ARTICLE | doi:10.20944/preprints201705.0099.v1
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: quantum discord; entanglement; negativity; violation of Bell inequalities; decoherence
Online: 11 May 2017 (16:05:33 CEST)
Quantum Correlations are studied extensively in quantum information domain. Entanglement Measures and Quantum Discord are good examples of these actively studied correlations. Detection of violation in Bell inequalities is also a widely active area in quantum information theory world. In this work, we revisit the problem of analyzing the behavior of quantum correlations and violation of Bell inequalities in noisy channels. We extend the problem defined in a recent study by observing the changes in negativity measure, quantum discord and a modified version of Horodecki measure for violation of Bell inequalities under amplitude damping, phase damping and depolarizing channels. We report different interesting results for each of these correlations and measures. All these correlations and measures decrease under decoherence channels, but some changes are very dramatical comparing to others. We investigate also separability conditions of example studied states.
ARTICLE | doi:10.20944/preprints201703.0223.v1
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: entanglement; relative entropy of entanglement; negativity; bell inequalities violation; quantum fisher information; optimization
Online: 31 March 2017 (08:14:32 CEST)
The violation of Bell's theorem is a very simple way to see that there is no underlying classical interpretation of quantum mechanics. The measurements made on the photons shows that light signal (information) could travel between them, hence completely eliminating any chance that the result was due to anything other than entanglement. Entanglement has been studied extensively for understanding the mysteries of non-classical correlations between quantum systems. It was found that violation of Bell's inequalities could be trivially calculated and for sets of nonmaximally entangled states of two qubits, comparing these entanglement measures may lead to different entanglement orderings of the states. On the other hand, although it is not an entanglement measure and not monotonic under local operations, due to its ability of detecting multipartite entanglement, quantum Fisher information (QFI) has recently received an intense attraction generally with entanglement in the focus. In this work, we visit violation of Bell's inequalities problem with a different approach. Generating a thousand random quantum states and performing an optimization based on local general rotations of each qubit, we calculate the maximal QFI for each state. We analyze the maximized QFI in comparison with violation in Bell's inequalities and we make similar comparison of this violation with commonly studied entanglement measures, negativity and relative entropy of entanglement. We show that there are interesting orderings for system states.
ARTICLE | doi:10.20944/preprints202004.0036.v1
Subject: Physical Sciences, Mathematical Physics Keywords: Bell theorem; Local function computer program; Violation CHSH
Online: 3 April 2020 (15:29:01 CEST)
If a clear and valid no-go for Einsteinian hidden parameters is real, it must in no way be possible to violate the CHSH with a local hidden variables based computer simulation. In the paper we show that with the use of a modied Glauber-Sudarshan method it is possible to violate the CHSH.The criterion value comes close to the quantum value and is approximately 2.4. The proof (POC) is presented with the use of an R computer program. The important snippets of the code are discussed and the complete code is presented in an appendix.
ARTICLE | doi:10.20944/preprints201705.0103.v1
Subject: Physical Sciences, General & Theoretical Physics Keywords: negativity; quantum discord; violation of bell inequalities; decoherence
Online: 12 May 2017 (05:30:28 CEST)
Quantum Correlations are studied extensively in quantum information domain. Entanglement Measures and Quantum Discord are good examples of these actively studied correlations. Detection of violation in Bell inequalities is also a widely active area in quantum information theory world. In this work, we revisit the problem of analyzing the behavior of quantum correlations and violation of Bell inequalities in noisy channels. We extend the problem defined in  by observing the changes in negativity measure, quantum discord and a modified version of Horodecki measure for violation of Bell inequalities under amplitude damping, phase damping and depolarizing channels. We report different interesting results for each of these correlations and measures. All these correlations and measures decrease under decoherence channels, but some changes are very dramatical comparing to others. We investigate also separability conditions of example studied states.
ARTICLE | doi:10.20944/preprints201708.0079.v2
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Bell polynomial; Bell number; Bell polynomial of the second kind; higher order derivative; generating function; Faa di Bruno formula; inversion theorem; Stirling number of the first kind; Stirling number of the second kind; explicit formula; inversion formula; logarithmically absolute monotonicity; logarithmically complete monotonicity; determinantal inequality; product inequality
Online: 25 August 2017 (08:41:30 CEST)
In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.
