Multiplicity Theory Research Articles

Multiplicity theory and its implications in education

Multiplicity theory and its implications in education

Architecture of intelligent SIEM for detecting cyber incidents in databases of military information and communication systems.

The article looks at the current supply of cyber defense to the databases of information and communication systems. The effectiveness of continuous visits to databases is analyzed. The analysis indicates that existing systems are not immune to security, which is a critical aspect in the field of cybersecurity. A new approach to the architecture of the SIEM system is being introduced, which is aimed at different parts of the protection circuit of the information and communication system. The fragmented architecture makes it possible to effectively detect and respond to cyber attacks at all levels of protection, from the operating system to databases. A new approach is being introduced to ensure cyber security of databases with the benefit of the intelligent capabilities of the SIEM system. The main aspect of this architecture is rich database protection, which allows you to effectively detect and respond to cyber attacks. The registration approach includes the addition of data from different levels to the security circuit of the information and communication system, the module for analyzing data about categories in the database, which operates on the basis of The use of fuzzy multiplicity theory methods, fuzzy logic inference and rule correlation module to improve the detection of cyber incidents. And also the integration of OLAP technologies to provide a deep and analytical view of the database security system. The architecture for identifying cyber incidents has been designed to improve the efficiency of identifying cyber incidents related to the functioning of the database of information and communication systems and. The result of the investigation is a reduction in the capabilities of the SIEM system in identifying and responding to cyber identities in the database of the military information system. Further, we directly monitor the operational model of the cybersecurity system of the information and communication system database.

An inequality in mixed multiplicities

The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of mR-primary ideals in a Noetherian local ring R, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a Minkowski-type inequality given in their paper. We also recover a result of Cutkosky, Srinivasan and Verma as a simple consequence of our inequality.

Theories of multiplicity: philosophical and theological conceptions of material and metaphysical entanglement

ABSTRACT Philosophical and theological treatments of difference and relation are often limited to traditional discursive boundaries of substance metaphysics and transcendent causality. Eschewing the historic desire to categorize substance and doctrinal investments in cosmological mechanism, multiplicity theory experiments with immanent and relational ontologies that are materially attentive and immersed in difference. Tracing the emergence of multiplicity and its theorizing across philosophical and theological registers, I begin with Henri Bergson and Alfred North Whitehead’s conceptually significant turn of the century work. Noting both Bergson and Whitehead’s profound influence on Gilles Deleuze, I examine the appearance and development of multiplicity in Deleuze’s individual writings and his collaborative work with Felix Guattari. Seeking resonances in the more recently theological theorizations of multiplicity, I find that Whitehead and Deleuze surface as vital interlocutors for the relational and incarnational theologies put forth, respectively, by Catherine Keller and Laurel Schneider. Each concerned with theological formulations of embodied difference and divine relation, I suggest that both Keller and Schneider draw on theories of multiplicity to explore immanence and interdependence as ontological postures. I conclude that across philosophical and theological discourse diverse conceptions of multiplicity offer resources for alternative theorizations of complex difference and relation.

Multiplicities and mixed multiplicities of arbitrary filtrations

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of m-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of r filtrations on an analytically unramified local ring R come from the coefficients of a suitable homogeneous polynomial in r variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of m-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.

Transport Calculation of the Multiplicity Moments for Cylinders

In a previous paper by Pázsit and Pál [“Multiplicity Theory Beyond the Point Model,” Ann. Nucl. Energy, Vol. 154 (2021)], a general transport theory calculation of the factorial moments of the number of neutrons emitted spontaneously from a sample was elaborated. In contrast to the original derivations by Hage and Cifarelli [“On the Factorial Moments of the Neutron Multiplicity Distribution of Fission Cascades,” Nucl. Instrum. Meth. Phys. Res. A, Vol. 236 (1985)] and Böhnel [“The Effect of Multiplication on the Quantitative Determination of Spontaneously Fissioning Isotopes by Neutron Correlation Analysis,” Nucl. Sci. Eng., Vol. 90 (1985)], also referred to as the point model, in the transport model the spatial and angular dependence of the internal fission chain is taken into account with a one-speed transport theory treatment. Quantitative results were given for a spherical item, and the bias of the point model regarding the estimation of the fission rate as compared to the more exact space-dependent model was estimated as a function of the size of the sphere and the factor. In the present paper the formalism and the quantitative work are extended to the treatment of items with cylindrical shapes, which are more relevant in many practical applications. Results are presented for both square cylinders () and for tall () and flat () cylinders. This way the differences between the cylinder and the sphere on one hand and those between the various cylinder shapes on the other hand can be estimated. The results show that the bias depends on the geometry of the cylinder quite moderately, but similarly to the case of the sphere, the bias of the point model is quite significant for larger item sizes and values, and it is nonconservative (underestimates the fissile mass) as well.

