Theorem States Research Articles

Improvements on the Leighton‐type oscillation criteria for impulsive differential equations and extension to non‐canonical case

We investigate the oscillatory behavior of solutions of second‐order linear differential equations with impulses. These equations can be categorized into two types: canonical and non‐canonical. This study examines the canonical and non‐canonical impulsive differential equations in connection with the famous Leighton‐type oscillation theorem. The well‐known theorem states that every solution of where and are continuous with , is oscillatory if and This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition fails, that is, when the equation is of non‐canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non‐canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results.

A strong Differentiable Absorption of Hilbert 𝑪∗-Modules with Connections, and Lifts of Unbounded Operators

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert 𝐶∗-module is isomorphic to a direct summand in the standard module of square summable sequences in the base 𝐶∗-algebra. This result be generalized byJens Kaad [Jka10] by incorporating a densely defined derivation on the base 𝐶∗-algebra. It following the perfect densely method of Jens Kaad[Jka10] leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correspondences) on Hilbert 𝐶∗-modules. These connections can in turn be used to define selfadjoint and regular ”lifts” of unbounded operators which act on an auxiliary Hilbert 𝐶∗- module.

Critical Spontaneous Breaking of U(1) Symmetry at Zero Temperature in One Dimension.

The Hohenberg-Mermin-Wagner theorem states that there is no spontaneous breaking of continuous internal symmetries in spatial dimensions d≤2 at finite temperature. At zero temperature, the quantum-to-classical mapping further implies the absence of such symmetry breaking in one dimension, which is also known as Coleman's theorem in the context of relativistic quantum field theories. One route to violate this "folklore" is requiring an order parameter to commute with a Hamiltonian, as in the classic example of the Heisenberg ferromagnet and its variations. However, a systematic understanding has been lacking. In this Letter, we propose a family of one-dimensional models that display spontaneous breaking of a U(1) symmetry at zero temperature, where the order parameter does not commute with the Hamiltonian. While our models can be deformed continuously within the same phase, there exist symmetry-preserving perturbations that render the observed symmetry breaking fragile. We argue that a more general condition for this behavior is that the Hamiltonian is frustration-free.

Rethinking Loss of Available Work and Gouy-Stodola Theorem

Abstract Exergy represents the maximum useful work possible when a system at a specific state reaches equilibrium with the environmental dead state at temperature To. Correspondingly, the exergy difference between two states is the maximum work output when the system changes from one state to the other, assuming that during the processes, the system exchanges heat reversibly with the environment. If the process involves irreversibility, the Guoy-Stodola theorem states that the exergy destruction equals the entropy generated during the process multiplied by To. The exergy concept and the Gouy-Stodola theorem are often used to optimize processes or systems, even when they are not directly connected to the environment. In the past, questions have been raised on if To is the proper temperature to use in calculating the exergy destruction. Here, we start from the first and the second laws of thermodynamics to unambiguously show that the useful energy loss (UEL) of a system or process should equal to the entropy generation multiplied by an equivalent temperature associated with the entropy rejected out of the entire system. For many engineering systems and processes, this entropy rejection temperature can be easily calculated as the ratio of the changes of the enthalpy and entropy of the fluid stream carrying the entropy out, which we call the state-change temperature. The UEL is unambiguous and independent of the environmental dead state, and it should be used for system optimization rather than the exergy destruction.

MODELING THE BENEFITS OF A MARRIAGE REVERSE ANNUITY CONTRACT WITH DEPENDENCY ASSUMPTIONS USING ARCHIMEDEAN COPULA

Social security benefits may not be enough for retirement. Equity release products like marriage reverse annuities can boost retirement income for older couples. Marriage reverse annuity’s contract convert all or part of the real estate value of elderly spouses while they are living (joint life status) or after one partner dies (last survivor status). Since husband and wife face the same death risk, the chance of death between spouses is believed to be dependent for realism. Thus, copula models the future dependency model of a husband and wife. Sklar's theorem states that copulas link bivariate distribution and marginal cumulative functions. One of the most common copulas is Archimedean copula. Clayton, Gumbel, and Frank are Archimedean copula that will be used in this investigation. The Indonesian Mortality Table IV data is used to obtain the marginal distribution of the male and female which will then be used to construct copulas (Clayton, Gumbel, and Frank) that combine two marginal distributions into a joint distribution. The marginal distribution of Indonesian Mortality Table IV is uncertain, hence Canonical Maximum Likelihood parameter estimation is utilized to estimate the parameter of copulas. Multiple-state models depict the marriage reverse annuity model for joint life and last survivor status. The probability structure is based on Sklar's theorem and copula survival function. The contract benefits calculation utilizing copulas (Clayton, Gumbel, and Frank) shows that joint life status benefits are higher than last survivor status. Joint life status uses the dependence assumption with Frank's copula to calculate the smallest annual benefit value of a marriage reverse annuity contract, while last survivor status uses the independence assumption (without copula).

