Analysis of sound radiation from a vibrating clamped thin rectangular plate without baffle and in the rigid baffle using exact formulas

We study double-circulant codes over a class of semi-local rings arising from the idempotent construction R=Zp2+uZp2, where u2=u, and p is an odd prime. Although both algebraic settings considered admit this presentation, they correspond to two distinct rings depending on whether the additional relation pu=0 is imposed or not. These two configurations induce different ideal lattices and symmetry properties, which play a decisive role in the structure and enumeration of codes. Exploiting the Chinese Remainder Theorem, we describe self-dual and linear complementary dual (LCD) double-circulant codes in a unified, componentwise manner. Exact enumeration formulas are derived by reducing the corresponding duality constraints to norm equations over finite fields and unramified Galois extensions of Zp2. We further construct explicit Fp-linear Gray maps from R2n to Fp6n in the degenerate case pu=0 and to Fp8n in the standard case pu≠0, and show that these maps preserve self-duality and the LCD property. Assuming a standard primitive-root hypothesis on the code length, as predicted by Artin’s primitive root conjecture, we establish asymptotic existence bounds for the Gray images of both LCD and self-dual double-circulant codes via a probabilistic argument. The degenerate case pu=0 yields a shorter Gray expansion and a stronger self-dual entropy threshold, while the case pu≠0 leads to a larger self-dual ensemble with distinct asymptotic characteristics.

For an increasing weighted tree G ω , we obtain an asymptotic value and a sharp bound on the stability index of the depth function of its edge ideal I ( G ω ) . Moreover, if G ω is a strictly increasing weighted tree, we provide the minimal free resolution of I ( G ω ) and an exact formula for the regularity of all powers of I ( G ω ) .

The (undirected) power graph $\mathcal{P}(G)$ of a finite group $G$ has vertex set $G$ and an edge $\{x,y\}$ whenever one of $x,y$ is a positive power of the other. Power graphs were introduced in directed form by Kelarev--Quinn and, in the undirected group setting, by Chakrabarty--Ghosh--Sen, and have since been studied widely .In this paper we give an explicit, computation-friendly treatment of $\mathcal{P}(C_n)$ and $\mathcal{P}(D_{2n})$ from a common viewpoint.For $C_n$ we show that adjacency is governed by divisibility of element orders, identify $\mathcal{P}(C_n)$ as a blow-up of the comparability graph of the divisor lattice, and derive closed formulas for degrees and edge counts. We also obtain exact expressions for the clique and chromatic numbers as the maximum totient-weight of a divisor chain, and for the independence number as the width of the divisor poset.For $D_{2n}$ we prove a sharp decomposition: $\mathcal{P}(D_{2n})$ is obtained from $\mathcal{P}(C_n)$ by attaching $n$ pendant leaves at the identity. This yields immediate transfer principles for many invariants and, in particular, an exact independence formula$\alpha(\mathcal{P}(D_{2n}))=n+W'(n)$, where $W'(n)$ is the width of the divisor poset of $n$ with $1$ removed.We conclude with algorithmic remarks showing how the main parameters can be computed efficiently from the prime factorization of $n$.

We study exceedance counts for order statistic intervals when boundary uncertainty is modeled through a fuzzy improved distribution function. In an ordinary setting, whether an observation falls below a threshold is decided by a crisp comparison, which can be unstable when specifications are vague, subject to tolerance bands, or expressed linguistically. We replace the crisp rule by a graded membership function and use the fuzzy improved cumulative distribution function Fμ. From an initial independent and identically distributed sample, with ordinary cumulative distribution function F, we form the random interval between the r-th and s-th order statistics, and we count how many of m independent newcomers fall inside this interval. Newcomers follow either the ordinary model (Q=F) or the fuzzy improved model (Q=Fμ). We derive exact finite-sample formulas, moments, and a distribution-free representation based on a probability integral transform, which yields the large-m limit law of the newcomer proportion. Numerical illustrations for exponential and uniform distributions show how fuzzification reshapes the distribution and can materially change predictive dispersion of exceedance counts.