ARTICLE | doi:10.20944/preprints202211.0120.v1
Subject: Physical Sciences, General & Theoretical Physics Keywords: Bell inequalities; Hidden variables; Contextuality; Entanglement; Local realism; Quantum nonlocality; Synchronization; Electrodynamics
Online: 7 November 2022 (11:25:03 CET)
We show that loophole-free Bell-type no-go theorems cannot be derived in theories involving local hidden fields. At the time of measurement, a contextuality loophole appears because each particle’s electromagnetic field interacts with the field of its respective apparatus, preventing the expression of the probability density as a function independent of the orientation of the measuring devices. Then, we use the dynamical evolution of the probability distribution to show that the spin-correlation integral can neither be expressed in terms of initial Cauchy data restricted to the particles. A correlation loophole ensues, which prevents the usage of the non-contextual correlation integrals required to demonstrate the CHSH-Bell inequality. We obtain a new inequality not violated by quantum correlation functions of entangled spin pairs, and propose that Maxwell’s electrodynamic field is the missing hidden variable triggering the coupled nonlinear oscillations of the particles, which bring about the synchronicities observed in the Einstein-Podolsky-Rosen-Bohm (EPRB) experiment.
ARTICLE | doi:10.20944/preprints201709.0034.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: multi-order logarithmic number; multi-order logarithmic polynomial; explicit formula; identity; recurrence relation; inversion theorem; Bell polynomial of the second kind; Stirling number; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity; Faa di Bruno formula
Online: 11 September 2017 (04:22:18 CEST)
In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.
ARTICLE | doi:10.20944/preprints202301.0023.v6
Subject: Physical Sciences, General & Theoretical Physics Keywords: Bell’s theorem; local realism; Bell-CHSH inequalities; quantum correlations; Bell-test experimentsexperiments
Online: 24 January 2023 (13:15:45 CET)
I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Consequently, what is ruled out by the Bell-test experiments is not local realism but the additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with.
ARTICLE | doi:10.20944/preprints202205.0274.v3
Subject: Physical Sciences, General & Theoretical Physics Keywords: Bell inequality; locality; nonlocality; realism; counterfactual definiteness
Online: 17 August 2022 (11:43:24 CEST)
We present a pragmatic analysis of the different meanings assigned to the term "local realism'' in the context of the empirical violations of Bell-type inequalities since its inception in the late 1970s. We point out that most of them are inconsistent and arise from a deeply ingrained prejudice that originated in the celebrated 1935 paper by Einstein-Podolski-Rosen. We highlight the correct connotation that arises once we discard unnecessary metaphysics.
ARTICLE | doi:10.20944/preprints202205.0015.v2
Subject: Physical Sciences, General & Theoretical Physics Keywords: Bell inequality; locality; nonlocality; local causality
Online: 11 July 2022 (08:49:57 CEST)
The alleged nonlocal character of quantum mechanics is inextricably related to the formulation of the Bell theorem. However, as we shall see, that relation is commonly incorrectly assessed. The departure from the clear line of reasoning that John Bell tried to convey has led to a polarization of part of the scientific community into radical irreconcilable positions. We show how the correct appreciation of Bell's work calls for reinterpreting the usual significance given to the Bell theorem yielding a more rational perspective of the problem. Given the relevance of the Bell-type inequalities in quantum information technology and quantum foundations, further clarification of their relation to the nonlocality conundrum deserves due attention. The exposition is also of didactic value. It shows the problems arising from incorrect inferences and superfluous metaphysical ideas.
