Mathematical Methods of Physics

Recent findings on biphoton entanglement in myelinated axons suggest quantum optical coherence in biological systems. This paper extends the Helix-Light-Vortex (HLV) Theory to interpret these results, integrating a quasicrystalline spacetime framework, a triadic spiral-time operator, and a QFT-compatible Lagrangian. Biphoton coherence is modeled as a transient perturbation in a discrete informational field, modulated by spiral time components (U1, U2, U3) and embedded in a 6D cut-and-project lattice. We derive an interaction Lagrangian with spiral-time corrections, propose informational gravitational backreaction, and introduce a symbolic grammar for state transitions. Experimental predictions include gravitational anomalies in neuronal systems, CMB low-ℓ modulations, and axion-like signatures. Technological implications, such as bio-quantum interfaces, are explored, unifying quantum neural phenomena with gravitational and consciousness dynamics.

Finite part integrals are core to the theory of regularizations, vital for modern physics by assigning finite values to divergent expressions, but they tend to require polynomial divergence or a nice analytic continuation. This paper has the purpose to derive a way to evaluate fp integrals of any growth using ideas from Fourier analysis \& distribution theory applied to asymptotic behavior.

In this paper we first present the known disagreement between the law of a second grade fluid and the statement concerning the asymptotic stability of the rest state. After that we put into evidence that for a third grade fluid (which is not considered in the framework described by Coleman's and Noll Approximation Theorem (ATh» the rest state is asymptotically stable, for any value of the constitutive modulus (tIl.

Here then we might say that as -j-is the fundamental ope- ration of the Differential Calculus, so -r-is the fundamental A# operation of the Calculus of Finite Differences. But there is a difference between the two cases which du ought to be noted. In the Differential Calculus -7-is not a true fraction, nor have du and dx any distinct meaning as symbols of quantity. The fractional form is adopted to express the limit to which a true fraction approaches. Hence -7-, and not d, there represents a real operation. But in the Aw Calculus of Finite Differences -r --is a true fraction. Its nu- Ax merator Au x stands for an actual magnitude. Hence A might itself be taken as the fundamental operation of this Calculus, always supposing the actual value of Ax to be given ; and the Calculus of Finite Differences might, in its symbolical character, be defined either as the science of the laws of the operation A, the value of Ax being supposed given, or as the science of the laws of the operation --. In consequence of the funda- LXX mental difference above noted between the Differential Calculus and the Calculus of Finite Differences, the term Finite ceases to be necessary as a mark of distinction. The former is a calculus of limits, not of differences. Though Ax admits of any constant value, the value usually given to it is unity. There are two reasons for this. First, the Calculus of Finite Differences has for its chief subject of application the terms of series. Now the law of a ART. 2.] OF FINITE DIFFERENCES. 3 series, however expressed, has for its ultimate object the deter- mination of the values of the successive terms as dependent upon their numerical order and position. Explicitly or im- plicitly, each term is a function of the integer which ex- presses its position in the series. And thus, to revert to language familiar in the Differential Calculus, the inde- pendent variable admits only of integral values whose com- mon difference is unity. In the series of terms I 2 9 2 q 2 A 2 the general or x th term is a? 2 . It is an explicit function of x. but the values of x are the series of natural numbers, and A#=l. Secondly. When the general term of a series is a function of an independent variable t whose successive differences are constant but not equal to unity, it is always possible to replace that independent variable by another, x, whose common difference shall be unity. Let <j> (t) be the general term of the series, and let At = h; then assuming t = hx we have At = hAx, whence Ax = 1. Thus it suffices to establish the rules of the calculus on the assumption that the finite difference of the independent variable is unity. At the same time it will be noted that this assumption reduces to equivalence the symbolsand A. We shall therefore in the following chapters develope the theory of the operation denoted by A and defined by the equation Au x = a x+1 -u x . But we shall where convenience suggests consider the more general operation