The Elwert exact formula for bremsstrahlung by non-relativistic electrons in a pure Coulomb field

Background. The absorption of radiation from an object in the environment has a great influence on the efficiency of wireless sensor networks. Taking into account also the need to transmit information from sensors to a central point delivering a final (complex) solution, it should be noted that the task of detecting a radiation object (target) in such systems depends significantly on the terrain. Indeed, often the direct transmission of information from sensors to the central node is not possible, and therefore the information has to be transmitted sequentially from sensor to sensor. Aim. Develop target detection algorithms used both in the sensors themselves and in the central node, with a consistent transmission of information from sensor to sensor taking into account the attenuation of radiation from the object when using sound and vibration type sensors. Methods. The synthesis of reception algorithms was carried out by classical methods of statistical processing of information, and the analysis of efficiency was carried out according to obtained exact recurrency formulas. Results. Two complex algorithms of object detection in wireless sensor networks were synthesized and their efficiency was determined for two types of sensors and different radiation propagation media. The former is shown to be more efficient than the latter, but loses out on the second algorithm due to the need to use sensors with higher power consumption. Conclusion. The results obtained during the studies can be effectively used in selecting the type of sensors depending on the characteristics of the propagation environment and the interference situation.

This article discusses the question of the exact formula for K–functional for a pair(λn1 (ω), ln∞(h)) and the equivalent formula for Kp–functional for a pair (λnp(ω), ln∞(h)).

This paper introduces and investigates the concepts of strong hub sets and the strong hub number in hypergraphs, extending the notion of hub sets defined for graphs. For a hypergraph H = (V, E), a subset S ⊆ V (H) is said to be a strong hub set if, for every pair of distinct vertices u, v ∈ V (H) − S, either u and v are adjacent or there exists a strong S-hyperpath joining them. The minimum cardinality of such a set is called the strong hub number of H, denoted by h∗(H). Fundamental properties of strong hub sets are established, and relationships between the strong hub number and various hypergraph parameters such as the domination number and connectivity are explored. The effects of vertex deletion and weak deletion on h∗(H) are studied, leading to several sharp bounds and recursive characterizations. In particular, it is shown that under weak deletion of a non cut vertex, the strong hub number remains invariant, while deletion of a cut vertex reduces it according to the structure of the resulting components. Special attention is devoted to hypertrees, where structural simplicity allows a precise characterization. This characterization provides an exact formula for h∗(H) in acyclic hypergraphs and establishes the foundational link between connectivity and hub-based path structure in hypertrees.

Longitudinal oscillations of an inhomogeneous chain of linear oscillators coupled by springs are investigated. Both outer springs of the chain are rigidly fixed to immovable supports. The system is subjected to external periodic forces. The inhomogeneity of the chain (the perturbed system) is due to the different stiffness coefficients of the springs. These coefficients deviate slightly from a certain nominal value and depend on dimensionless deviation parameters. Zero values of these parameters correspond to a homogeneous (unperturbed) system. The resonant case is considered when the frequency of the external periodic force coincides with one of the eigenfrequencies of the unperturbed system. To construct an exact periodic solution of the perturbed system, the Lyapunov–Schmidt method is applied. As the problem is linear, this method allows to reduce it to a finite-dimensional algebraic problem of constructing a generalized Jordan chain for a degenerate linear operator. Necessary and sufficient conditions on the dimensionless deviation parameters are obtained, under which the length of such a chain is equal to 1 or 2. For each case, explicit exact formulas for the chain are derived, providing a complete description of the periodic solution. It is shown that for a generalized Jordan chain of length 1, the periodic solution of the perturbed system continuously transforms into a certain periodic solution of the unperturbed system as the small parameter $\varepsilon$ tends to zero. If the length of the generalized Jordan chain is $2$, the periodic solution of the perturbed system possesses a first-order pole at $\varepsilon=0$ and, reduces to a one-parameter family of periodic solutions of the unperturbed system. Numerical simulation was performed for a chain of eight oscillators. Plots of periodic solutions and phase trajectories of the perturbed system are constructed for various values of the small parameter.

Abstract We investigate projection constants within classes of multivariate polynomials over finite‐dimensional real Hilbert spaces. Specifically, we consider the projection constant for spaces of spherical harmonics and spaces of homogeneous polynomials as well as for spaces of polynomials of finite degree on the unit sphere. We establish a connection between these quantities and certain weighted ‐norms of specific Jacobi polynomials. As a consequence, we present exact formulas, computable expressions, and asymptotically accurate estimates for them.

