On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

We present several formulae for the large t t asymptotics of the Riemann zeta function ζ ( s ) \zeta (s) , s = σ + i t s=\sigma +i t , 0 ≤ σ ≤ 1 0\leq \sigma \leq 1 , t > 0 t>0 , which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum ∑ a b n − s \sum _a^b n^{-s} for certain ranges of a a and b b . In addition, we present precise estimates relating this sum with the sum ∑ c d n s − 1 \sum _c^d n^{s-1} for certain ranges of a , b , c , d a, b, c, d . We also study a two-parameter generalization of the Riemann zeta function which we denote by Φ ( u , v , β ) \Phi (u,v,\beta ) , u ∈ C u\in \mathbb {C} , v ∈ C v\in \mathbb {C} , β ∈ R \beta \in \mathbb {R} . Generalizing the methodology used in the study of ζ ( s ) \zeta (s) , we derive asymptotic formulae for Φ ( u , v , β ) \Phi (u,v, \beta ) .

Towards the development of an automated electrical self-potential sensor of melt and rainwater flow in snow

To understand snow structure and snowmelt timing, information about flows of liquid water within the snowpack is essential. Models can make predictions using explicit representations of physical processes, or through parameterization, but it is difficult to verify simulations. In situ observations generally measure bulk quantities. Where internal snowpack measurements are made, they tend to be destructive and unsuitable for continuous monitoring. Here, we present a novel method for in situ monitoring of water flow in seasonal snow using the electrical self-potential (SP) geophysical method. A prototype geophysical array was installed at Col de Porte (France) in October 2018. Snow hydrological and meteorological observations were also collected. Results for two periods of hydrological interest during winter 2018–19 (a marked period of diurnal melting and refreezing, and a rain-on-snow event) show that the electrical SP method is sensitive to internal water flow. Water flow was detected by SP signals before it was measured in conventional snowmelt lysimeters at the base of the snowpack. This initial feasibility study shows the utility of the SP method as a non-destructive snow sensor. Future development should include combining SP measurements with a high-resolution snow physics model to improve prediction of melt timing.

Structural Properties of Faces of the Cone of Copositive Matrices

In this paper, we study the properties of faces and exposed faces of the cone of copositive matrices (copositive cone), paying special attention to issues related to their geometric structure. Based on the concepts of zero and minimal zero vectors, we obtain several explicit representations of faces of the copositive cone and compare them. Given a face of the cone of copositive matrices, we describe the subspace generated by that face and the minimal exposed face containing it. Summarizing the results obtained in the paper, we systematically show what information can be extracted about the given copositive face in the case of incomplete data. Several examples for illustrating the main findings of the paper and also for justifying the usefulness of the developed approach to the study of the facial structure of the copositive cone are discussed.

A card game for designing activities for technology-enhanced learning in higher education

The importance of providing mechanisms and tools that effectively support the transition from implicit to explicit representations of Learning Design has been emphasised by previous research in the field of Technology-Enhanced Learning (TEL). In addition, the benefits of Game-based learning approaches have been long documented in the educational research literature. The paper presents the design, implementation and evaluation of a card game that aims to support the design process of TEL activities in higher education. The game was tested by a group of 36 students and tutors (n = 36) in higher education during an interactive workshop. Feedback was asked by the participants using an anonymous survey. The results reveal that the participants a) are satisfied with the game process, b) appreciate the groupwork and interaction taking place, and c) believe that they used their communication and collaboration skills. The paper includes the description of the outputs of a group (i.e., the cards selected for their TEL scenario and their actual TEL scenario) to exemplify that it is possible to use the game in order to elicit or diagnose existing LD knowledge from the game participants. The paper concludes on the usefulness of the approach suggested, limitations, and plans for future work.

Degenerate poly-Bell polynomials and numbers

Numerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when $\lambda \rightarrow 0$ λ → 0 . Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if $k = 1$ k = 1 . We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

2-Variable Fubini-degenerate Apostol-type polynomials

This work deals with the mathematical inspection of a hybrid family of the degenerate polynomials of the Apostol-type. The inclusion of the derivation of few series expansion formulas, explicit representations and difference equations for this hybrid family brings a novelty to the existing literature. Moreover, certain connection formulas and several novel identities for these polynomials are established and investigated. The graphical representations of certain degenerate polynomials are explored and several new interesting pattern of the zeros are observed.

Spontaneous Representations of Disability and Attitudes toward Inclusive Educational Practices: a Mixed Approach

The present study's primary aims were a) to explore non-disabled adults' spontaneous representation of disability and the specific associations related to adults and children with disabilities; to investigate participants' general perception of specific inclusive educational practices and the potential impact of contact with disabled individuals on children. We used a mixed (qualitative and quantitative) approach in a sample of 628 participants aged 18 to 82 (M=28.59, SD=11.50). Our results suggested that most explicit representations of disability were negatively valenced, i.e., people generally used pessimistic and detrimental related words. Psychomotor deficiencies comprised the most frequent disability category associated with disabled adults, while autism was the most frequent disability related to disabled children. Participants considered that the inclusion of physically disabled children (compared to children with intellectual disabilities) in public schools has a more positive effect on non-disabled children. The previous contact with a friend or a family member with a disability significantly and positively impacted the general attitude toward disability and inclusive educational practices. Results are discussed regarding their practical implications for the educational system and specific strategies related to inclusive public policies.

On the Irreducibility of Polynomials Associated with the Complete Residue Systems in any Imaginary Quadratic Fields

For a Gaussian prime π and a nonzero Gaussian integer β = a + b i ∈ ℤ i with a ≥ 1 and β ≥ 2 + 2 , it was proved that if π = α n β n + α n − 1 β n − 1 + ⋯ + α 1 β + α 0 ≕ f β where n ≥ 1 , α n ∈ ℤ i \ 0 , α 0 , … , α n − 1 belong to a complete residue system modulo β , and the digits α n − 1 and α n satisfy certain restrictions, then the polynomial f x is irreducible in ℤ i x . For any quadratic field K ≔ ℚ m , it is well known that there are explicit representations for a complete residue system in K , but those of the case m ≡ 1 mod 4 are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.

Least-thickness symmetric circular masonry arch of maximum horizontal thrust

This analytical note shall provide a contribution to the understanding of general principles in the Mechanics of (symmetric circular) masonry arches. Within a mainstream of previous research work by the authors (and competent framing in the dedicated literature), devoted to investigate the classical structural optimization problem leading to the least-thickness condition under self-weight (“Couplet-Heyman problem”), and the relevant characteristics of the purely rotational five-hinge collapse mode, new and complementary information is here analytically derived. Peculiar extremal conditions are explicitly inspected, as those leading to the maximum intrinsic non-dimensional horizontal thrust and to the foremost wide angular inner-hinge position from the crown, both occurring for specific instances of over-complete (horseshoe) arches. The whole is obtained, and confronted, for three typical solution cases, i.e., Heyman, “CCR” and Milankovitch instances, all together, by full closed-form explicit representations, and elucidated by relevant illustrations.