Fractals, Monotiles, and the Geometry of Aperiodic Space.
What if a single tile could fill all of three-dimensional space without ever repeating? In my recent work, I construct a countably infinite family of such tiles—each uniquely shaped, non-convex, and decorated with fractal antennas. These tiles obey matching rules that enforce aperiodicity—not through brute force, but through the elegance of mathematical structure.
Using tools from fractal geometry, symbolic dynamics, and operator algebras, I rigorously prove that these tilings are genuinely aperiodic. The spectral fingerprint of each tiling reveals a singular continuous structure, reminiscent of the strange long-range order found in quasicrystals.
This research opens new avenues for understanding spatial order, with implications for physics, materials science, and the deep geometry of space itself.
This pioneering work on aperiodic monotiles transcends pure mathematics, unveiling a hidden layer of structural possibility in three-dimensional space. The universe is neither a periodic crystal nor a chaotic foam—it is a hierarchical mosaic of infinite aperiodic variety. Suddenly, the universe just became more intriguing.

The Kittim Projection presents a geometric theory of spacetime and matter grounded in harmonic projection structures derived from the golden ratio (φ). Evolving from earlier work published as “Quasicrystalline Black Holes” (May 2025), this paper redefines dimensional layering and time itself as dual projection modes: T⁺ (forward coherence) and T⁻ (structural memory). The model identifies a 4D coherence field and a 2D influence boundary as core features of how form, resonance, and measurement emerge. A companion paper on fermion mass prediction using φ + 1 harmonic scaling has been published separately. Together, these works aim to advance a unified understanding of mass, time, and structure from first principles of projection geometry.

As “Stern-Gerlach first” becomes increasingly popular in the undergraduate quantum mechanics curriculum, we show how one can extend the treatment found in conventional textbooks to cover some exciting new quantum phenomena. Namely, we illustrate how one can describe a delayed choice variant of the quantum eraser which is realized within the Stern-Gerlach framework. Covering this material allows the instructor to reinforce notions of changes in basis functions, quantum superpositions, quantum measurements, and the complementarity principle as expressed in whether we know “which-way” information or not. It also allows the instructor to dispel common misconceptions of when a measurement occurs and when a system is in a superposition of states.

Within the framework of self-adjoint operator of time in non-relativistic quantum mechanics some properties of solutions of Schroedinger equation, related to Hilbert space formalism, are investigated for two types of time dependent Hamiltonian.

Algebraic digital signature algorithms with a commutative hidden group, which are based on the computational difficulty of solving large systems of power equations, are promising candidates for post-quantum cryptoschemes, especially in securing applications like the internet of things (IoT) and other information technologies. Associative finite non-commutative algebras are used as an algebraic support of the said algorithms. Among such algebras, finite quaternion-type algebras have been identified as strong candidates for providing algebraic support. This paper investigates the decomposition of these algebras into commutative subrings and explores their multiplicative groups, which can serve as potential hidden groups. The analysis reveals the existence of three distinct types of subrings, with derived formulas for the number of subrings and the orders of their multiplicative groups. These findings align with previous studies on fourdimensional algebras defined by sparse basis vector multiplication tables. Using the finite quaternion-type algebras as algebraic support, a novel post-quantum signature algorithm characterized in using two mutually non-commutative hidden groups has been developed.

We propose a reformulation of the Bekenstein bound in which entropy is not fundamentally constrained by geometric surface area, but by the symmetry class of the boundary-encoded in its coset structure Σ ∼ = G/H. Using Haar measure on G/H, we show that both the volume and maximum entropy of a radially symmetric region arise from the same invariant object, independent of embedding geometry. This leads to a testable divergence: when symmetry and area disagree (e.g., in non-smooth or irregular boundaries), the entropy bound scales with symmetry, not surface extent. This reformulation suggests that information constraints in physics are fundamentally governed by symmetry, not geometry-offering a new group-theoretic foundation for gravitational thermodynamics.

