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Let ℤ n be the ring of integer modulo n with two binary operators, addition (+) and multiplication (.) , where n is a positive integer. The special set is defined as = { a ∈ ℤ n : (∃ b ∈ ℤ n)   b a ≡ a ,   b ≢ a ,   b ≢ 1 }. Our purpose in the present paper is to propose a new family of interconnection networks that are Cayley graphs on this special set and denote it by Ω − (Z n). In this paper, we define a relationship between G and G e ∗ , G e ∗ is a derived graph from G by removing r edges, where r is a known fixed value. We also give the spectrum of absorption Cayley graph, unitary addition Cayley graph, and Ω − (Z n). We also provide values of n for which the graph Ω − (ℤ n) is hyperenergetic and discuss the structural properties of this graph, such as planarity and connectedness. [ABSTRACT FROM AUTHOR]

A graph G is singular if its adjacency matrix is singular. The starting vertices of two paths P b 1 and P b 2 are simultaneously bound to the ending vertex of the path P s 1 , and the ending vertices of the paths P b 1 and P b 2 are bound to the starting vertex of path P s 2 . Meanwhile, the starting vertex of the path P s 1 is bound to a vertex of the cycle C a 1 , and the ending vertex of the path P s 2 is bound to a vertex of the cycle C a 2 . Thus, the resulting graph is written as ξ (a 1 , a 2 , b 1 , b 2 , s 1 , s 2) . This is denoted by ζ (a 1 , a 2 , b 1 , b 2 , s) = ξ (a 1 , a 2 , b 1 , b 2 , 1 , s) and ε (a 1 , a 2 , b 1 , b 2) = ζ (a 1 , a 2 , b 1 , b 2 , 1) , which are referred to as the ξ -graph, ζ -graph and ε -graph for short, respectively. It is known that there are 15 kinds of tricyclic graphs. The purpose of this paper is to study the necessary and sufficient conditions for ξ -graphs, ζ -graphs and ε -graphs to be singular graphs. We analyzed the structure of the elementary spanning subgraphs of the graph G = ξ (a 1 , a 2 , b 1 , b 2 , s 1 , s 2) . By calculating the determinant of the adjacency matrix of the graph G, the necessary and sufficient conditions for the determinant of the graph G to be zero is obtained, and so the necessary and sufficient conditions for graph ξ (a 1 , a 2 , b 1 , b 2 , s 1 , s 2) to be singular are obtained. As the corollaries, the necessary and sufficient conditions for graphs ζ (a 1 , a 2 , b 1 , b 2 , s) and ε (a 1 , a 2 , b 1 , b 2) to be singular are also obtained. [ABSTRACT FROM AUTHOR]

An atypical situation, rarely encountered in the area of offenders who exercise the means of appeal of the contraventional complaint in the field of traffic on public roads, has generated controversies regarding the interpretation of Article 118 of the Government Emergency Ordinance No 195/2002, including of the rules of application contained in the Order No 141/2014, to which there are added other problems of interpretation and suspicions of unconstitutionality of Article 103 (2) of the Government Emergency Ordinance No 195/2002, which regulate extension of the duration of the suspension of exercising the right to drive vehicles on public roads by another 30 days, if, within a period of 6 months from committing a contravention, a driver commits another contravention to the traffic rules. Although it is well known that the overwhelming majority of offenders who are subject to the complementary sanction of suspension of the right to drive motor vehicles, agricultural or forestry tractors or trams, this being the full name of this complementary sanction, file a complaint in order to benefit from the suspension of the execution of this complementary sanction, the judicial practice has revealed that there are also cases in which such offenders, for well-justified reasons, renounce this benefit and begin the execution of the complementary contraventional sanction, even if they have exercised the mentioned means of appeal. As the practice of the police units that manage the IT application for keeping records of withheld licenses and sanctions applied to the drivers of motor vehicles or trams is to register ex officio that they are operating the suspension of execution of the complementary sanction, when they receive the alert that the contraventional complaint has been filed, although Article 118 (3) of the Government Emergency Ordinance No 195/2002 establishes that the offender applicant is the one who must submit/send to the police unit of which the fact-finding agent is a part the document certifying the situation of filing the complaint and only after this step it will be made the corresponding mention of the suspension of the execution of this sanction, the question of violation of the right of access to justice arises. Such a violation derives from the situation that the offender in question faces the situation to waive the right to exercise the means of appeal of the contraventional complaint, in order to „immediately" execute the sanction, due to the uncertainty of the duration of the related judicial proceedings as well. If such offenders will take the risk and will file a complaint, also formulating a request with the aim of obtaining in due time a court decision ordering the entry of the mention that they started the execution simultaneously with the filing of the complaint, as it will result from the court decisions used as judicial practice, they will face the exception of inadmissibility. Finally, the text of Article 103 (2) of the Government Emergency Ordinance No 195/2002, which regulates the extension of the duration of the suspension of the right to drive vehicles on public roads by another 30 days in the case of committing a new contravention, evokes the institution of „contraventional recidivism" and it must be analyzed whether this institution finds its place in the matter of contraventions, under the conditions in which the institution in question has applicability only in the field of traffic on public roads, raising discussions as to whether such regulation is constitutional. Controversial is also the nature of the 15-day term established by the legislator for the communication of the extension provision implied by this text in the context in which the legislator did not regulate the consequences of its non-compliance, as it did in the case of the communication of the contraventional fact-finding report, for the latter establishing the prescription for the execution of contraventional sanctions. Last but not least, it arouses interest to what extent the principle of loyalty of evidence finds its place in the context of traffic contraventions. [ABSTRACT FROM AUTHOR]