Multiplicity theory beyond the point model

Passive methods of nuclear safeguards determine the important parameters of an unknown sample from the statistics of the detection of the neutrons emitted from the item. These latter are due to spontaneous fissions and (α,n) reactions, enhanced by internal multiplication before leaking out. Based on the original work of Böhnel, the methodology of traditional multiplicity counting is based on the first three factorial moments of the number of neutrons, emitted from the sample due to one source event. These “Böhnel moments” were derived in the so-called “point model”, in which no space-dependence is accounted for, rather a uniform first collision probability is assumed for each neutron, irrespective of the position of its birth and its velocity direction, and, more important, it is assumed to be the same for all generations in the fission chain as for the source neutrons. The purpose of the present work is to derive the same factorial moments in a one-speed space-dependent model, in which the position and direction of the neutrons is accounted for, but (similarly to the original Böhnel model), no energy dependence is assumed. The integral equations for the moments are solved numerically with a collision number expansion. It is shown that compared to the space-dependent calculations, the unfolding method using the point model underestimates the fissile mass and the underestimation increases with increasing both of fissile mass and the value of α.

Further generalization of symmetric multiplicity theory to the geometric case over a field

Abstract Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a field. A characterization of generalized stars in the case of combinatorially symmetric matrices is given.

Change in vertex status after removal of another vertex in the general setting

In the theory of multiplicities for eigenvalues of symmetric matrices whose graph is a tree, it proved very useful to understand the change in status (Parter, neutral, or downer) of one vertex upon removal of another vertex of given status (both in case the two vertices are adjacent or non-adjacent). As the subject has evolved toward the study of more general matrices, over more general fields, with more general graphs, it is appropriate to resolve the same type of question in the more general settings. “Multiplicity” now means geometric multiplicity. Here, we give a complete resolution in three more general settings and compare these with the classical case (216 “Yes” or “No” results). As a consequence, several unexpected insights are recorded.

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

In this note, we develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's “polygon rule”.

리좀의 개념을 통해 본 릭 오웬스의 디자인 특성

The purpose of this study is to analyze Rick Owens’ design characteristics using the concept of rhizome and consider rhizome as a new paradigm for design analysis. To do so, the concept and principle of rhizome as characterized by Deleuze and Guattari was reviewed and linked to the philosophy of Deleuze to produce based on which Rick Owens’ design was analyzed. Rhizome is a term used philosophically by Deleuze to refer to underground stem plants and includes the principles of connection and heterogeneity, multiplicity, asignifying rupture, and cartography and decalcomanie. Among Deleuze’s philosophical concepts, the concept that can be considered most related to rhizome refers to deterritorialization and reterritorialization, multiplicity theory, and devenir theory. The study examined from 2006 F/W to 2019 S/S men’s and women’s ready-to-wear fashion collections as well as the furniture and interior works Rick Owens designed himself. After analysis, the design characteristics from the concept of rhizome were determined as follows. The characteristics of blurring boundaries and decoded design were present in the creative activities regardless of the area of creation. Further, fluidity and openness were revealed by the rhizome- like acceptance of inspiration and the mixture and of heterogeneous elements. The features of transformation and expansion were shown by the expression of creative “becoming” and experimental fashion shows.

Positivity of mixed multiplicities of filtrations

The theory of mixed multiplicities of filtrations by m-primary ideals in a ring is introduced in [Cutkosky, P. Sarkar and H. Srinivasan, Trans. Amer. Math. Soc. 372 (2019) 6183–6211]. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When R is analytically irreducible, and I ( 1 ) , … , I ( r ) are filtrations of R by m R -primary ideals, we show that all of the mixed multiplicities e R ( I ( 1 ) [ d 1 ] , … , I ( r ) [ d r ] ; R ) are positive if and only if the ordinary multiplicities e R ( I ( i ) ; R ) for 1 ⩽ i ⩽ r are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module.

MICHEL SERRES’S LEIBNIZIAN STRUCTURALISM

In this article I examine Michel Serres’s seminal study of Leibniz: Le Système de Leibniz et ses modèles mathématiques (1968), a book which, in spite of its significance, has never been discussed at length. I analyse Serres’s structural reconstruction of Leibniz’s philosophy by focusing on the way in which the notion of structure operates in his reading, simultaneously at the level of method and at a metaphysical level (through a general theory of multiplicities). I then argue that Serres foregrounded expression and analogy as the keystones of Leibniz’s systematicity and formalized Leibniz’s perspectivism as a general heuristics into his oeuvre. I finally contrast Serres’s conception with Jean-Claude Milner’s, and contend that Serres’s contribution to 1960s French structuralism does not lie in his emphasis on mathematics so much as in his reformulation of the entire structuralist project through epistemological pluralism and translation.

