4,924 papers in this slice of arXiv.
Jasem Hamoud, Duaa Abdullah
Trees of maximum σ-irregularity subject to a prescribed maximum degree Δ have been characterized for Δ=4
Jonathan Washburn, Sebastian Pardo-Guerra, Milan Zlatanović
We study nonnegative solutions of the reciprocal cost law on the positive rationals and determine which of them admit regular extensions to the positive reals. Using the substitution H=1+F, we reduce the reciprocal composition law to d'Alembert's functional equation. We prove that every nonnegative solution on Q>0 is determined by one real weight αp for each prime p, with only a global sign identification. Thus the rational solution space is infinite dimensional. We prove that every nonnegative rational solution has an algebraic extension to R>0, but a regular extension exists exactly when the prime weights satisfy αp=λlogp for some λ∈R. In this case the extension is unique and belongs to the one-parameter family Fλ(x)=cosh(λlogx)−1. Otherwise the rational solution is unbounded on every nonempty open subset of Q>0, and the regular locus is closed and nowhere dense. We also extend the result to arbitrary nontrivial subgroups of R>0, where the alternative is governed by rational rank. Finally, we show that unit logarithmic curvature selects the canonical reciprocal cost J(x)=(x+x−1)/2−1, while, for a carrier G whose logarithm is dense in R, the single asymptotic condition F(es)∼s2/2 implies both regularity and calibration.
Pavel B. Dubovski, Jeffrey A. Slepoi
The basic result in the theory of linear ODEs is that the dimension of the fundamental set of linearly independent solutions is equal to the order of equation. This view is typically extended from integer-order to fractional differential equations. The purpose of this research is to demonstrate that this expectation is incorrect and explain why. We show that linear fractional differential equations may admit additional linearly independent solutions beyond their order. This phenomenon occurs when the coefficients of the equation cross certain threshold values. This phase transition produces a hyper-dimensional fundamental set of solutions and offers new insight into fractional differential equations and leads to the revisions of the statements of initial and boundary value problems.
Vitaliy Tsyganok, Oksana Mulesa
This paper considers combinatorial and sampling approaches to deriving a priority vector from an incomplete pairwise comparison matrix. It is shown that the informationally significant combinations of matrix elements are in one-to-one correspondence with spanning trees of the graph induced by the matrix. The combinatorial approach is based on the enumeration of all spanning trees and the componentwise aggregation of the priority vectors derived from them. To reduce computational complexity, a sampling Monte Carlo approach with a sequential stopping criterion is proposed, ensuring a prescribed accuracy for estimating the components of the resulting vector.
Nilotpal Kanti Sinha
For integers 1≤m≤n, let s(n,m) denote the number of m-tuples (k1,…,km) of nonnegative integers satisfying n=j=1∑mjkj. We obtain a complete asymptotic expansion for logs(n,m), uniformly for all n≥m as m→∞. The coefficients are given explicitly in terms of limiting prime-block functions arising from the residue structure of the problem. We also determine the full hierarchy of multiplicative corrections in the sparse regime, identifying explicit constants at every fixed order and, in particular, the first correction constants 1/4 and (1−log2)/4. The results give uniform two-parameter asymptotics as both the target and the number of allowed reciprocal parts grow.
Alessandro Razeto, Nicola Rossi
At the end of 2025, a record computation of π reached 314 trillion decimal digits, providing the largest numerical dataset ever generated for this constant. We exploit this unprecedented dataset to investigate whether the digits of π themselves can serve as a source of pseudorandom numbers for estimating π through the simplest Monte Carlo method. Our results go beyond the normality hypothesis by providing empirical evidence of a high degree of statistical randomness in the available digits, although not of digit independence, which cannot hold for a deterministic sequence. By optimizing the mapping of the digit sequence into Monte Carlo samples, we obtain the highest precision allowed by the dataset. As predicted, the method successfully reproduces the first sequence of decimal digits, demonstrating that the largest available dataset of π digits can be used to recover π≈3.141593 through Monte Carlo simulation.