A complex unit gain graph, or T -gain graph, is a triple Φ = (G , T , φ) comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers T = { z ∈ C : | z | = 1 } , and a gain function φ : E → → T with the property that φ (e i j) = φ (e j i) − 1. A cactus graph is a connected graph in which any two cycles have at most one vertex in common. In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n (G) − 2 m (G) + 2 c (G) − 1 , where n (G) , m (G) and c (G) are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30]. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for invertibility of symmetric three-by-three block matrices having a double saddle-point structure that guarantee the unique solvability of double saddle-point systems. We consider various scenarios, including the case where all diagonal blocks are allowed to be rank deficient. Under certain conditions related to the nullity of the blocks and intersections of their kernels, an explicit formula for the inverse is derived. [ABSTRACT FROM AUTHOR]

In spite of its countless amendments, a new bill to reform the State Land and Urban Rehabilitation Law is currently being processed by Parliament that aims, among other things, to introduce reforms that academic opinion and the real estate sector have been demanding for several legislatures now. The measures aim to correct the detrimental effects of strictly applying to urban planning instruments the invalidity regime established for administrative regulations; and to better define who can challenge these instruments and when. Their ultimate purpose is to avoid the legal uncertainty that has been hanging over urban plans in recent years, especially when they are challenged for reasons other than those hypothetically intended by the applicable regulations.. [ABSTRACT FROM AUTHOR]

A graph G is called triangle-free if G does not contain any triangle as its induced subgraph. Let G n be the set of triangle-free graphs of order n each of which has three positive eigenvalues. In this paper, we find 20 specific graphs in G n , each of which has nullity no more than 2, and we show that in terms of three graph transformations all the other graphs of G n can be constructed from these 20 specific graphs. Hence, we completely characterize the triangle-free graphs with exactly three positive eigenvalues. [ABSTRACT FROM AUTHOR]

Copyright of University of Sharjah Journal of Law Sciences (JLS) is the property of University of Sharjah - Scientific Publishing Unit and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider the adjacency spectrum of cycle-spliced signed graphs (CSSG), i.e., signed graphs whose blocks are (independent) signed cycles. For a signed graph S, the nullity η(S) is the multiplicity of the 0-eigenvalue. The adjancency spectrum of cycle-spliced (signed) graphs is studied in the literature for the relation between the nullity η and the cyclomatic number c, in particular, it is known that 0 = η(S) = c(S) + 1. In this paper, nonsingular cycle-spliced bipartite signed graphs are characterized. For cycle-spliced signed graphs S having only odd cycles, we show that η(S) is 0 or 1. Finally, we compute the nullity of CSSGs consisting of at most three cycles. [ABSTRACT FROM AUTHOR]

In this paper we develop the theory of cyclic flats of q-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of F q m -independence of an F q -subspace of F q n and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field. [ABSTRACT FROM AUTHOR]

Restituting the dogmatic elaborations of the authors of the ius commune sometimes involves a systematic approach that seems unfaithful to the sources. Using the example of the nullitas of juridical acts, this article considers the identification of theories in the works of learned law. The link between the common conception of nullity as an ontological defect and the technical discourse on the nullity of prohibited acts led to constructions that can be described as theories. They address the question of how norms fit into the legal order, and thus the coherence of the law.