ARTICLE | doi:10.20944/preprints201809.0205.v1
Subject: Physical Sciences, General & Theoretical Physics Keywords: Bell-inequalities; quantum nonlocality; computer simulations of Bell tests; local causality; contextuality loophole; photon identification loophole
Online: 12 September 2018 (01:06:46 CEST)
Bell type inequalities are proven using oversimplified probabilistic models and/or counterfactual definiteness (CFD). If setting-dependent variables describing measuring instruments are correctly introduced none of these inequalities may be proven. In spite of this a belief in a mysterious quantum nonlocality is not fading. Computer simulations of Bell tests allow studying different scenarios how the experimental data might have been created. They allow also to generate outcomes of various counterfactual experiments such as repeated or simultaneous measurements performed in different settings on the same ‘’ photon-pair” etc. They allow reinforcing or relaxing CFD- compliance and /or to study the impact of various “photon identification procedures” mimicking those used in real experiments. Using a specific setting- dependent identification procedure data samples consistent with quantum predictions may be generated. It reflects an active role of instruments during the measurement process. Each setting dependent data samples are consistent with specific setting –dependent probabilistic models which may not be deduced using non-contextual local realistic or stochastic hidden variables. In this paper we discuss the results of these simulations. Since the data samples are generated in a locally causal way, these simulations provide additional strong arguments for closing the door on quantum nonlocality
ARTICLE | doi:10.20944/preprints201708.0090.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Bell number; Bell polynomial; generalization; explicit formula; inversion formula; inversion theorem; Stirling number; Bell polynomial of the second kind; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity
Online: 26 August 2017 (09:12:05 CEST)
In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.
HYPOTHESIS | doi:10.20944/preprints201809.0456.v1
Subject: Chemistry, Physical Chemistry Keywords: glass transitions, universality, Bell-Evans-Polanyi principle
Online: 24 September 2018 (12:44:06 CEST)
The Vogel-Fulcher-Tammann equation is exposed as a particular example of the mean field theory. It is generalized by taking into account an arbitrary critical exponent of susceptibility, discriminating between different classes of universality. The Bell-Evans-Polanyi principle is employed to estimate the difference between the activation energies of flows in crystals and glasses, which appears to coincide with the excess Gibbs energy of the glass compared to the crystal.
ARTICLE | doi:10.20944/preprints202106.0703.v1
Subject: Physical Sciences, Acoustics Keywords: Feynman, Bell, Ballentine, Koopman, two slit experiment, Bell type experiments, classical probability theory, Kolmogorov, conditional versus unconditional probability
Online: 29 June 2021 (12:50:11 CEST)
We start with the discussion on misapplication of classical probability theory by Feynman in his analysis of the two slit experiment (by following the critical argumentation of Koopman, Ballentine, and the author of this paper). The seed of Feynman's conclusion on the impossibility to apply the classical probabilistic description for the two slit experiment is treatment of conditional probabilities corresponding to different experimental contexts as unconditional ones. Then we move to the Bell type inequalities. Bell applied classical probability theory in the same manner as Feynman and, as can be expected, he also obtained the impossibility statement. In contrast to Feynman, he formulated his no-go statement not in the probabilistic terms, but by appealing to nonlocality. This note can be considered as a part of the author's attempts for getting rid off nonlocality from quantum physics.
ARTICLE | doi:10.20944/preprints201908.0213.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Central Bell polynomials; central factorial numbers; degenerate central Bell polynomials; Stirling numbers of the first kind; special numbers; special polynomials.
Online: 20 August 2019 (11:13:05 CEST)
In this paper, we firstly consider extended degenerate central factorial numbers of the second kind and provide some properties of them. We then introduce unified degenerate central Bell polynomials and numbers and investigate many relations and formulas including summation formula, explicit formula and derivative property. Moreover, we derive several correlations for the fully degenerate central Bell polynomials associated with the degenerate Bernstein polynomials and the degenerate Bernoulli, Euler and Genocchi numbers.
ARTICLE | doi:10.20944/preprints201801.0118.v1
Subject: Physical Sciences, General & Theoretical Physics Keywords: Maximally multiqubit entangled state; Bell-pair state; CNOT gates
Online: 12 January 2018 (17:19:05 CET)
We propose a novel protocol to build a maximally entangled state based on controlled-not (CNOT) gates. In particular, we give detailed steps to construct maximally entangled state for 4-, 5-, and 6-qubit systems. The advantage of our method is the simple algebraic structure which can be realized via current experimental technology.
