Computer Science and Mathematics

Abstract: For an n × n matrix over the algebra of ordered pairs of normalized real numbers (OPNs), the labelled nullities of shifted powers through order ⌈n/2⌉, at all complex scalar parameters, determine its real similarity class. Neither a characteristic polynomial nor a minimal polynomial is supplied. The order bound is optimal in every dimension. The proof recovers the short primary blocks by classical Weyr differences and determines the remaining lengths from the total dimension. We also characterize exactly which truncated profiles are realizable. The OPNs algebra is identified with ℝ × ℝ through its two real characters. Smith chains and Fitting ideals give the corresponding global invariants, while explicit natural tensor and Tor identifications interpret the local data as higher evaluation fibers. These invariants are preserved under transport between normalizations.

Abstract: We study the Shannon entropy increment generated by the self-power escort transformation on a finite probability simplex, in which each coordinate is raised to its own value and the resulting weights are normalized. With the continuous boundary convention 0⁰ = 1, we prove that entropy increases strictly at every nonuniform probability vector, in every dimension. The proof establishes the stronger fact that the entropy increment is strictly Schur-convex in the interior. Its main ingredient is a monotonicity estimate for the coordinate derivatives, based on an aggregation inequality for self-powers and a uniform two-point weight bound. A facewise argument extends strict entropy increase to all boundary points. We also prove binary output majorization and construct interior counterexamples to output majorization in every dimension at least five. Finally, we determine the sharp quadratic entropy-production coefficient near the uniform distribution. All map-specific ingredients are derived here; the entropy theorem does not assume a separate convergence or contraction theorem for the transformation.

Abstract: For a fixed integer \( n\ge2 \), let \( \Delta_{n-1} \) denote the standard probability simplex in $\( \mathbb R^n \)$. We introduce and analyze the self-power escort transformation \( T_n:\Delta_{n-1}\to\Delta_{n-1} \) defined by \( \begin{equation*}T_n(p)_i=\frac{p_i^{p_i}}{\sum_{j=1}^n p_j^{p_j}},\qquad i=1,\ldots,n,\end{equation*} \) with the continuous boundary convention \( 0^0=1 \). Unlike the usual power escort transformation, whose exponent is an external parameter, the exponent here is the coordinate itself; equivalently, each coordinate is reweighted by \( \exp(p_i\log p_i) \). The paper proves that this elementary self-feedback rule has a rigid global dynamics. Every boundary point is immediately activated, every orbit enters an explicitly described compact core after one step, the uniform distribution is the unique fixed point, and the Hilbert log-diameter contracts exponentially after core entrance. Consequently every orbit converges exponentially to the uniform distribution, and no nontrivial periodic orbit exists. We then compute the complete linear spectrum and second-order normal form at the uniform state, obtaining a sign transition between the binary and higher-dimensional systems. For \( n\ge4 \), the post-core Hilbert contraction constant is sharpened to a two-variable min--max problem involving only the smallest and largest coordinates. We also classify the entire one-step uniformization fiber \( T_n^{-1}(\nu_n) \): besides the vertices and the uniform point, each dimension \( n\ge3 \) has exactly one nonuniform interior orbit type, up to permutation. Finally, in the binary case we prove global contraction in logit coordinates and an arithmetic escape phenomenon: algebraic irrational inputs become transcendental after one step, while rational nonuniform inputs become transcendental after two steps.

Abstract: We introduce a Hausdorff-type fuzzy partial metric in the family of compact nonempty subsets of a fuzzy partial metric space (FPMS) and investigate its basic properties. The construction allows the usual set-valued Hausdorff approach to be formulated in a setting in which both fuzziness and nonzero self-distances are present. Using this structure, we establish a FP theorem for multivalued fuzzy (η)-contractive mappings on 1-complete FPMSs. The proof is based on the construction of an iterative sequence, the derivation of a geometric estimate through the function (η), and the completeness of the underlying space. An example is provided to verify the hypotheses of the main FP theorem and to illustrate the resulting FP. We further study a parameterized multivalued mapping and prove a homotopy FP theorem under suitable contractive, parameter-continuity, boundary, and local stability conditions. The result shows that the existence of a FP is preserved between the two endpoints of the parameter interval. These results extend the study of multivalued FP problems from partial metric and fuzzy metric settings to the combined framework of FPMSs.

