Quantum q-series identities

As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.

Matchings under Preferences: Strength of Stability and Tradeoffs

We propose two solution concepts for matchings under preferences: robustness and near stability . The former strengthens while the latter relaxes the classical definition of stability by Gale and Shapley (1962). Informally speaking, robustness requires that a matching must be stable in the classical sense, even if the agents slightly change their preferences. Near stability, however, imposes that a matching must become stable (again, in the classical sense) provided the agents are willing to adjust their preferences a bit. Both of our concepts are quantitative; together they provide means for a fine-grained analysis of the stability of matchings. Moreover, our concepts allow the exploration of tradeoffs between stability and other criteria of social optimality, such as the egalitarian cost and the number of unmatched agents. We investigate the computational complexity of finding matchings that implement certain predefined tradeoffs. We provide a polynomial-time algorithm that, given agent preferences, returns a socially optimal robust matching (if it exists), and we prove that finding a socially optimal and nearly stable matching is computationally hard.

On the generalized fractional Laplacian

The objective of this paper is, for the first time, to extend the fractional Laplacian (−△) s u(x) over the space Ck (Rn ) (which contains S(Rn ) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in Rn . We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.

Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.

Product Lifecycle: Social and Political Reflections from the Digital and Sustainable Perspectives

Digitalization and sustainability are the drivers of the global development of the future that have slowly conquered the agendas of governments and organizations on every continent. In this context, the pandemic has proved to be a powerful technological accelerator, helping to give a greater boost to these drivers, “guiding” leading the productive and economic sector throughout the world. Today the sustainability and digitalization represent the indispensable prerequisites to add economic, environmental, and social sovereignty. In fact, the scenario that the Coronavirus is leaving us foreshadows the need not to be satisfied with reaching targets for reducing greenhouse gas emissions, but to imagine “global” governance for the development of business models based on the new digital frontiers. Thus, what are the challenges for achieving the paradigms of sustainability and digitization in this new era? And what are the tools for a “digicircular” transformation? The aim of this chapter is to investigate these issues. To this end, it should be noted that, in this chapter, our aim is not to present an analysis of literature in the classical sense but rather political and social reflections.

Convexity-preserving properties of set-valued ratios of affine functions

"The aim of this paper is to introduce some classes of set-valued functions that preserve the convexity of sets by direct and inverse images. In particular, we show that the so-called set-valued ratios of a ne functions represent such a class. To this aim, we characterize them in terms of vector-valued selections that are ratios of a ne functions in the classical sense of Rothblum."

Библиски алузии во современиот македонски роман

Biblical Allusions in Macedonian Contemporary Novel. This paper aims to show some models of biblical intertext in Macedonian contemporary novels. The concept “allusion” in this frames is not considered in classical sense, but it means more an inter-play, a ludicrous strategy of postmodern writing. The biblical allusions are interpreted as explicate quotes, implicit references, inclusions, re-writings, parodies, travesties, echoes etc. In the focus of interest are the novels: Novel for Noah by Danilo Kocevski, On the Road to Damascus by Elizabeta Bakovska, The Prophet of Diskantrija by Dragi Mihajlovski and Witch by Venko Andonovski.

A′ constructions

This chapter applies the theory of the preceding chapters to A’ constructions, such as wh-questions and relative clauses. The main result of this chapter is that there is a range of ways in which the conceptual ‘work’ associated with these constructions can be expressed in the correspondence between syntax, phonology, and meaning. None of them involve ‘movement’ in the classical sense, although some constructional devices do express links between constituents not in canonical position relative to their governing heads, giving the illusion of movement.

Digital Transformation of National Security in the Context of Global Hybrid Threats

In this article, the authors raise the issues of compliance (adequacy) of the ideology, approaches and infrastructure of the national security system in the light of global changes in the economy, de facto hybrid wars and ongoing natural and environmental emergencies. As a central element of preparation for such events and reaction to them, the authors considers mobilization in the classical sense, enshrined in the legislation, analyzes it for compliance with current hybrid threats, and also suggests new approaches to the digital transformation of the complex system of national security and mobilization, in particular.

On Uniform Stability with Growth Rates of Stochastic Skew-Evolution Semiflows in Banach Spaces

The main purpose of this paper is to study a more general concept of uniform stability in mean in which the uniform behavior in the classical sense is replaced by a weaker requirement with respect to some probability measure. This concept includes, as particular cases, the concepts of uniform exponential stability in mean and uniform polynomial stability in mean. Extending techniques employed in the deterministic case, we obtain variants of some results for the general cases of uniform stability in mean for stochastic skew-evolution semiflows in Banach spaces.