We study central simple algebras in various ways, focusing on the role of $p$-central subspaces. The first part of my thesis is dedicated to the study of Clifford algebras. The standard Clifford algebra of a given form is the generic associative algebra containing a $p$-central subspace whose exponentiation form is equal to the given form. There is an old question as for whether these algebras have representations of finite rank over the center, and jointly with Daniel Krashen and Max Lieblich we managed to provide a positive answer. Different generalizations of the structure of the Clifford algebra are presented and studied in that part too. The second part is dedicated to the study of $p$-central subspaces of given central simple algebras, mainly tensor products of cyclic algebras of degree $p$. Among the results, we prove that $5$ is the upper bound for the dimension of 4-central subspaces of cyclic algebras of degree 4 containing pairs of standard generators. The third part is d...

We study the subfields of quaternion algebras that are quadratic extensions of their center in characteristic $2$. We provide examples of the following: Two nonisomorphic quaternion algebras that share all their quadratic subfields, two quaternion algebras that share all the inseparable but not all the separable quadratic subfields, and two algebras that share all the separable but not all the inseparable quadratic subfields. We also discuss quaternion algebras over global fields and fields of Laurent series over a perfect field of characteristic $2$ and show that the quaternion algebras over these fields are determined by their separable quadratic subfields.

In this paper, we prove that for a given biquaternion algebra over a field of characteristic two, one can move from one symbol presentation to another by at most three steps, such that in each step at least one entry remains unchanged. If one requires that in each step two entries remain the same then their number increases to fifteen. We provide even more basic steps that in order to move from one symbol presentation to another one needs to use up to forty-five of them.

Starting from a static spherically symmetric solution of the Einstein's field equations in the second approximation in the perfect fluid scheme, in the exterior of the source, the corresponding metrics depend on a dimensionless parameter α . Taking the Sun for the source, the influence of α on the classical Solar System tests of the general relativity is estimated, the only α-dependent one being the Shapiro radar-echo delay experiment. Performing such a test in given conditions within a precision better than 10 -2 , it is possible to obtain an experimental value for α . If α = 1, the Einstein's equations in the weak field approximation take the D'Alembert form, which attests the existence of gravitational waves 1 .

Despite the mathematical elegance and experimental success of quantum mechanics, its foundational understanding remains incomplete. In particular, the probabilistic interpretation governed by Born's rule and the mystery of quantum superposition continue to spark debate. This note introduces a new perspective, the Hidden Deterministic Interpretation (HDI), which identifies the often-disregarded global phase of the Schrödinger wavefunction as a hidden variable, fixed by a second physical condition beyond normalization. This approach not only resolves the mystery of quantum superposition, but also offers deterministic dynamics for single particles, avoids the measurement problem, and naturally explains the classical-to-quantum transition. A statistical derivation of Born's rule emerges from this framework. This brief note outlines the central insights and invites interested readers to explore further details in the associated full-length manuscript.

Let us recall a well-known school task: In the (Euclidean E 2 ) plane of a triangle ABC we draw regular triangles outward on sides of ABC, say ABC, BCA, CAB, respectively. Prove that the segments AA, BB, CC intersect each other in a point K, that is the isogonal point of ABC (also called Fermat point or Fermat-Torricelli point) and the distance sum AK + BK + CK is minimal for K among all points of the plane. Professor Kárteszi noticed that instead of regular triangles we can draw isosceles ones with all equal base angles, and the above K (called Kárteszi point) exists also in the Bolyai-Lobachevsky hyperbolic plane H 2 (in the sphere S 2 as well, (see also Kálmán, 1989 and Sect. 2), the orthocentre, barycentre are specific cases. There is a more general extremum problem of (Yaglom, 1968, problem 83, with modified notation): In the plane (E 2 ) of a given triangle ABC find a point K such that the quantity αKA + βKB + γ KC, where α, β, γ are given positive numbers, has the smallest possible value. This problem leads to a more general triangle configuration and to an analogous extremal point K. Moreover, as a new result of this paper, an extension onto "absolute plane" (S 2 , E 2 , H 2 , M 2 Minkowski plane, G 2 Galilei (or isotropic) plane) can be formulated and solved by three reflections theorem (see e.g. Molnár, 1978 and Sect. 4, Weiss, 2018), and geometric (Grassmann-Clifford type) algebra (Perwass & Hildenbrand, 2004 and Sect. 3). Open problems arise as well. By this we want to follow F. Kárteszi's didactical credo (see also his wonderful book (Kárteszi, 1976) of great international success): Start with a natural, elementary, visually well understandable task! Then follow the manipulations, tools, new mathematical concepts, the technical machinery; then the solution, occasional theory, further applications, extensions. . .