The purpose of this paper is to analyze the legal framework on amendments to Public-Private Partnership Contracts as provided by the Public-Private Partnership Law No. 18,786. To this end, within the scope of unilateral amendments, the consequences of imposing such amendments in an illegitimate manner and the possibility of unilaterally imposing a limitation of the contract's subject matter are examined. Additionally, aspects regarding bilateral amendments to Public-Private Participation Contracts and possible limitations applicable to the subject matter on which they may be based, are addressed. This paper also studies the possible consequences that non-compliance with the formal requirements established in the applicable law may have on the validity of the amendment in question. Finally, the grounds for renegotiation and, especially, the case of illegitimate unilateral amendments as an additional ground not expressly provided for in the applicable law, are analyzed. [ABSTRACT FROM AUTHOR]

Copyright of Jordanian Journal of Law & Political Science is the property of Mutah University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Revista Digital de Derecho Administrativo is the property of Universidad Externado de Colombia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The purpose of this article is to give one of the possible answers to the interpretative question posed in interpretative case No. 2/2023 of the General Assembly of the Civil College (OSGK) of the Supreme Court of Cassation (SCC), namely „is it nullity on the grounds Art. 42, b. „c“ of the Inheritance Act (IA) a will which is made both for the care already taken of the testator and with a view to future care to be taken of him for the rest of his life; can in such a case, when interpreting the will of the testator, it be assumed that the only motive for making the will is contrary to the law, since it violates the principle of gratuity of the testamentary disposition?‘ In order to justify the answer proposed in this article, both the current judicial practice related to the posed question and the achievements of the Bulgarian civilist doctrine were used. Undoubtedly, the answer to this question will have a significant practical significance, as it will predetermine the direction of development of future judicial practice in this matter [ABSTRACT FROM AUTHOR]