Abstract: The purpose of this paper is to establish a systematic connection between multi-valued topological groups and neutrosophic multi-valued functions. Multi-valued algebraic structures generalize ordinary algebraic operations by allowing the product of two elements to be a nonempty subset rather than a single element. Hypergroups in the sense of Marty constitute one of the principal frameworks for such structures. On the other hand, neutrosophic theory incorporates truth, indeterminacy, and falsity components, while multi-valued neutrosophic sets permit these components to take sets of values rather than single numbers. We introduce a class of neutrosophic multi-valued functions adapted to compact-valued mappings and construct the neutrosophic lift of a multi-valued topological group. We define neutrosophic translations, neutrosophic hyperoperations, neutrosophic inverse maps, neutrosophic subgroups, and neutrosophic continuous homomorphisms. Several structural results are proved. In particular, we show that a Vietoris-continuous multi-valued group operation induces a natural neutrosophic multi-valued group-valued function whenever its three neutrosophic components are Vietoris-continuous. Conversely, suitable level-set conditions allow one to recover the underlying multi-valued topology from the neutrosophic representation. A principal result establishes an equivalence, under natural regularity conditions, between continuity of a neutrosophic multi-valued operation and simultaneous continuity of its truth, indeterminacy, and falsity components. We also investigate neutrosophic α-cuts, level hypergroups, neutrosophic subgroups, homomorphisms, quotient structures, and convergence. The resulting theory provides a bridge between topological hyperstructure theory, set-valued analysis, and neutrosophic mathematics.

Abstract: Recursive interval algebras are built from \(R = \mathbb{Z}[\Omega^{-1}]\), where \(\Omega > 1\) is an integer, by repeated formation of ordered pairs. The component operations \(\operatorname{pos}\) and \(\operatorname{len}\) define arithmetic between arbitrary finite depths by descent to scalars. The finite levels realize commutative algebras with square-zero generators, and their path coordinates connect algebraic structure with a recursively weighted norm. A sharp one-level multiplication estimate yields exact constants for prescribed, possibly unequal, depths, with all equality cases and their attainability over the original coefficient ring. Both tensor orientations have fixed-block norm factorization and an exact criterion for mixed-depth inner-product factorization. Integral exponential and logarithmic operations determine normalized consecutive-block decompositions of units. Recursive derivative evaluation characterizes coefficient-ring preservation and gives sharp bounds on homogeneous inputs. Across unrestricted finite depths, local boundedness on a fixed-scalar fiber holds exactly when all derivative data of order at least two vanish. Degree scaling makes products contractive precisely on a Euclidean parameter ball, and outside that ball no finite depth-independent bound exists. Tensor concatenation further gives sharp commutator and Jordan associator estimates, including block-length-dependent constants and arithmetic conditions for attaining their endpoints.

Abstract: Class ordinals and class cardinals are first defined as an extension of ordinals and cardinals, which are called set ordinals and set cardinals. In order to create an unambiguous definition of a maximally large class, this definition needs to quantify over all set theories. Accordingly, the Axiom of Maximal Strength (AMS) asserts that there exists a class cardinal that does not Ω⁺-provably exist in any MK-extending T_M-fragment that has an injective theorem-preserving Ω⁺-interpretation into a ZFC-extending T_M-fragment that is definable by a formula without parameters in the infinitary first-order language L_∞,∞, where Ω⁺-provability and Ω⁺-interpretability are defined in a logic that can make a number of deductive steps equal to the number of sentences in L_∞,∞. This logic is called Ω⁺-logic. The meta-theory T_M that interprets AMS is presupposed to be Ω⁺-sound, Ω⁺-complete, and an infinitary extension of MK. It is then proven that T_M is Ω⁺-consistent with AMS, that T_M is maximally strong, and that the class Ord of all the ordinals T_M proves to exist, is an Ω⁺-consistent version of Cantor's absolute infinite Ω.