The radical of the Brauer algebra B (x) f is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f ). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of f . The proof is by direct approach for x = 0 , and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable B (x) f -modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley-Lieb algebra -realised inside B (1) f -acting on it. "Ahi quanto a dir che sia è cosa dura lo radical dell'algebra di Brauer pur se'l pensier già muove a congettura" N. Barbecue, "Scholia"

We consider the consequences of describing the metric properties of space-time through a quartic line element ds 4 = G µνλρ dx µ dx ν dx λ dx ρ . The associated "metric" is a fourth-rank tensor G µνλρ . We construct a theory for the gravitational field based on the fourth-rank metric G µνλρ which is conformally invariant in four dimensions. In the absence of matter the fourth-rank metric becomes of the form G µνλρ = g (µν g λρ) therefore we recover a Riemannian behaviour of the geometry; furthermore, the theory coincides with General Relativity. In the presence of matter we can keep Riemannianicity, but now gravitation couples in a different way to matter as compared to General Relativity. We develop a simple cosmological model based on a FRW metric with matter described by a perfect fluid. Our field equations predict that the entropy is an increasing function of time. For k obs = 0 the field equations predict Ω ≈ 4y, where y = p ρ ; for Ω small = 0.01 we obtain y pred = 2.5 × 10 -3 . y can be estimated from the mean random velocity of typical galaxies to be y random = 1 × 10 -5 . For the early universe there is no violation of causality for t > t class ≈ 10 19 t P lanck ≈ 10 -24 s. Short title: Conformal fourth-rank gravity. Classification Number: 0450 Unified field theories and other theories of gravitation. "The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth-root of a quartic differential expression." As pointed out by Riemann, the next possibility is r = 4. In this case the line element is given by (1.5) Riemann went no farther in exploring the above geometry, and gave no justification for that omission. Of course, at first sight, a space with a line element of the form (1.5) may seem bizarre. However, such geometry cannot be excluded a priori and its exclusion must be done in a mathematically educated way. This was partially done by Helmholtz. 2 He showed that the existence of rigid bodies, which do not change their shapes and therefore the metric relations under translations and rotations, leaves us with Riemannian geometry as the only possibility. The Helmholtz result seemed quite satisfactory and therefore no more concern for higher-rank geometries appeared. It seems that the arrival of General Relativity, with its underlying Riemannian geometry, caused this important problem to be forgotten. However, the problem merits further attention, not only from a mathematical point of view, but also for the applications it found in theoretical and mathematical physics. It is here that the introductory considerations come into play. In fact, the difficulty of conceiving geometries other than the Riemannian limited their developments. We are therefore going to develop that chapter of differential geometry in which Riemann and Helmholtz stopped their scientific enquiries. To close these historical comments. An indirect verification of the Riemannian structure of the universe at our daily life scales was performed by Gauss 3 in 1826. The experiment was intended to verify departures from flatness, but as a side result he also verified no departures from Riemannianicity. At the scale of distances of our daily life fourth-rank geometry is not realised in nature and the only place where it can play some role is in high-energy, or short distances, physics. In fact, at high energies, a regime to which we do not have direct experimental access, the very concept of rigid body may be no longer valid and the Helmholtz argumentation no longer applicable. The natural question now is: why we would like to work with fourth-rank geometry, and no other of the infinitely many possible generalisations of Riemannian geometry, to describe the physics at high energies. The answer is provided by experiments, such as deep inelastic scattering, which show that, at very high energies, physical processes are scale, or conformally, invariant. Therefore, high-energy physics is associated to a geometry exhibiting, in a model independent way, conformal invariance in 4 dimensions. In another work 4 we show that the critical dimension, for which field theories are integrable, is equal to the rank of the metric. Therefore, if we want to construct an integrable field theory in 4 dimensions showing agreement with the observed conformal invariance at high energies we must take recourse to fourth-rank geometry. This result also explains why, if one relies only on Riemannian geometry, integrable conformal models can be constructed only in 2 dimensions (strings). We arrive therefore to the following scheme: at short distances, high-energies, the geometry is of fourth-rank while at large distances, low energies, the geometry is of secondrank, Riemannian. It is clear furthermore that the Riemannian behaviour of the geometry must be recovered as the low-energy limit of the high-energy theory. This would be possible if at low-energies the fourth-rank metric tensor G µνλρ becomes of the form