In the Iranian legal system, transactions involving foreign currencies have consistently been subject to governmental supervision and legislative scrutiny. This is due to the intrinsic relationship between foreign currencies and the value of the national currency, as well as their impact on matters of export and import. A further factor is the desire of the general public to invest in foreign currencies in Iran. This is motivated by a desire to maintain the value of their money, which is achieved by purchasing and holding foreign currencies, or by engaging in currency speculation and brokering. So many regulations have been passed regarding currency transactions including: circulars and monetary-banking regulations; Acts regarding the terms and conditions of foreign currency transaction; method of currency's price discovering; competent institutions; maximum volume of permitted exchangeable currencies, etc. The last legislative step regarding currency transactions in Iran, is the amendment Act of the "Combating Goods and Currency Smuggling Act" enacted on 1 January 2022. In these regulations, some requirements and formalities are prescribed for concluding and validating foreign currencies transactions. In the meantime, two legal descriptions of currency transactions are subject of this article: "Solemn", and "Real" description. So, we should examine that if these legal formalities make the currency transactions as a solemn contract? And moreover, is the "Delivery", a condition for validity of the currency transactions, and therefore it should be considered as a real contract? The method of this research is library and documentary, and the research approach is descriptive and analytical. Materials, includes Acts and regulations, as well as legal theories. The technical analysis will also refer to the legal principles and rules, including the general principles of contracts, as well as analogy and induction, and logical interpretation. The research plan will commence with an explanation of the concept of foreign currency and currency transactions. This will be followed by a separate analysis of each objective and formal statutory condition for concluding such a transaction, based on legal regulations and theories, as well as principles and rules. Thereafter, the legal description will be discussed, and finally, the research proposal will be presented. The research questions are: what formalities are prescribed in Iranian law, and especially in the latest legal amendments, for conducting currency transactions? Are the formalities as such a way that currency transactions become a solemn contract? Is the delivery a condition for the validity of such transactions and therefore currency transactions should be considered as a real contract? Assuming these two descriptions, what are the effects of them? And finally, whether these two works reasonable, acceptable and enforceable in Iran's current legal system or not? It seems that in the current Iran's legal system, and considering the set of foreign currency regulations and especially the recent amendments in the "Combating Goods and Currency Smuggling Act ", currency transactions should be considered as both solemn and real contract. As in the aforementioned Act, at the first, foreign currency transactions must be done with the intervention of imposed intermediary institutions; i.e. Banks and Exchanges and Financial & Credit Institutions (Paragraph (C) of the Supplementary Article 2 bis), that is considered a form of imposition on the contract parties, which is against the consensual principle. Also, it is necessary to register currency transactions in some governmental "Currency Registration Systems" (Paragraph (F) of Supplementary Article 2 bis), which is also an unusual formality for concluding a contract. Secondly, from the point of view of real contracts, not only currency transactions are a type of "Bay al-Sarf" (the classic contract for exchanging the gold and silver against each other, or exchanging the money for money) which is an obvious example of real contracts in Iran (Article 346 of the Civil Code); the Act of Combating Goods and Currency Smuggling, considers the delivery of currency, as an essential factor in all currency transactions (Paragraph (D) of the Supplementary Article 2 bis). Based on this definition, not only the statutory formalities in currency transactions are as legal condition for their concluding, and therefore in case of non-compliance with these formalities the transaction should be considered "Null"; the delivery is also a condition for validity of the contract. Moreover, the mechanism of currency's ownership proof has changed. So, despite the fact that currency is movable property, in many cases, possession does not prove the holder's ownership; Rather, proving the ownership will require the registration's receipt in the governmental currency systems (Paragraph (G) of the Supplementary Article 2 bis). Also, the ownership of currencies which have been obtained before or after the recent amendments Act, is subjected to two different types of proving mechanisms (Paragraph (H) of the Supplementary Article 2 bis): based on possession (old system), and based on governmental currency system's receipts (new system), which theoretically and practically, contains many difficulties. Finally, despite what said as a rule about the nullity of currency transactions without complying the statutory formalities, it seems that the "inopposabilité" (non-invocability, i.e. the contract is valid between its parties on one hand, and is invalid to the government and other third parties on the other hand), is fairer, more efficient, and more compatible with the practical position of foreign currencies in Iran, and therefore it is suggested to the legal doctrine, judicial precedent, and legislator. [ABSTRACT FROM AUTHOR]

The legal will of the private law moral persons is structured and formed according to the same mechanism legally recognised for the natural person. The inner dimension of the legal will of moral persons is not a cognitive process, but the consequence of a set of legal and statutory procedures that replace the cognitive process. This institutional process also benefits from the legal assumptions of existence and legality of the cause, and these assumptions can be overturned by the analysis of the statutory process. The sanction of relative nullity for the lack of cause cannot be applied even in the case of the total absence of the statutory process relevant for the formation of the internal dimension of the legal will, while the act concluded with third parties is in most situations opposable to the moral person and cannot be challenged in court with nullity. The bases for this solution are the acknowledgement of the multilayer formation of the legal will of moral persons and the adoption in the Romanian legal system of the Repräsentationstheorie ideology.

Nullities represent procedural sanctions that intervene if the fundamental principles of the criminal process are not respected. What is imperative to mention is the fact that these procedural sanctions operate only judicially, which means that they must be ascertained by the judicial body. Unlike the old regulation, the new criminal procedure code regulates new cases of relative nullity and essentially changes the procedure for the application of nullities, the new legislator wanting to impose a new technique for approaching the drafting of the rules. Thus, we find the rules that regulate the conduct of the criminal process in the code of criminal procedure, Constitution, special laws, CEDO s.a. With regard to the procedural phase in which these rules may be violated, the cited provisions are applicable from the start of the criminal prosecution in rem, throughout the criminal prosecution, the preliminary chamber, in court, appeal, extraordinary appeals, including in the execution of court decisions.