Abstract: We introduce a Gaussian spectral-packet detector for horizontal displacement of the non-trivial zeros of the Riemann zeta function and connect it quantitatively to the prime side of the explicit formula. For distinct zeros \(\rho=\beta_\rho+i\gamma_\rho\) with multiplicity \(m_\rho\), let \(S_\lambda(t)=\sum_\rho m_\rho e^{-\lambda(z_\rho-t)^2},\qquad z_\rho=\gamma_\rho-i(\beta_\rho-\tfrac12),\) and for a compact interval \(I\), \(Q_I=\max\left\{0,\sup_\rho\left[(\beta_\rho-\tfrac12)^2-\mathrm{dist}(\gamma_\rho,I)^2\right]\right\}\). We prove that the logarithmic exponential rate of \(\sup_{t\in I}|S_\lambda(t)|\) is exactly \(Q_I\): a limsup for arbitrary compact \(I\), and a genuine limit when \(I\) has positive length. The interval lower bound uses an \(O(1/\lambda)\) micro-shift and a finite-dimensional norm-separation argument, preventing dominant-cluster phase cancellation from destroying the exponential rate. This yields an exact RH-equivalent subexponential-growth criterion. Using an even shifted Gaussian and the Guinand–Weil formula in a fixed \(2\pi\) Fourier convention, we derive \(S_\lambda=B_\lambda+G_\lambda-P_\lambda\) and prove an explicit uniform prime-tail estimate. In particular, a finite arithmetic reconstruction preserves the same exponential rate when \(\log N(\lambda)\ge(2+\varepsilon)\lambda\). We then examine the pre-asymptotic regime computationally: finite-prime peak reconstruction, blind synthetic off-critical-pair recovery, multi-pair interference, and nuisance-projected waveform identifiability. These computations test finite-scale identifiability only; they do not prove RH or identify an actual off-critical zero.

Abstract: We prove that every four-interval partition of the unit interval admits a partition of the integers such that each frequency union is a Riesz spectrum for the corresponding union of intervals, and also for every interval with the same measure.

Abstract: The quantum measurement problem asks how the smooth, unitary evolution of a superposition gives way to a single, definite outcome when a macroscopic detector interacts with a quantum system. This paper presents a geometric toy model of this interaction, with a deterministic factorization stage and a probabilistic selection step. The entangled state of a quantum system and a detector is embedded into semi-structured complex (ℍ) state space, where newly introduced projection operators, called trigger transformations, factorize the entangled state into a tensor product of two identical subsystems: one describing the quantum system and one describing what the detector registers. This mirrored factorization is shown to reduce exactly to the original entangled state on projection back to standard Hilbert space. Branch-indexed selector operators R, also introduced here, then return the state to Hilbert space and retain a single branch (or measurement outcome), with the standard quantum probabilities acting as structural weights of the branches. The model thereby confines the non-unitary step of measurement to a single, explicitly defined selection operation, offering an alternative to an ad-hoc wave function collapse. In its present form, the outcome rule and the Born weights are stated as an explicit postulate rather than derived, and a possible coupling to background vacuum fluctuations is identified as a route to future experimental tests.

Abstract: We prove that every partition of \( [0,1) \) into five intervals admits a partition of \( \mathbb Z \) whose nonempty unions are Riesz spectra for the corresponding unions of intervals. No arithmetic assumption on the endpoints is required. The proof combines folding along the primes with local constructions in the rational affine space determined by the lengths. The rational-rank \( 3 \) case is based on a computer-assisted verification of a finite family of Fourier minors by exact integer arithmetic.

Abstract: In this study, the numerical comparison is performed in solving first-order neutrosophic initial value problems employing two algorithms, namely, the Euler and fourth-order Runge-Kutta (RK4) methods in the neutrosophic environment. As an example of such kind of problem, the model y^' (t)=0.5∙y(t) on [0,2] is chosen, but in the case of the additional neutrosophic initial condition y(0)=⟨(0.8,1.0,1.3)⟩. This problem is represented as crisp-endpoint systems by means of ( α,β,γ )-cut approach, and then, it is solved numerically at each cut level while using exact solution as a reference. In accordance with the iteration results, the maximum absolute error of the Euler algorithm at h=0.2 is 1.619×10^(-1) at point t=2, whereas for the RK4 approach, this value is 2.7×10^(-6), meaning that the improvement degree reaches about five orders of magnitude. The analysis of step refinement confirms the convergence orders of the methods that are 1.00 and 4.00 for the Euler and RK4 approaches, respectively.