The quantum transport is investigated in minimal Kitaev chain coupled to normal metal leads. The exact formulas of the charge current I and the conductance G are exhibited using the nonequilibrium Greens function approach. For simplify, the cases of asymmetric and symmetric bias voltage, the charge current, and the conductance G are presented at zero temperature. For asymmetric bias, due to the crossed Andreev reflection, the current and conductance are asymmetric. However, there is not crossed Andreev reflection in the case of symmetrically applied bias voltage. The current and conductance are symmetric about bias voltage eV = 0. In both cases, as the system's parameters ε L , ε R , and t , Δ satisfy the relation ε L ε R = t 2 −Δ 2 , there are zero‐bias peaks, which is known as poor man's Majoranas. A zero magnitude of nonlocal conductance Δ G NL , along with a flip in the sign of Δ G NL , is observed. The gap of Δ G NL must close and reopen, signaling a topological phase transition.

Exact formula and spectral decomposition of the heat flux in molecular dynamics for arbitrary many-body potentials

Abstract. This paper derives a closed-form representation for the characteristic function of the sum , where and are Poisson random variables connected through an arbitrary bivariate copula. While the sum of independent Poisson variables remains Poisson-distributed, dependence—introduced via a discrete copula—fundamentally alters the distributional structure of . Using the discrete version of Sklar’s theorem and a difference-operator representation of copulas for count data, we establish an exact series formula for the characteristic function of under completely general dependence. Furthermore, we develop explicit analytic expressions for three Archimedean copulas (Clayton, Frank, and Gumbel) and provide a numerically tractable representation for the Gaussian copula using the distributional transform approach. Numerical experiments confirm that different forms of dependence create distinct signatures in the characteristic function, highlighting its usefulness for modeling dependent count data in insurance, hydrology, and reliability applications.

The concept of dynamic symmetry in art and extensive measurements on Greek vases suggest that a vase and its parts can be inscribed into similar rectangles, with all rectangles having the same ratio of lengths of their side. Such an observation is often used in describing self-similarity and fractal geometry. This work proposes a hypothesis that a logarithmic spiral describes the equation of the cross-section of a Greek vase. From extensive measurements, the parameters of such spirals are computed, and explicit formulae are derived for volume based on a few size measurements. The exact formula is quite complex and cannot be easily used, certainly not in antiquity. Therefore, a simple approximation formula is proposed for amphorae, the most important type of vase. This formula expresses the volume of the vase in terms of its diameter and the height of the corresponding solid. The approximation is compared with some exact volume computation results reported for amphorae, and it is shown that the proposed approximation is fairly close to the exact value. The simplicity of the proposed formula suggests an efficient method of calculating volume that was probably known in antiquity.

A uniformly spaced coherent train of pulses is commonly used in radar systems for improving Doppler resolution. The Doppler resolution is defined by the total coherent duration of the processed signal and can be very high for such sequences. But the ambiguity function of the pulse train still suffers from significant sidelobes. Transmitting the same signal periodically and then processing the returns coherently introduces large ambiguities in the matched filter delay-Doppler response, or the pulse train ambiguity function, which occur at multiples of pulse repetition interval along the delay axis and at multiples of pulse repetition frequency along Doppler. Thus, modifications are usually applied to reduce these sidelobes. Different parameters of the pulse train can be varied to optimize the ambiguity function and improve resolution properties. The most common way is by implementing interpulse amplitude weighting. In many works, exact analytical formulas for the ambiguity function of pulse sequences have been obtained, but they are quite cumbersome, so it is difficult to determine from them how exactly the sequence parameters affect the shape of the ambiguity function. In this paper, a simple analytical expression for the cross-section of the pulse train ambiguity function along the Doppler axis is obtained. It valid for zero time offset. It allows us to simply understand and clearly determine the effect of intra- and inter-pulse weighting on the shape of the ambiguity surface along the central Doppler frequency axis. It is shown by simulation that this effect is preserved for non-zero time offsets. The Hamming window was used in this paper to process the pulses. In the case of inter-pulse weighting, it applies to all pulses in the sequence. In the case of intra-pulse weighting, the amplitude of each pulse was varied using this window.

