ is of class $${\mathscr {C}}^\infty $$ . The boundary of the maximal open subset having all such properties is also explicitly characterized. The result is a direct consequence of a general theorem on abstract complex manifolds with boundary, with Monge-Ampere exhaustions of regularity $$\mathscr {C}^{k}$$ for some $$k \ge 5$$ . In fact, analogues of the above properties hold for each bounded strongly pseudoconvex complete circular domain with boundary of such weaker regularity.

The previous chapters showed, in some detail, how to make amplifiers using a single transistor. Further detailed analysis for any particular amplifier is left to those who find it necessary. Here some additional ideas applicable to amplifiers are presented more generally. To start, yet another general theorem for linear circuits is considered. After that, some useful configurations that rely on multiple transistors are presented, as well as some more general concepts that apply to any linear amplifier.

We give a significantly simplified proof of the exponential separation between regular and general resolution of Alekhnovich et al. (2007) as a consequence of a general theorem lifting proof depth to regular proof length in resolution. This simpler proof then allows us to strengthen the separation further, and to construct families of theoretically very easy benchmarks that are surprisingly hard for SAT solvers in practice.

Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small scales. Iterated function systems satisfying these properties were shown to exist by the author and by B\'{a}r\'{a}ny and K\"{a}enm\"{a}ki. In this paper we prove a general theorem on the existence of such iterated function systems within a parameterised family. This theorem shows that if a parameterised family contains two independent subfamilies, and the set of parameters that cause exact overlaps satisfies some weak topological assumptions, then the original family will contain an iterated function system satisfying the desired properties. We include several explicit examples of parameterised families to which this theorem can be applied.

In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in $1982$ and the second proposed by Xu in $1998$, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.

Let $\mathcal{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R}, 0$., Comment: Filled a gap and improved exposition. Comments are welcome

This paper presents the accessibility and small-time local controllability (STLC) results for $N$-link horizontal planar manipulators with only one unactuated joint. STLC is important in controls, both for design considerations and because large and swinging maneuvers may be avoided for close reconfiguration if a system is STLC. Despite the fact that controllability of underactuated horizontal planar manipulators has been extensively studied, most work focused only on three-link and global controllability. This paper thus has two contributions: 1) using Lie brackets to study the accessibility and STLC for underactuated two-link manipulators with different actuator configurations, and illustrating the results from a perspective of system dynamics, 2) obtaining the accessibility and STLC results for $N$-link manipulators with one unactuated joint by considering realistic models and different actuator configurations. It is found that an $N$-link ($N\geq 3$) with the first joint actuated is STLC for a subset of equilibrium points based on Sussmann's general theorem for STLC. Moreover, with the dynamics of $N$-link considered in the controllability analysis, it gives relatively simple forms for the nontrivial vector fields, which make it easy to determine at which configurations a model loses full rank condition for accessibility.

For graphs H and F, the generalized Turan number ex ( n , H , F ) is the largest number of copies of H in an F-free graph on n vertices. We say that H is F-Turan-good if ex ( n , H , F ) is the number of copies in the ( χ ( F ) − 1 ) -partite Turan graph, provided n is large enough. We present a general theorem in case F has an edge whose deletion decreases the chromatic number. In particular, this determines ex ( n , P k , C 2 l + 1 ) and ex ( n , C 2 k , C 2 l + 1 ) exactly, if n is large enough. We also study the case when F has a vertex whose deletion decreases the chromatic number.

The problem of the existence of complex \begin{document}$\ell_1$\end{document} solutions of two difference equations with exponential nonlinearity is studied, one of which is nonautonomous. As a consequence, several information are obtained regarding the asymptotic stability of their equilibrium points, as well as the corresponding generating function and \begin{document}$z-$\end{document} transform of their solutions. The results, which are obtained using a general theorem based on a functional-analytic technique, provide also a rough estimate of the region of attraction of each equilibrium point for the autonomous case. When restricted to real solutions, the results are compared with other recently published results.

The problem of constructing polynomials with all coefficients $1$ or $-1$ and large merit factor (equivalently with small $L^4$ norm on the unit circle) arises naturally in complex analysis, condensed matter physics, and digital communications engineering. Most known constructions arise (sometimes in a subtle way) from difference sets, in particular from Paley and Singer difference sets. We consider the asymptotic merit factor of polynomials constructed from other difference sets, providing the first essentially new examples since 1991. In particular we prove a general theorem on the asymptotic merit factor of polynomials arising from cyclotomy, which includes results on Hall and Paley difference sets as special cases. In addition, we establish the asymptotic merit factor of polynomials derived from Gordon-Mills-Welch difference sets and Sidelnikov almost difference sets, proving two recent conjectures., 22 pages, this revision contains a more general version of Thm. 2.1