Abstract: This paper studies the existence and uniqueness of solutions for an irregular fractional boundary value problem (IFBVP) of order \(q\in(3/2,2]\). In order to apply Rus's fixed point theorem with the maximum and \(L^2\)metrics, we derive new estimates for the associated Green's function. We evaluate the squared Green’s function in a double integral to obtain the constant in the \(L^2\)estimate. Evaluating the mixed terms of this integral makes the appearance of the hypergeometric function ₃F₂(1), in which we apply classical transformation formulas to obtain finite sum evaluations of these terms, yielding the explicit constant \(\mathcal{K}_2(q,\rho,\theta)\). Using these estimates, we establish existence and uniqueness of solutions to the IFBVP under a Lipschitz condition on the nonlinear function \(f\)with constant \(L>0\), provided that \(L\sqrt{\mathcal{K}_2(q,\rho,\theta)}<1\). Numerical comparisons with an existing condition involving \(\Lambda(q,\rho,\theta)\)show that our condition is less restrictive for some parameter values, while the existing condition is less restrictive for others.

Abstract: We develop a differential-geometric framework for connectingmulti-valued topological groups with classical Lie groups and Lie algebras. A multi-valued group operation is interpreted as a hyperoperation \( \star:G\times G\longrightarrow \mathcal P^*(G), \) where \( \mathcal P^*(G) \) denotes the family of nonempty subsets of \( G \). The principal difficulty is that the usual tangent-space construction for a Lie group cannot be applied directly to an arbitrary hypergroup: the product of two points is no longer a point, and hence there is no canonical differential of the multiplication map. To overcome this obstruction we introduce the notions of smooth multi-valued topological groups, locally single-valued branches, infinitesimal-collapse structures, and branch-compatible multi-valued Lie groups. We prove that a smooth local branch defines a local Lie group and consequently a Lie algebra. We establish conditions under which different branches induce the same Lie algebra, and we construct a canonical quotient Lie group whenever the ambiguity of the multi-valued operation is generated by a normal subgroup. We prove that the tangent structure of a branch-compatible multi-valued group is a Lie algebra and that the first-order part of the multi-valued product is governed by the ordinary Lie bracket. Several examples are studied in detail. These include ordinary Lie groups viewed as degenerate multi-valued groups, quotient-induced multi-valued operations, finite-group extensions, matrix examples, a two-branch deformation of a Lie group, and a class of multi-valued structures arising from normal subgroup ambiguity. A counterexample shows why continuity alone is insufficient for the existence of a Lie algebra. Finally, we formulate a multi-valued Baker--Campbell--Hausdorff principle and describe possible directions toward a genuine theory of Lie hypergroups, multi-valued Lie groupoids, and neutrosophic Lie hypergroups.

Abstract: Let G be a finite group admitting an epimorphism onto the dihedral group D2p, where p is an odd prime. We study near-factorizations \( G\setminus\{1\}=S\times T \)through the quotient images of the two factors in the integral group ring Z[D_2p]. The quotient images satisfy the identity \( \widehat f(S)\widehat f(T)=|Ker(f)|H-1 \), and the irreducible representations of D_2p turn this identity into necessary conditions on the quotient multiplicities: the nontrivial linear character forces the two coset counts of each factor to differ by exactly one, and the two-dimensional representations impose determinant-unit conditions in cyclotomic integer rings. We also derive a Fourier energy identity, a joint energy inequality, and an exact finite system of equations for the quotient multiplicities. Under an explicit orbit-commutativity hypothesis we obtain closed formulas for the orbit norms of the quotient images. Finally, we prove a lifting criterion for near-factorizations of D_2p×C_m and give an exact verification procedure.

Abstract: This article provides an advanced overview of the structural, analytical, and geometric bridges connecting multidimensional linear algebra and multilinear harmonic analysis. We explore the foundational duality between metric tensor products and multilinear forms, examine modern multilinear Calderón–Zygmund theory, review phase-space wave packet decompositions underlying the bilinear Hilbert transform, detail the role of multilinear Kakeya and Brascamp–Lieb inequalities in ℓ²(L^p) decoupling theory, and synthesize their applications to nonlinear partial differential equations through fractional Leibniz rules and multi-parameter analysis.