Abstract This paper presents an efficient numerical method for solving variable-order fractional differential equations (VO-FDEs) by using the fractional-order Chelyshkov functions (FCHFs). The variable-order fractional derivative and the variable-order fractional integral are considered in the Caputo sense and the Riemann-Liouville sense, respectively. The exact formula for the Riemann-Liouville variable-order fractional integral of the FCHFs is obtained. This value, together with the spectral collocation method are used to transform the VO-FDE into a system of algebraic equations. The convergence of the proposed method is demonstrated. The performance of the introduced method is tested through various examples of VO-FDEs. A comparison with recent numerical methods shows that the proposed method can be successfully used to solve VO-FDEs with accuracy and efficiency.

The weakly asymmetric exclusion process (WASEP) in one dimension is a paradigmatic system of interacting particles described by the macroscopic fluctuation theory (MFT) in the presence of driving. We consider an initial condition with densities ϱ_{1},ϱ_{2} on either side of the origin, so that for ϱ_{1}=ϱ_{2} the gas is stationary. Starting from the microscopic description, we obtain exact formulas for the cumulant generating functions and large deviation rate functions of the time-integrated current and the position of a tracer. As the asymmetry/driving is increased, these describe the crossover between the symmetric exclusion process and the weak-noise regime of the Kardar-Parisi-Zhang (KPZ) equation: we recover the two limits and describe the crossover from the WASEP cubic tail to the 5/2 and 3/2 KPZ tail exponents. Finally, we show that the MFT of the WASEP is classically integrable, by exhibiting the explicit Lax pairs, which are obtained through a novel mapping between the MFT of the WASEP and a complex extension of the classical anisotropic Landau-Lifshitz spin chain. This shows integrability of all MFTs of asymmetric models with quadratic mobility as well as their dual versions.

ABSTRACT This paper focuses on solving the Fredholm–Volterra integro‐differential equations (IDEs) using a numerical method based on function approximation with generalized fractional‐order Bell wavelets (GFOBWs). Previous studies have shown that the Legendre polynomials are highly effective for approximating solutions to fractional differential equations, while the Bell polynomial collocation method provides significantly better results than the Legendre collocation method for IDEs. Building on these findings, we propose a novel collocation technique using fractional‐order Bell wavelets to address the Fredholm–Volterra IDEs. An exact formula is derived for the Riemann–Liouville fractional integral operator (RLFIO) of the GFOBW. This formula is combined with the collocation method to convert the IDEs into a system of algebraic equations, enabling an efficient numerical solution method. Additionally, the absolute error of the numerical solution obtained from the proposed method is estimated. Various examples are provided to demonstrate the accuracy of the method.

Abstract Let $\{X_{i}\}_{i\geq1}$ be a sequence of independent and identically distributed random variables and $T\in\{1,2,\ldots\}$ a stopping time associated with this sequence. In this paper, the distribution of the minimum observation, $\min\{X_{1},X_{2},\ldots,X_{T}\}$ , until the stopping time T is provided by proposing a methodology based on an appropriate change of the initial probability measure of the probability space to a truncated (shifted) one on the $X_{i}$ . As an application of the aforementioned general result, the random variables $X_{1},X_{2},\ldots$ are considered to be the interarrival times (spacings) between successive appearances of events in a renewal counting process $\{Y_{t},t\geq0\}$ , while the stopping time T is set to be the number of summands until the sum of the $X_{i}$ exceeds t for the first time, i.e. $T=Y_{t}+1$ . Under this setup, the distribution of the minimal spacing, $D_{t}=\min\{X_{1},X_{2},\ldots,X_{Y_{t}+1}\}$ , that starts in the interval [0, t ] is investigated and a stochastic ordering relation for $D_{t}$ is obtained. In addition, bounds for the tail probability of $D_{t}$ are provided when the interarrival times have the increasing failure rate / decreasing failure rate property. In the special case of a Poisson process, an exact formula, as well as closed-form bounds and an asymptotic result, are derived for the tail probability of $D_{t}$ . Furthermore, for renewal processes with Erlang and uniformly distributed interarrival times, exact and approximation formulae for the tail probability of $D_{t}$ are also proposed. Finally, numerical examples are presented to illustrate the aforementioned exact and asymptotic results, and practical applications are briefly discussed.