ARTICLE | doi:10.20944/preprints202006.0214.v1
Subject: Mathematics & Computer Science, Information Technology & Data Management Keywords: COVID-19; Prediction model; Pandemic bell curve; India; Different scenarios
Online: 17 June 2020 (09:40:23 CEST)
This paper is an attempt to present a COVID-19 prediction model for India. Lockdown plays an important role in the arrest of community spread of the disease. This was evident from the study of other countries such as Russia, Belgium and Germany, where peak cases were recorded within a month of the imposition of lockdown, that it showed an immediate positive effect. However, in India, even after 65 days of lockdown, there is no decrease in the number of daily new cases reported. There were many models prepared for India and almost all of them were proven wrong by the increase in the number of cases. The model in this paper is prepared using the COVID-19 trend in other countries, population density and the pandemic bell curve. Based on the available data until 24th May 2020, two scenarios have been presented. In one, the peak shall be obtained when the number of daily new cases per million reaches 190 and in the second when the daily new cases per million reach 724. One model predicts the number of cases to reach 1 million by mid-July 2020. The other model predicts the number of cases to peak by mid-July with the total cases reaching 20 million. The predicted cases were compared with the actual cases recorded for the period 25th May to 11th June 2020. It was observed that the actual values matched quite reasonably with the predicted values.
Subject: Physical Sciences, General & Theoretical Physics Keywords: Tsirelson bound; Bell-CHSH inequality; superquantum correlations; quantum information theory
Online: 2 July 2019 (04:24:45 CEST)
To answer Wheeler's question "Why the quantum?" via quantum information theory according to Bub, one must explain both why the world is quantum rather than classical and why the world is quantum rather than superquantum, i.e., "Why the Tsirelson bound?" We show that the quantum correlations and quantum states corresponding to the Bell basis states, which uniquely produce the Tsirelson bound for the Clauser-Horne-Shimony-Holt quantity, can be derived from conservation per no preferred reference frame (NPRF). A reference frame in this context is defined by a measurement configuration, just as with the light postulate of special relativity. We therefore argue that the Tsirelson bound is ultimately based on NPRF just as the postulates of special relativity. This constraint-based/principle answer to Bub's question addresses Fuchs' desideratum that we "take the structure of quantum theory and change it from this very overt mathematical speak ... into something like [special relativity]." Thus, the answer to Bub's question per Fuchs' desideratum is, "the Tsirelson bound obtains due to conservation per NPRF."
ARTICLE | doi:10.20944/preprints202009.0463.v1
Subject: Physical Sciences, General & Theoretical Physics Keywords: quantum mechanics; probability; quantum logic; uncertainty relation; Bell-Kochen-Specker theorem
Online: 20 September 2020 (14:01:11 CEST)
Max Born's statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. While the lat- ter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the gen- eral definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg's and others' un- certainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.
ARTICLE | doi:10.20944/preprints201708.0044.v1
Subject: Mathematics & Computer Science, Analysis Keywords: identity; Bell polynomial; unnamed polynomial; explicit formula; inversion theorem; Stirling number; binomial coeﬃcient
Online: 11 August 2017 (13:51:05 CEST)
In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.
TECHNICAL NOTE | doi:10.20944/preprints202001.0045.v3
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: detection-loophole; coincidence-loophole; Bell experiments; quantum entanglement; event-based simulation; EPR-B experiments
Online: 15 April 2021 (13:37:27 CEST)
In this note, I analyze the data generated by M. Fodje's (2013) simulation programs "epr-simple" and "epr-clocked". They were written in Python and published on Github. Inspection of the program descriptions shows that they make use of the detection-loophole and the coincidence-loophole respectively. I evaluate them with appropriate modified Bell-CHSH type inequalities: the Larsson detection-loophole adjusted CHSH, and the Larsson-Gill coincidence-loophole adjusted CHSH (NB: its correctness is conjecture, we do not have proof). The experimental efficiencies turn out to be approximately eta = 81% (close to optimal) and gamma = 55% (far from optimal). The observed values of CHSH are, as they should be, within the appropriately adjusted bounds. Fodjes' detection-loophole model turns out to be very, very close to Pearle's famous 1970 model, so the efficiency is close to optimal. The model has the same defect as Pearle's: the joint detection rates exhibit signaling. Fodje's coincidence-loophole model is actually a clever modification of his detection-loophole model. Because of this, however, it cannot lead to optimal efficiency.