For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differential equations, the equation for the density of so-called "resting cells" was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view.

We give a detailed account of the Hamiltonian GNH analysis of the parametrized unimodular extension of the Holst action. The purpose of the paper is to derive, through the clear geometric picture furnished by the GNH method, a simple Hamiltonian formulation for this model and explain why it is difficult to arrive at it in other approaches. We will also show how to take advantage of the field equations to anticipate the simple form of the constraints that we find in the paper.

We give a detailed account of the Hamiltonian GNH analysis of the parametrized unimodular extension of the Holst action. The purpose of the paper is to derive, through the clear geometric picture furnished by the GNH method, a simple Hamiltonian formulation for this model and explain why it is difficult to arrive at it in other approaches. We will also show how to take advantage of the field equations to anticipate the simple form of the constraints that we find in the paper.

In the episode of “How To Prove It” that appeared in the March 2018 issue of At Right Angles, we studied a number of characterisations of a parallelogram; they were also listed in the article on parallelograms elsewhere in that issue. Three assertions had been made in the article without proof. We provide the proofs in this article.

The work is devoted to recent investigations of the Lax-Sato compatible linear vector field equations, especially to the the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Plebański and later analyzed in a series of articles. The AKS-algebraic and related R-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equation being considered. It is shown that all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of the very interesting Lagranged'Alembert type mechanical interpretation. of the devised integrability scheme with the Lax-Sato equations is also discussed.

Definizione 2.1. Siano (X, || • || X) e (Y, || • || Y) spazi di Banach e T : X → Y un operatore lineare. L'operatore T si dice compatto se T (B X)è un sottoinsieme relativamente compatto di Y. Esempio 2.2. Si consideri un compatto Ω ⊆ R N e una funzione K : Ω×Ω → R continua. Sia T : L p (Ω) → L p (Ω) l'operatore integrale così definito: T f (x) = Ω K(x, y)f (y)dy, f ∈ L p (Ω), x ∈ Ω. Dal teorema della convergenza dominata segue facilmente che T (L p (Ω)) ⊆ C(Ω). Proviamo che Tè compatto. Osserviamo che per ogni f ∈ L p (Ω) con f p ≤ 1, applicando la disuguaglianza di Hölder, si ottiene T f ∞ ≤ ||K|| p ′ f p ≤ ||K|| ∞ |Ω| 1/p ′. Dunque l'insieme T (B L p (Ω))è equilimitato in C(Ω). Inoltre, per l'uniforme continuità di K, per ogni ε > 0 esiste δ > 0 tale che per ogni x 1 , x 2 , y ∈ Ω: ||x 1 − x 2 || < δ ⇒ |K(x 1 , y) − K(x 2 , y)| < ε|Ω| −1/p ′. Pertanto per ogni f ∈ B L p (Ω) , se ||x 1 − x 2 || < δ, applicando nuovamente la disuguaglianza di Hölder si deduce: |T f (x 1) − T f (x 2)| ≤ Ω |K(x 1 , y) − K(x 2 , y)||f (y)|dy ≤ ε. Quindi T (B L p (Ω))è anche equiuniformemente continuo e dunque, per il teorema di Ascoli-Arzelà,è relativamente compatto in C(Ω). Per l'immersione continua di C(Ω) in L p (Ω), otteniamo che T (B L p (Ω))è relativamente compatto in L p (Ω). Poiché i sottoinsiemi relativamente compatti di uno spazio di Banach sono anche limitati, ogni operatore compatto Tè anche continuo (cf. Propositione 1.1). Inoltre, vale la seguente caratterizzazione.