Nullities represent procedural sanctions that intervene if the fundamental principles of the criminal process are not respected. What is imperative to mention is the fact that these procedural sanctions operate only judicially, which means that they must be ascertained by the judicial body. Unlike the old regulation, the new criminal procedure code regulates new cases of relative nullity and essentially changes the procedure for the application of nullities, the new legislator wanting to impose a new technique for approaching the drafting of the rules. Thus, we find the rules that regulate the conduct of the criminal process in the code of criminal procedure, Constitution, special laws, CEDO s.a. With regard to the procedural phase in which these rules may be violated, the cited provisions are applicable from the start of the criminal prosecution in rem, throughout the criminal prosecution, the preliminary chamber, in court, appeal, extraordinary appeals, including in the execution of court decisions. [ABSTRACT FROM AUTHOR]

Let Γ be a finite simple graph with adjacency matrix A. Γ is said to be singular if and only if A has an eigenvalue 0. The nullity of Γ, denoted by null $ (\Gamma) $ (Γ) , is the multiplicity of the eigenvalue 0 in the spectrum of Γ. In this paper, we completely determine the singularity of graphs Γ for which the dicylic or semi-dihedral group acts transitively and faithfully on V as a group of automorphisms. Moreover, we give formulas to calculate the nullities for such graphs. [ABSTRACT FROM AUTHOR]

The second part of the topic generated by the approach to the specifics of the nullity of the fiscal procedure acts, as they were delimited in the first part, is intended for the application of the annulment cases, delimited, in particular, by the French doctrine of administrative law, to the fiscal administrative act. Beforehand, there are required references to the problems generated by the approaches from the French doctrine aimed at identifying the generic cases of annulment in relation to the way in which nullity is regulated in the rules of the Fiscal Procedure Code. In the context of the discussions concerning the relative nullity, the correct relationship between the notions of „nullity”, „annulability” and „annulment” must be clarified. Thus, in the comments concerning Article 49 of the Fiscal Procedure Code, differentiating between nullity and annulability, in the claim that the latter would constitute the „attribute” of the administrative acts affected by flaws of unlawfulness arising from the non-compliance with some validity conditions of lesser importance, it is evoked the thesis of the supporters of the presence of relative nullity including in the matter of administrative acts. However, the relative nullity is not compatible with the public interest on the basis of which any rule of administrative law is enacted. In order to avoid any confusions, the term „nullity” must denote the organic state of the act affected by serious defects of unlawfulness arising from the non-compliance with the conditions of validity specific to each category of legal acts, the term „annulability”, particularized, as a rule, in the wording „are annulable”, must be used to denote virtual nullities, and the word „annulment”, only to evoke the approach of the authorities to proceed with the annulment of legal acts affected by the flaws/defects of unlawfulness likely to generate their nullity. Given that, in the tax law doctrine, there are opinions evoking the support for the theory of the presence of legal non-existence and of relative nullity, it must be checked to what extent they can find their place in the matter of fiscal administrative acts. Although the doctrine in the matter contains tangential references to the non-existence, when commenting on the fiscal procedural texts referring to the communication of the fiscal administrative act, there is no regulation that expressly refers to the non-existence of such acts, as the legislator proceeded when specifying the sanction applicable to some administrative acts (the decisions and ordinances of the Government, as well as the decisions of the Prime Minister) not published in the Official Gazette of Romania. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [18] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.

Given a bipartite graph G with bipartition (U , W) , we denote by Q (G) the set of all real U × W matrices B = [ b u , w ] with b u , w = 0 if u and w are non-adjacent, b u , w ≠ 0 if u and w are connected by a single edge, and b u , w ∈ R if u and w are connected by multiple edges. We denote by N (G) the set of all U × W matrices X = [ x u , w ] with x u , w = 0 if u and w are adjacent. We say that a matrix B ∈ Q (G) has the Asymmetric Strong Arnold Property (ASAP) if for all X ∈ N (G) , if X T B = 0 and B X T = 0 , then X = 0. If G is a bipartite graph for which there exists a matrix B ∈ Q (G) that has the Asymmetric Strong Arnold Property, we define the stable minimum bipartite rank mbr S (G) as the smallest rank of any matrix B ∈ Q (G) having the ASAP. We show that if H is a matching minor of G , then mbr S (G) ≤ mbr S (H) + 1 / 2 (| V (G) | − | V (H) |). If G has a bipartition with parts of the same size, we define the stable maximum bipartite nullity M b S (G) as the largest nullity of any matrix B ∈ Q (G) having the ASAP. Then M b S (H) ≤ M b S (G). We give a characterization in terms of forbidden matching minors of the classes of graphs G with M b S (G) = 0 and with M b S (G) ≤ 1. [ABSTRACT FROM AUTHOR]