On soft factors and transmutation operators

The well known soft theorems state the specific factorizations of tree level gravitational (GR) amplitudes at leading, sub-leading and sub-sub-leading orders, with universal soft factors. For Yang-Mills (YM) amplitudes, similar factorizations and universal soft factors are found at leading and sub-leading orders. Then it is natural to ask if the similar factorizations and soft factors exist at higher orders. In this note, by using transformation operators proposed by Cheung, Shen and Wen, we reconstruct the known soft factors of YM and GR amplitudes, and prove the nonexistence of higher order soft factor of YM or GR amplitude which satisfies the universality.

Hereditarily decomposable continua have non-block points

In this note we expand upon our results from [1] to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e. points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.

Tauberian theorems on ℝ⁺ and applications

Let f f be a bounded and uniformly continuous function from R \mathbb {R} to a Banach space X X and s p ( f ) \mathbf {sp}(f) be its Carleman spectrum. A classical Tauberian theorem states that f f is constant if and only if s p ( f ) ⊂ { 0 } \mathbf {sp}(f) \subset \{ 0 \} , and f f is ω \omega -periodic if and only if s p ( f ) ⊂ 2 π ω Z \mathbf {sp}(f) \subset \frac {2\pi }{\omega } \mathbb {Z} for some ω > 0 \omega >0 . However, one cannot expect analogous results on R + \mathbb {R}^+ since there is a counterexample showing that the case of R + \mathbb {R}^+ contrasts dramatically with the case of R \mathbb {R} . In this paper, we succeed in extending the above classical Tauberian theorem to R + \mathbb {R}^+ and obtain an extension of the well-known Ingham theorem. We also apply our Tauberian theorems to abstract Cauchy problems and improve a result in [Russian Math. 58 (2014), pp. 1–10]. Moreover, as an application, we present an extension of a Katznelson-Tzafriri theorem in [J. Funct. Anal. 103 (1992), pp. 74–84] with weaker assumptions. In addition, it is interesting to note that several of our results and examples show that S \mathcal {S} -asymptotically ω \omega -periodic functions on R + \mathbb {R}^{+} is just the “natural” analogue of periodic functions on R \mathbb {R} .

A Sharp Threshold for a Random Version of Sperner's Theorem

ABSTRACTThe Boolean lattice consists of all subsets of partially ordered under the containment relation. Sperner's Theorem states that the largest antichain of the Boolean lattice is given by a middle layer: the collection of all sets of size , or also, if is odd, the collection of all sets of size . Given , choose each subset of with probability independently. We show that for every constant , the largest antichain among these subsets is also given by a middle layer, with probability tending to 1 as tends to infinity. This is best possible, and we also characterize the largest antichains for every constant . Our proof is based on some new variations of Sapozhenko's graph container method.

Universal approximation theorem for vector- and hypercomplex-valued neural networks

The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications, including regression and classification tasks. Furthermore, it is valid for real-valued neural networks and some hypercomplex-valued neural networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural networks. However, hypercomplex-valued neural networks are a type of vector-valued neural network defined on an algebra with additional algebraic or geometric properties. This paper extends the universal approximation theorem for a wide range of vector-valued neural networks, including hypercomplex-valued models as particular instances. Precisely, we introduce the concept of non-degenerate algebra and state the universal approximation theorem for neural networks defined on such algebras.

Loophole-Free Test of Local Realism via Hardy's Violation.

Bell's theorem states that the quantum mechanical description of physical quantities cannot be fully explained by local realistic theories, laying a solid basis for various quantum information applications. Hardy's paradox is celebrated as the simplest form of Bell's theorem concerning its "All versus Nothing" approach to test local realism. However, due to experimental imperfections, existing tests of Hardy's paradox require additional assumptions of the experimental systems, and these assumptions constitute potential loopholes for faithfully testing local realistic theories. Here, we experimentally demonstrate Hardy's nonlocality through a photonic entanglement source. By achieving a detection efficiency of 82.2%, a quantum state fidelity of 99.10%, and applying high-speed quantum random number generators for the measurement setting switching, the experiment is implemented in a loophole-free manner. During 6h of running, a strong violation of P_{Hardy}=4.646×10^{-4} up to 5 standard deviations is observed with 4.32×10^{9} trials. A null hypothesis test shows that the results can be explained by local realistic theories with an upper bound probability of 10^{-16348}. These testing results provide affirmative evidence against local realism, and establish an advancing benchmark for quantum information applications based on Hardy's paradox.