In this paper, we present two new results to the classical Floquet theory, which provides the Floquet multipliers for two classes of the planar periodic system. One these results provides the Floquet multipliers independently of the solution of system. To demonstrate the application of these analytical results, we consider a cholera epidemic model with phage dynamics and seasonality incorporated.

Quantum Hall systems are a suitable theme for a case study in the general area of nanotechnology. In particular, it is a good framework in which to search for universal principles relevant to nanosystem modeling, and nanosystem-specific signal processing. Recently, we have been able to construct a PDE model of a quantum Hall system, which consists of the Schrödinger equation supplemented with a special type nonlinear feedback loop. This result stems from a novel theoretical approach, which in particular brings to the fore the notion of quantum information. In this article we undertake to modify the original model by substituting the dynamics based on the Dirac operator. This leads to a model that consists of a system of three nonlinearly coupled first order elliptic equations in the plane.

In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) = k(x . y) satisfy Mercer's condition and thus may be used in Support Vector Machines (SVM), Regularization Networks (RN) or Gaussian Processes (GP). In particular, we show that if the kernel is analytic (i.e. can be expanded in a Taylor series), all expansion coefficients have to be nonnegative. We give an explicit functional form for the feature map by calculating its eigenfunctions and eigenvalues.

The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The great adaptibility of this string model with respect to various regularization methods is pointed out. We survey several regularization methods: the cutoff method, the complex contour integration method, and the zeta-function method. The most powerful method in the present case is the contour integration method. The Casimir energy turns out to be negative, and more so the larger is the number of pieces in the string. The thermodynamic free energy F is calculated for a two-piece string in the limit when the tension ratio x = T I /T II approaches zero.

Abstract. We show how to construct the simple exceptional Lie algebra of type G2 by explicitly constructing its 7 dimensional representation. No knowledge of Lie theory is required, and all relevant coefficients can be remembered by use of the simple mnemonic βαββαβ. 1.

The first term describes the energy levels of a two-level system, and the second describes a simple harmonic oscillator. The two quantum systems are coupled linearly, as accounted for by the remaining term. The transition energy of the two-level system is ΔE, and the oscillator energy is �ω0. In the multiphoton regime the oscillator energy is much less than the two-level system energy

Rings are built from abelian groups. While ideals (I), special subrings of Rings (R), with Rings, form Quotient Rings (R/I) isomorphic to existing Rings (¯R). Thus, this process creates new Rings. Here, we use
this platform to explore ideals such as prime, principal and Maximal, based on certain theories to enhance this creativity to our advantage. Examples are considered, examined and explored for applicability of these
theories and creativity of New Rings.

Special orthogonal group of the space of pure quaternions in a quaternion algebra over a quadratic field Unit groups of orders in quaternion algebras over number fields provide important examples of non-commutative arithmetic groups. Let K = Q( √ d ) be a quadratic field with d < 0 a squarefree integer such that d ≡ 1(mod 8), and let R be its ring of integers. In this note we study, through its representation in SO 3 (R), the group of units of several orders in the quaternion algebra over K with basis {1, i, j, k} satisfying the relations i 2 = j 2 = -1, ij =ji = k.

Structured light fields to understand quantum entanglement involve the combination of superluminal and subluminal components that help to probe undiscovered quantum phenomena. Dark state excitations of a localized nature based on light matter interactions can have tachyonic modes and subluminal Higgs bosons that conform to dark energy models. Such Vacuum energy states with highly entangled information can correspond cosmological mind and memories associated with bursts of light radiations as part of universal energy substrates and decay processes.

We give the basic structure of the multivariable Ore extensions S = A[t; σ, δ]
introduced in the work of Martínez-Peñas and Kschischang. The Pseudo multilinear transformations (PMT’s) are introduced and correspond to modules over S. These maps are strongly connected to the evaluation of polynomials in S. A general product formula is obtained. PMT’s help to put some structure on the set of roots of a polynomial f (t) ∈ S.