RB Yadav, Arpan Sharma
In this paper, we introduce the notion of crossed homomorphisms on associative superalgebras. We show that crossed homomorphisms are precisely the Maurer--Cartan elements of a naturally associated graded Lie algebra. This characterization enables us to construct a cohomology theory of crossed homomorphisms. We then investigate formal deformations of crossed homomorphisms and derive the corresponding deformation equations. It is proved that the infinitesimal part of a formal deformation is a 1-cocycle in the associated cohomology, while the obstruction to extending a deformation of finite order is a 2-cocycle.
Nilotpal Kanti Sinha
Let P+(n) be the largest prime factor of n, and let χ=χ−4 be 1 on primes 1(mod4) and −1 on primes 3(mod4). For fixed k≥2 we study Dk(x)=n≤xΩ(n)=k∑χ(P+(n)). Thus Dk(x) compares the two residue classes according to the largest prime factor of integers having exactly k prime factors, counted with multiplicity. Assuming RH for ζ(s) and L(s,χ)=β(s), we prove a pointwise explicit formula. The main term is a fixed negative contribution plus an absolutely convergent oscillating sum over the zeros ρ=21+iγ of L(s,χ). The ordering of the prime factors produces k Perron denominators, and the principal coefficient of a zero is Ok((1+∣γ∣)−k). We then show that the total size of all zero terms is strictly smaller than the fixed contribution. Hence Dk(x)<0 for all sufficiently large x. The analogous problem without fixing k is still open.
K. Mahesh Krishna
Let K be a non-Archimedean valued field. Let n∈N. For every (aj)j=1n,(bj)j=1n∈Kn, we show that align* |∑_j=1ⁿa_jb_j|²≤ {|∑_j=1ⁿ a_j²||∑_k=1ⁿb²_k|, ₁≤ j<k ≤ n|a_jb_k-a_kb_j|²}, align* align* |∑_j=1ⁿ a_j²||∑_k=1ⁿb²_k|≤ {|∑_j=1ⁿ a_jb_j|², ₁≤ j<k ≤ n|a_jb_k-a_kb_j|²}, align* align* |AM(a_j)_j=1ⁿ||AM(b_j)_j=1ⁿ|≤ { |AM(a_jb_j)_j=1ⁿ|, (1/|n|²)₁≤ j < k ≤ n|a_j-a_k||b_j-b_k|}, align* align* |AM(a_jb_j)_j=1ⁿ|≤ { |AM(a_j)_j=1ⁿ||AM(b_j)_j=1ⁿ|, (1/|n|²)₁≤ j < k ≤ n|a_j-a_k||b_j-b_k|}, align* where AM denotes the arithmetic mean. First and second are non-Archimedean versions of Cauchy-Schwarz angle-length and third and fourth are Chebyshev arithmetic mean inequalities. Unlike in the Archimedean case, no conditions are required to derive non-Archimedean Chebyshev arithmetic mean inequalities.
Ganesh Bahadur Basnet, Narayan Prasad Pahari, Feng Qi +1
Let Z−=∖{−1,−2,…}. In 2023, Qi and Lim gave two claims for summing the infinite series k=1∑∞(k2k)α+k1(4±1)k,α∈C∖Z−. In present paper, the authors establish several sum functions of the infinite and finite series k=1∑∞(k2k)α+k1(4z)kandk=1∑n(k2k)α+k1(4z)k for α∈C∖Z− and n∈N={1,2,…} in terms of the Gauss hypergeometric functions 2F1 and the generalized hypergeometric functions 3F2 for α∈C∖Z− and n∈N. In light of the Euler integral representation of the Gauss hypergeometric function 2F1, the author present several closed forms of two Gauss hypergeometric functions 2F1, two generalized hypergeometric functions 3F2, and the classical incomplete beta functions Bz(21,21+n) and Bz(21,1+n). With the help of the Euler hypergeometric transform, the authors derive closed forms of five Gauss hypergeometric functions. In addition, the authors also obtain a closed form of the differential operator [(1−z)dzd(1−z)]nz(1−z)arcsinz for n∈N0={0}∪N.