SHORT NOTE | doi:10.20944/preprints201703.0055.v1
Subject: Mathematics & Computer Science, Analysis Keywords: closed form; Stirling polynomial; Stirling number; Bernoulli number; Faá di Bruno's formula; Bell polynomial
Online: 10 March 2017 (10:28:19 CET)
In the paper, by virtue of the Faá di Bruno formula and two identities for the Bell polynomial of the second kind, the authors find a closed form for the Stirling polynomials in terms of the Stirling numbers of the first and second kinds.
ARTICLE | doi:10.20944/preprints201807.0193.v2
Subject: Physical Sciences, Applied Physics Keywords: bell states; BSM; EPR pairs; LOCC; no-cloning theorem; quantum communications; quantum entanglement; quantum teleportation
Online: 26 July 2018 (04:59:03 CEST)
A simplified version of the quantum teleportation protocol is presented in here. Its experimental confirmation will have deep implications for a better understanding of Quantum Entanglement with a particular projection on Quantum Communications.
ARTICLE | doi:10.20944/preprints201611.0092.v2
Subject: Keywords: semantic spatial trajectory; role based access control; Bell-Lapadula model; multi-policy; Web Ontology Language
Online: 17 November 2016 (15:19:51 CET)
With the proliferation of locating devices, more and more raw spatial trajectories are formed, and many works enrich these raw trajectories with semantics, and mine patterns from both raw and semantic trajectories, but access control of spatial trajectories is not considered yet. We present a multi-policy secure model for semantic spatial trajectories. In our model, Mandatory Access Control, Role Based Access Control and Discretionary Access control are all enforced, separately and combined, and we represent the model semi-formally in Ontology Web Language.
ARTICLE | doi:10.20944/preprints201908.0214.v1
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: Poisson distribution; raw moments; Bell polynomials; degenerate exponential function; unsigned Stirling number of the first kind.
Online: 20 August 2019 (11:14:22 CEST)
The main purpose of this paper is to introduce and investigate degenerate Poisson distrib- ution which is a new extension of the Poisson distribution including the degenerate expo- nential function. We then provide several properties of the degenerate Poisson distribution such as the first and the second raw moments and di¤erence operator property. Moreover, we acquired the skewness and the kurtosis for the degenerate Poisson distribution. We also derive its moment generating function by which we define the degenerate Bell polynomials and give a connection for these polynomials related to the unsigned Stirling numbers of the rst kind.
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: discrete degenerate random variables; degenerate binomial random variable; degenerate Poisson random variable; new type degenerate Bell polynomials
Online: 15 November 2019 (16:43:03 CET)
In this paper, we introduce two discrete degenerate random variables, namely the degenerate binomial and degenerate Poisson random variables. We deduce the expectations of the degenerate binomial random variables. We compute the generating function of the moments of the degenerate Poisson random variables, which leads us to define the new type degenerate Bell polynomials, and hence obtain explicit expressions for the moments of those random variables in terms of such polynomials. We also get the variances of the degenerate Poisson random variables. Finally, we illustrate two examples of the degenerate Poisson random variables.
ARTICLE | doi:10.20944/preprints201708.0017.v1
Subject: Mathematics & Computer Science, Analysis Keywords: simplification; coefficient; ordinary differential equation; higher order Frobenius–Euler number; Fa`a di Bruno formula; Bell polynomial of the second kind; inversion formula
Online: 4 August 2017 (15:47:44 CEST)
In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.
ARTICLE | doi:10.20944/preprints201711.0120.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: simplification; coefficient; nonlinear ordinary differential equation; generating function; Catalan number; inverse matrix; lower triangular integer matrix; Faά di Bruno formula; Bell polynomial of the second kind; inversion theorem
Online: 20 November 2017 (07:20:26 CET)
In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.
ARTICLE | doi:10.20944/preprints201708.0026.v1
Subject: Mathematics & Computer Science, Analysis Keywords: simplification; coefficient; ordinary differential equation; higher order Bernoulli number of the second kind; Stirling number of the first kind; Stirling number of the second kind; inversion formula; Bell polynomial of the second kind; Faà di Bruno formula.
Online: 8 August 2017 (07:59:57 CEST)
In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.