In quantum formalism, a discrepancy exists between Schrödinger's wavefunction and Born's rule, which is reconciled through a geometric derivation of Born's rule as a limiting case based on the relative frequency of detection. This derivation leverages the relationship between the global phase of the state vector and the specific eigenstate observed during quantum experiments. The naturally available Born's picture within Born's probabilistic interpretation not only demonstrates the equivalence of classical and quantum mechanical times but also elucidates various quantum enigmas. It provides a physical explanation for wavefunction collapse, leading to the statistical emergence of the Copenhagen interpretation, and facilitates the derivation of Malus' law for the transmission of a single photon through a polarizer. The longstanding measurement problem is addressed through the resolution of Schrödinger's cat paradox, subsequently resolving Wigner's friend and the Frauchiger-Renner paradoxes at the single-quantum level. Furthermore, Born's picture offers a clear resolution to the conundrum of quantum superposition, with a proposed experimental verification using a modified Mach-Zehnder interferometer. Significantly, this approach provides a causal explanation for Wheeler's delayed-choice experiment and resolves the fundamental mystery of quantum mechanics inherent in Young's double-slit experiment.

A novel approach based upon vertex operator representation is devised to study the AKNS hierarchy. It is shown that this method reveals the remarkable properties of the AKNS hierarchy in relatively simple, rather natural and particularly effective ways. In addition, the connection of this vertex operator based approach with Lie-algebraic integrability schemes is analyzed and its relationship with τ-function representations is briefly discussed. An approach based on the spectral and Lie-algebraic techniques for constructing vertex operator representation for solutions to a Riemann type hydrodynamical hierarchy is devised. A functional representation generating an infinite hierarchy of dispersive Lax type integrable Hamiltonian flows is obtained.

The complete integrability of a generalized Riemann type hydrodynamic system is studied by means of a novel combination of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are constructed.

The complete integrability of a generalized Riemann type hydrodynamic system is studied by means of a novel combination of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are constructed.

We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration space.

The Pythagorean theorem is the most famous theorem in the world. It was discovered more than two thousand years. Currently there are hundreds of proofs. In this proof the area of a right-angled triangle and the ratio of the areas of similar triangles depending on of the similarity coefficient it will be used. It will also be important to prove that they are area of similar triangles is directly proportional to the square of the similarity coefficient.

In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, the electro-hydro-dynamical model for local electric field, signal processing waves through optical fibers, etc. We determine the useful and further general exact traveling wave solutions of the above mentioned NLDEs by applying the How to cite this paper:

In the m-dimensional affine space AG(m, q) over the finite field F q of odd order q, the analog of the Euclidean distance gives rise to a graph G m,q where vertices are the points of AG(m, q) and two vertices are adjacent if their (formal) squared Euclidean distance is a square in F q (including the zero). In 2009, Kurz and Meyer made the conjecture that if m is even then G m,q is a strongly regular graph. In this paper we prove their conjecture.

We choose such boundary conditions for open IIB superstring theory which preserve N = 1 SUSY. The explicite solution of the boundary conditions yields effective theory which is symmetric under world-sheet parity transformation Ω : σ → −σ. We recognize effective theory as closed type I superstring theory. Its background fields,beside known Ω even fields of the initial IIB theory, contain improvements quadratic in Ω odd ones.

The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for second class constraints are computed. Finally, canonical quantization is performed leading to infinitesimal form of irreducible representations of Poincare group.

In this paper, we present new exact solution sets of nonlinear conformable time-fractional coupled Drinfeld-Sokolov-Wilson equation which arise in shallow water flow models, when special assumptions are used to simplify the shallow water equations by means of Sine-Gordon expansion method. We also present an analyticalapproximate method namely perturbation-iteration algorithm (PIA) for the system. Basic definitions of fractional derivatives are described in the conformable sense. An example is given and the results are compared to exact solutions. The results show that the presented methods are powerful, reliable, simple to use and ready to apply to wide range of fractional partial differential equations.