On the recent-$k$-record of discrete random variables

Let $X_1,~X_2,\cdots$ be a sequence of independent and identically distributed random variables which are supposed to be observed in sequence. The $n$th value in the sequence is a $k-record~value$ if exactly $k$ of the first $n$ values (including $X_n$) are at least as large as it. Let ${\bf R}_k$ denote the ordered set of $k$-record values. The famous Ignatov's Theorem states that the random sets ${\bf R}_k(k=1,2,\cdots)$ are independent with common distribution. We introduce one new record named $recent-k-record$ in this paper: $X_n$ is a $j$-recent-k-record if there are exactly $j$ values at least as large as $X_n$ in $X_{n-k},~X_{n-k+1},\cdots,~X_{n-1}$. It turns out that recent-k-record brings many interesting problems and some novel properties such as prediction rule and Poisson approximation which are proved in this paper. One application named "No Good Record" via the Lov{\'a}sz Local Lemma is also provided. We conclude this paper with some possible connection with scan statistics.

Counting rainbow triangles in edge‐colored graphs

AbstractLet be an edge‐colored graph on vertices. The minimum color degree of , denoted by , is defined as the minimum number of colors assigned to the edges incident to a vertex in . In 2013, Li proved that an edge‐colored graph on vertices contains a rainbow triangle if . In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in . As a consequence, we prove counting results for rainbow triangles in edge‐colored graphs. One main theorem states that the number of rainbow triangles in is at least , which is best possible by considering the rainbow ‐partite Turán graph, where its order is divisible by . This means that there are rainbow triangles in if , and rainbow triangles in if when . Both results are tight in the sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph (i.e., rainbow triangles sharing a common vertex).

Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the "source data" I, and the other is for the "interface data" O. Such a specification is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a transformation (or "query" below) that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability "up to unique isomorphism". The latter connection uses a variation of a result of Gaifman concerning "relatively categorical" theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

Nelson algebras, residuated lattices and rough sets: A survey

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder.

More variations on Nagel and Gergonne analogues of the Steiner-Lehmus theorem

The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal then the corresponding sides have equal lengths. That is to say if P is the incentre of ΔABC and if BP and CP meet the sides AC and AB at B′ and C′, respectively, then An elegant proof of this theorem appeared in [1] and is reproduced in [2].

Generalized Hölder Inequality in Herz-Morrey Spaces with Variable Exponent

The paper investigates the conditions for the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The main results are based on the exponent functions p(·) and α(·) . The proof of the first main result using the generalized Hölder’s inequality in Lebesgue spaces. The second main result of the paper is related to the weak space of the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The theorems state the equivalence of certain conditions for the inequality. Mathematical proofs and analysis are providing to support the presented results for findings contribute to the understanding of Hölder’s inequalities in variable exponent spaces and their applications in Herz-Morrey spaces.

Snapshot disjointness in temporal graphs

In the study of temporal graphs, only paths respecting the flow of time are relevant. In this context, many concepts of walk disjointness have been proposed over the years and the validity of Menger's Theorem, as well as the complexity of related problems, have been investigated. Menger's Theorem states that the maximum number of vertex/edge disjoint paths between two vertices equals the minimum number of vertices/edges required to disconnect them. This paper introduces and investigates a type of disjointness that is only time-dependent. Two walks are said to be snapshot disjoint if they do not use edges active in the same snapshot. The related paths and cut problems are then defined and proved to be W[1]-hard and XP-time solvable when parameterized by the solution size. Additionally, in the light of the definition of Mengerian graphs given by Kempe, Kleinberg, and Kumar in their seminal paper (STOC'2000), we define a snapshot Mengerian graph as a graph G for which there is no time labeling for its edges where Menger's Theorem does not hold in the context of snapshot disjointness. We then characterize Mengerian graphs in terms of forbidden structures and provide a polynomial-time recognition algorithm. All our proofs work on the strict and the non-strict cases. Finally, we also present results about problems defined by similar types of disjointness, namely, multiedge disjointness and departure time disjointness.

Corrigendum: Transcendental Brauer groups of products of CM elliptic curves

AbstractThis is a corrigendum to the paper Transcendental Brauer groups of products of CM elliptic curves, published in (J. Lond. Math. Soc. 93 (2016), 397–419). In this note, we point out a mistake affecting the statements of Theorems 1.3, 5.2, 5.3 and Proposition 5.1 in op. cit., and provide a correction.

Observation of nonlinear response and Onsager regression in a photon Bose-Einstein condensate

The quantum regression theorem states that the correlations of a system at two different times are governed by the same equations of motion as the single-time averages. This provides a powerful framework for the investigation of the intrinsic microscopic behaviour of physical systems by studying their macroscopic response to a controlled external perturbation. Here we experimentally demonstrate that the two-time particle number correlations in a photon Bose-Einstein condensate inside a dye-filled microcavity exhibit the same dynamics as the response of the condensate to a sudden perturbation of the dye molecule bath. This confirms the regression theorem for a quantum gas, and, moreover, demonstrates it in an unconventional form where the perturbation acts on the bath and only the condensate response is monitored. For strong perturbations, we observe nonlinear relaxation dynamics which our microscopic theory relates to the equilibrium fluctuations, thereby extending the regression theorem beyond the regime of linear response.