Arjun Kumar Rathie, Feng Qi, Dongkyu Lim
The objective of this paper is to develop an extension of Kummer's second theorem and to establish generalized forms of four classical combinatorial identities---Knuth's old sum (also known as Reed--Dawson's combinatorial identity), Riordan's combinatorial identity, Gould's combinatorial identity, and Touchard's combinatorial identity---using a hypergeometric-series approach. Several new identities also arise as special cases of our main results.
Pang Ern Thang, Gerard Sayson
For the classic bridge and torch problem with crossing times {1,…,n}, we derive a closed-form expression for the optimal crossing time T(n) using a recurrence relation derived from the problem's optimal substructure, thus obtaining T(n)=4n2+3n−5+8(−1)n−1 which holds for all n≥2. We generalize the problem to a bridge of capacity 3 and obtain the optimal crossing time T3(n)=6n2+2n−36181+4(−1)n−92cos(32nπ) which holds for all n≥7. As such, we obtain a new sequence A392834 in the On-Line Encyclopedia of Integer Sequences. Lastly, we also explore this problem for star graphs, and see how we can recover some classic identities involving the sum of floor functions.
Daniel Parrochia
Starting with some historical considerations, we examine the various attempts to model old age that fall within the realm of mathematical physics. For example, we study the mathematical theory of declining functions, which, since Laurent Schwartz's work on distributions, are generally defined on what is called a "Schwartz space." We explain how this essentially analytic formalization applies to the representation of the decline of vitality (or of a number of vital functions). We then show that there exist many other possible formalizations that utilize the resources of algebra, in particular the theory of "inclines" of Cao, Kim, and Roush and that of relational structures, on which an algebraic notion of "age," in Cameron's sense, can be defined. But we maintain, in conclusion, that a more positive representation of old age, consistent, moreover, with what a number of cultural sources from Antiquity to the 20th century tell us, should mobilize another type of formalization, capable of selecting, within the overall debate between the living being and its environment, persistent structures that do not decline amidst the decline of others.
Grant Molnar
We study the entire interpolation G(z)=∫01(zx)dx of the Gregory coefficients. Its completion satisfies the positive Markov-transform identity sin(πz)πzG(z)=n=1∑∞n−zn∣Gn∣. Consequently, every zero is real and simple; the negative zeros are the integers −1,−2,…, and one zero ρn lies in each (n,n+1). We derive complete logarithmic asymptotics for ρn−n, determine the Cartwright growth and canonical products of G, and realize 1/ρn spectrally. The resulting relative determinant yields γ=n=1∑∞(n1−ρn1).
Mario DeFranco
Fix an integer L≥1, and a function φ:Z≥1→[0,L]∩Z with φ−1({0})={1}. We define a multi-point variant of the Newton-Raphson-Simpson, which we call the max-phi method, as follows. We use φ to define the iteration number of a rooted plane tree. Then we construct formal series that are weighted generating functions of rooted plane trees with iteration number at most N. Finally we use these formulas to define the max-phi method applied to an arbitrary L-differentiable function.
Alberto Cabada, Paula Cambeses-Franco, Lucía López-Somoza
In this paper we study a functional differential equation subject to two-point boundary value conditions, where the functional dependence is introduced through an operator of the form equation* ∑_k=1^lγ_k(t)C_k(u), equation* where Ck:C(I)→R, k=1,…l, (I:=[a,b]) are linear continuous operators and γk∈L1(I) for all k=1,…,l. This formulation encompasses, among others, equations with piecewise constant arguments and those with integral-type dependence. We analyze this class of equations by deducing and characterizing their Green's function, as well as by computing the related adjoint problem. This approach enables us to establish connections among different types of functional equations and to relate equations with piecewise constant arguments to impulsive differential equations and non local boundary value problems. Next, we develop a series of comparison principles and results that allow us to characterize the regions where the Green's function of the original problem and of its adjoint maintain a constant sign. Finally, we illustrate the theoretical finding with representative examples.