Mixed multiplicities of divisorial filtrations

Suppose that R is an excellent local domain with maximal ideal mR. The theory of multiplicities and mixed multiplicities of mR-primary ideals extends to (possibly non Noetherian) filtrations of R by mR-primary ideals, and many of the classical theorems for mR-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for mR-primary ideals do not hold for filtrations.In this article, we consider multiplicities and mixed multiplicities of R by mR-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of mR-primary ideals, that are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of mR-primary ideals I⊂I′ with the same multiplicity then I and I′ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. The classical Minkowski inequalities for mR-primary ideals I1 and I2 hold quite generally for filtrations. If R has dimension two and there is equality in the Minkowski inequalities, then Teissier and Rees and Sharp have shown that there are powers I1a and I2b that have the same integral closure. This theorem does not hold for arbitrary filtrations. The Teissier-Rees-Sharp theorem has been extended by Katz to mR-primary ideals in arbitrary dimension. We show that the Teissier-Rees-Sharp theorem does hold for divisorial filtrations in an excellent local domain of dimension two.We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme X birationally dominating R, when R is an algebraic local domain (essentially of finite type over a field).

A narrative approach to coaching multiple selves

According to many philosophers and psychologists, we are each not one, but many. The origins of multiplicity theories are briefly outlined before several contemporary theories are described in more detail. In this conceptual paper, it is argued that a multiplicity perspective of self, challenges us to think differently about coaching. Six specific implications are considered, and a pragmatic link between multiplicity theory and post-modernist principles explored. It is argued that a narrative approach to coaching may be particularly useful for the coach interested in experimenting with multiplicity theory, and some initial approaches to applying multiplicity theory in practice are suggested.

Diagonalizable matrices whose graph is a tree: the minimum number of distinct eigenvalues and the feasibility of eigenvalue assignments

Abstract Considered are combinatorially symmetric matrices, whose graph is a given tree, in view of the fact recent analysis shows that the geometric multiplicity theory for the eigenvalues of such matrices closely parallels that for real symmetric (and complex Hermitian) matrices. In contrast to the real symmetric case, it is shown that (a) the smallest example (13 vertices) of a tree and multiplicity list (3, 3, 3, 1, 1, 1, 1) meeting standard necessary conditions that has no real symmetric realizations does have a diagonalizable realization and for arbitrary prescribed (real and multiple) eigenvalues, and (b) that all trees with diameter < 8 are geometrically di-minimal (i.e., have diagonalizable realizations with as few of distinct eigenvalues as the diameter). This re-raises natural questions about multiplicity lists that proved subtly false in the real symmetric case. What is their status in the geometric multiplicity list case?

The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree

It is known that an n-by-n Hermitian matrix, n≥2, whose graph is a tree necessarily has at least two eigenvalues (the largest and smallest, in particular) with multiplicity 1. Recently, much of the multiplicity theory, for eigenvalues of Hermitian matrices whose graph is a tree, has been generalized to geometric multiplicities of eigenvalues of matrices over a general field (whose graph is a tree). However, the two 1's fact does not generalize. Here, we give circumstances under which there are two 1's and give several examples (without two 1's) that limit our positive results.

Direct integrals of matrices

It is shown that each linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitary inequivalent irreducible matrices. This leads to a simplification of the so-called prime (or central) decomposition and the multiplicity theory for such operators. The concept of so-called p-isomorphisms between special classes of such operators is discussed. All results are formulated in more general settings; that is, for tuples of closed densely defined operators affiliated with finite type I von Neumann algebras.

Koszul factorization and the Cohen–Gabber theorem

We present a sharpened version of the Cohen–Gabber theorem for equicharacteristic, complete local domains $(A,\mathfrak{m},k)$ with algebraically closed residue field and dimension $d>0$. Namely, we show that for any prime number $p$, $\operatorname{Spec}A$ admits a dominant, finite map to $\operatorname{Spec}k[[X_{1},\ldots,X_{d}]]$ with generic degree relatively prime to $p$. Our result follows from Gabber’s original theorem, elementary Hilbert–Samuel multiplicity theory, and a “factorization” of the map induced on the Grothendieck group $\mathbf{G}_{0}(A)$ by the Koszul complex.

On polyharmonic regularizations of [formula omitted]-Hessian equations: Variational methods

This work is devoted to the study of the boundary value problem (−1)αΔαu=(−1)kSk[u]+λf,x∈Ω⊂RN,u=∂nu=∂n2u=⋯=∂nα−1u=0,x∈∂Ω, where the k-Hessian Sk[u] is the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum f obeys suitable summability properties. We prove the existence of at least two solutions, of which at least one is isolated, strictly by means of variational methods. We look for the optimal values of α∈N that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results.