Abstract: Fuzzy membership and the standard Zadeh truth-value operations are usually introduced as mathematical primitives. This paper derives their full truth-value algebra from native physical contents supplied independently by five theories: Newtonian mechanics, Schrödinger quantum mechanics, perturbative string theory, the M2/M5 brane sectors of M-theory, and general relativity. The corresponding contents are Liouville volume, Born density, perturbative S-matrix channel content |M_α|²dΦ_α obtained from unitarity and Cutkosky cuts, the positive geometric/tension density derived from the M2/M5 worldvolume actions, and relativistic proper volume. Normalization gives values in [0,1] and physical complementation gives 1−μ. An exact characterization proves that the Zadeh minimum and maximum identities hold for a pair of physical properties if and only if the pair is comparable modulo a content-null set; strict positivity reduces this to ordinary nesting. The physical integral contents are nonatomic and divisible, so a nested physical chain can be constructed whose normalized contents realize every u ∈ [0,1]. Hence min, max, and 1−u close on the physically realized truth values, giving the full Zadeh algebra ([0,1],min,max,1−·) at the truth-value level. A common localization theorem, with the common carrier and Boolean domain made explicit, recovers Boolean indicators under concentration away from the property boundary.

Abstract: Meritocratic lotteries seek to implement prescribed selection probabilities reflecting merit or priority while meeting diversity quotas. We study this implementation problem on a hypergraph, with arbitrary target marginals and integer lower and upper quotas. Using network-flow integrality and mean-preserving rounding, we give an explicit lottery satisfying every quota when the traits admit a partition into two disjoint-set families. For general trait hypergraphs, a floating-variable construction preserves all marginals and the group size exactly, while violating each quota by at most \( D-1 \), where \( D \) is the maximum number of traits possessed by an agent. The construction runs in polynomial time and outputs at most as many selected groups as there are agents. We further obtain trait-specific bounds when a small set of agents has high trait overlap and a concentration-based guarantee allowing small marginal errors. Applications to representative selection illustrate the trade-off between exact marginal implementation and exact quota compliance. The resulting framework combines explicit lottery construction, overlap-sensitive bounds, and fixed-cardinality selection.

Abstract: Let \( A=(a_{ij}) \) be an \( n\times n \) positive semidefinite Hermitian matrix and let \( A(i| j) \) be the submatrix obtained from \( A \) by deleting row \( i \) and column \( j \) of \( A \), where \( 1\leq i, j\le n \). We prove that, for all \( i \) and \( j \), \( \begin{split} &a_{ii}\ per A(i| i)-a_{ij}\ perA(i|j)-a_{ji}\ per A(j| i) +a_{jj}\ per A(j| j)\leq 2\ per A. \end{split} \) This inequality was first posed by Pate [10] and later stated as Conjecture 11 in Wanless’s survey article [11] on permanents.

Abstract: This paper develops a class of anisotropic spaces of eight-dimensional numbers with real components. Starting from a general bilinear multiplication law, we impose commutativity and associativity and derive the resulting relations among the structure constants. Each eight-dimensional number is represented by an $8\times 8$ matrix, and this representation preserves addition and multiplication. Analysis of the determinant of the representing matrix identifies 112 spaces for which it is a nonnegative homogeneous form and therefore defines a modulus. The zero divisors are characterized explicitly and shown to form subspaces of dimensions two, four, and six, with prescribed relations among their components. We introduce the spectrum of an eight-dimensional number through the eigenvalues of its matrix representation and establish the principal algebraic properties of the spectral mapping, including its compatibility with addition, multiplication, conjugation, and inversion. A spectral norm is then defined and shown to satisfy the norm axioms. We also identify a distinguished subspace that is closed under multiplication and on which the spectral and Euclidean norms coincide. For unit elements of this subspace, the corresponding matrix representations are orthogonal. The resulting number systems retain associative and commutative multiplication while allowing zero divisors and provide a natural extension of real, complex, and four-dimensional number systems.