The congruence speed formula

We solve a few open problems related to a peculiar property of the integer tetration ^{b}a, which is the constancy of its congruence speed for any sufficiently large b = b(a). Assuming radix-10 (the well known decimal numeral system), we provide an explicit formula for the congruence speed V(a) ∈ ℕ_0 of any a ∈ ℕ − {0} that is not a multiple of 10. In particular, for any given n ∈ ℕ, we prove to be true Ripà’s conjecture on the smallest a such that V(a) = n. Moreover, for any a ≠ 1 ∶ a ≢ 0 (mod 10), we show the existence of infinitely many prime numbers, p_j = p_j(V(a)), such that V(p_j) = V(a).

Modal Analysis of Pseudo-Schell Model Sources

All pseudo-Schell model sources have been shown to possess the same continuous set of circularly symmetric modes, all of them presenting a conical wavefront. For keeping energy at a finite level, the mode amplitude along the radial coordinate is modulated by a decreasing exponential function. A peculiar property of such modes is that they exist in the Laplace transform’s realm. After a brief discussion of the near-zone, we pass to the far-zone, where the field can be evaluated in closed form. The corresponding features of the intensity distribution are discussed.

Half-Auxeticity and Anisotropic Transport in Pd Decorated Two-Dimensional Boron Sheets

<p></p><p>If one strains a material, most materials shrink normal to the direction of applied strain. Similarly, if a material is compressed, it will expand in the direction orthogonal to the pressure. Few materials, those of negative Poisson ratio, show the opposite behavior. Here, we show an unprecedented feature, a material that expands normal to the direction of stress, regardless if it is strained or compressed. Such behavior, namely, half-auxeticity, is demonstrated for a borophene sheet stabilized by decorating Pd atoms. We explore Pd-decorated borophene, identify three stable phases of which one has this peculiar property of half auxeticity. After carefully analyzing stability, mechanical and electronic properties we explore the origin of this very uncommon behavior and identify it as a structural feature that may also be employed to design further 2D nanomaterials.</p><br><p></p>

Half-Auxeticity and Anisotropic Transport in Pd Decorated Two-Dimensional Boron Sheets

<p>If one strains a material, most materials shrink normal to the direction of applied strain. Similarly, if a material is compressed, it will expand in the direction orthogonal to the pressure. Few materials, those of negative Poisson ratio, show the opposite behavior. Here, we show an unprecedented feature, a material that expands normal to the direction of stress, regardless if it is strained or compressed. Such behavior, namely half-auxeticity, is demonstrated for a borophene sheet stabilized by decorating Pd atoms. We explore Pd-decorated borophene, identify three stable phases of which one has this peculiar property of half auxeticity. After carefully analyzing stability, mechanical and electronic properties we explore the origin of this very uncommon behavior, and identify it as a structural feature that may also be employed to design further 2D nanomaterials.</p><br>

Half-Auxeticity and Anisotropic Transport in Pd Decorated Two-Dimensional Boron Sheets

<p>If one strains a material, most materials shrink normal to the direction of applied strain. Similarly, if a material is compressed, it will expand in the direction orthogonal to the pressure. Few materials, those of negative Poisson ratio, show the opposite behavior. Here, we show an unprecedented feature, a material that expands normal to the direction of stress, regardless if it is strained or compressed. Such behavior, namely half-auxeticity, is demonstrated for a borophene sheet stabilized by decorating Pd atoms. We explore Pd-decorated borophene, identify three stable phases of which one has this peculiar property of half auxeticity. After carefully analyzing stability, mechanical and electronic properties we explore the origin of this very uncommon behavior, and identify it as a structural feature that may also be employed to design further 2D nanomaterials.</p><br>

Half-Auxeticity and Anisotropic Transport in Pd Decorated Two-Dimensional Boron Sheets

If one strains a material along a direction, most materials shrink normal to that direction. Similarly, if you compress the material, it will expand in the direction orthogonal to the pressure. Few materials, those of negative Poisson ratio, show the opposite behavior. Here, we show an unprecedented feature, a material that expands normal to the direction of force regardless if it is strained or compressed. Such behavior, called half-auxeticity, has been found for a borophene sheet stabilized by decorating Pd atoms. Herein, we explore Pd-decorated borophene, identify three stable phases of which one has this peculiar property of half auxeticity. After carefully analyzing stability, mechanical and electronic properties we explore the origin of this very uncommon behavior.<br>

A brief review on the techniques used for the enhancement of luminescence of red emitting thin film

In the past few years, red emitting thin film activated by Eu3+ has received much attention. Europium has a peculiar property that it exhibits both types of emission on the basis of their valencies. In this review, we try to make a full list of known methods which may be useful for the enhancement of luminescence. It has been found that luminescence can be enhanced by increasing substrate temperature, film roughness, changing the morphology etc. Finally we discuss the mechanism of co-doping with different elements for the enhancement of luminescence.