Peter Chocian
Let χ be a primitive odd Dirichlet character of conductor f. For the universal projector polynomials Pm introduced in arXiv:2607.23177, defined by ∑mPm(X)Ym=−log(1−X(1−e−Y)), we evaluate the character Fourier spectrum at ht=ζft/(ζft−1): for every odd m, ∑tχˉ(t)Pm(ht)=τ(χˉ)Bm,χ/(mm!). After reduction at any prime above p∤f, the case m=p−j identifies this spectrum, including its exact nonzero scalar, with the divided generalized Bernoulli value attached to χω−j; the spectral-zero and Bernoulli-zero criteria therefore agree over every residue field, with no splitting hypothesis on the coefficient field. We connect the identity with the local Kummer spectrum of the circular unit 1−ζfζp and carry the programme through in the first non-real case: for the two primitive quartic characters modulo 5 and primes p<500, p≡1(mod20), exactly eleven zero lines occur. On each line an integral character projection of 1−ζ5ζp is everywhere locally unramified, a finite split-prime Artin computation proves it is not a global p-th power, and the character-wise Main Conjecture shows the radical generates the complete order-p Hilbert class component. Six of the eleven components occur at classically regular primes. At p=61 the two conjugate characters contribute on different indices. A deterministic integer-arithmetic program (ancillary file) verifies the enumeration, the divided digits, and every certificate.
Andrey P. Jivkov, Muhammad Azeem, Aneirin Griffiths +1
Combinatorial Mesh Calculus (CMC) formulates conservation laws directly on cell complexes using combinatorial differential forms and their cochain representations. This note develops an incidence-based geometric extension of that framework for directional quantities and local geometric structure on explicit cellular organisation. The central construction is a family of local fibres generated intrinsically by the incidence structure of the complex. Each vertex carries a vector space spanned by its incident edge directions, providing a combinatorial analogue of a tangent space whose dimension reflects local topology. These fibres define local coefficient spaces for incidence-based vector-, covector- and endomorphism-valued cochains. Directed transport maps compare states attached to neighbouring fibres, while a canonical solder form relates fibre directions to the underlying cell-complex structure. Together they give rise to finite combinatorial analogues of transport, torsion, curvature and metric structure. Evaluation cup products provide pairings between kinematic and force-like quantities, while bundle Hodge operators induced by the fibre metric relate vector- and covector-valued cochains, and weighted covariant incidence operators define degree-raising transport-corrected operations on incidence cochains. The framework does not assume a smooth manifold, fixed-rank bundle, cellular sheaf or local system. Instead, geometric structure is generated directly from the organisation represented by the cell complex. The resulting theory establishes a combinatorial scaffold for geometry on explicit cellular organisation and identifies the principal mathematical questions required for its further development, including admissible transport classes, metric compatibility, Cartan-type structure equations and locality-dependent algebraic structures.
Tomislav Burić, Lenka Mihoković, Toni Milas
This paper studies asymptotic expansions of the Gaussian and Archimedean compounds of two arbitrary bivariate means that need not be symmetric. Recursive algorithms for the coefficients of such expansions were previously obtained only in the symmetric case. Here, this restriction is removed and recursions are derived for the general nonsymmetric setting. All coefficient calculations are carried out in the algebra of formal power series. This introduces an additional case in the recursion, as well as singular configurations in which the formal invariance equation does not determine the coefficients recursively. The formal problem is kept separate from the existence and convergence of the iterative procedures. We also connect the first formal coefficients with Farhi's metric, recall its known global convergence criterion, and derive the corresponding local rates. The algorithms are illustrated using weighted power means and several known identities. Finally, the method is extended to expansions with sign-dependent coefficients and is applied to the neo-Pythagorean means and to a recently introduced two-parameter family.
Paolo Leonetti
Zsolt Páles asked whether there exists a function f:R→R such that f(3x+2y)≤3f(x)+2f(y) for all x≤y which is not midconvex. We show that the answer is negative.