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The paper discusses the impact of causa on the validity of obligations in contract law. Given that many institutions in modern law trace their origins to Roman law, the paper begins by presenting the role of the cause of obligation in Roman law. The authors then analyzes the causal and anti-causal positions of legal theory, emphasizing that the cause of contractual obligation is a highly debatable institute of the law of obligations. The final part of the paper is dedicated to the cause of contractual obligation in our contract law. Considering that the Law on Obligations accepts both subjective and objective conceptions of the cause of contractual obligation. [ABSTRACT FROM AUTHOR]

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Let T be a tree with p ≥ 3 pendant edges and let L (T) be its line graph. In this paper, we prove the multiplicity of an arbitrary adjacency eigenvalue of L (T) is no larger than p − 1. Furthermore, the line graphs L (T) of trees which contain an adjacency eigenvalue with multiplicity p − 1 are completely determined. [ABSTRACT FROM AUTHOR]

A signed graph is a pair (G , Σ) , where G = (V , E) is a graph (in which parallel edges are permitted, but loops are not) with V = { 1 , ... , n } and Σ ⊆ E. The edges in Σ are called odd and the other edges of E even. If there are parallel edges, then only two edges in each parallel class are permitted, one of which is even and one of which is odd. By S (G , Σ) we denote the set of all symmetric n × n matrices A = [ a i , j ] with a i , j < 0 if i and j are connected by an even edge, a i , j > 0 if i and j are connected by an odd edge, a i , j ∈ R if i and j are connected by both an even and an odd edge, a i , j = 0 if i ≠ j and i and j are non-adjacent, and a i , i ∈ R for all vertices i. The maximum nullity M (G , Σ) of a signed graph (G , Σ) is the maximum nullity attained by any A ∈ S (G , Σ). Arav et al. gave a combinatorial characterization of 2-connected signed graphs (G , Σ) with M (G , Σ) = 2. In this paper, we give a complete combinatorial characterization of the signed graphs (G , Σ) with M (G , Σ) = 2. [ABSTRACT FROM AUTHOR]

The Federal Rules of Civil Procedure permit litigants to make changes to the substance of their initial pleading. Those changes raise a constitutional question when the initial pleading fails to establish a constitutionally required element of a federal court's jurisdiction: May the court permit the change, or must it dismiss the complaint as a nullity? The federal circuit courts are split in their answers to that question, with some circuits even issuing internally inconsistent holdings under different procedural rules. But regardless of the procedural rule at issue, the answer should be the same: Article III's jurisdictional requirements do not prohibit procedural moves from curing a jurisdictional defect. Taking that position, this Note contributes the only thorough analysis of the so-called "nullity doctrine" and its vices and, in the process, clarifies the relationship between Article III's jurisdictional requirements and the procedural rules that effectuate them. [ABSTRACT FROM AUTHOR]

In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler's Construction which is a local enlargement applied to a vertex in a graph is introduced to generate nut graphs of higher order. Let N(ρ) denote the set of integers n such that there exists a regular nut graph of degree ρ and order n. It is proven that N(3) = {12} ∪ {2k: k ≥ 9} and that N(4) = {8, 10, 12} ∪ {n: n ≥ 14}. The problem of determining N(ρ) for ρ > 4 remains completely open. [ABSTRACT FROM AUTHOR]

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In this paper, we completely characterize all the trees on n vertices with diameter d for which there is a symmetric matrix with nullity n − d and n − d − 1 , respectively. These characterizations cover all recent results proved for the standard 0 − 1 adjacency matrices. Here a new technique is developed for the general case, breaking the limitation of star complement technique for the standard 0 − 1 adjacency matrices. [ABSTRACT FROM AUTHOR]

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Let $ G $ be a finite simple graph and let $ A(G) $ be its adjacency matrix. Then $ G $ is $ singular $ if $ A(G) $ is singular. Suppose $ P_{b_{1}}, P_{b_{2}}, P_{b_{3}} $ are three paths with disjoint vertices, where $ b_i\geq 2 (i = 1, 2, 3) $, and at most one of them is 2. Coalescing together one of the two end vertices of each of the three paths, and coalescing together the other end vertex of each of the three paths, the resulting graph is called the $ \theta $-graph, denoted by $ \theta(b_{1}, b_{2}, b_{3}) $. Let $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ be the graph obtained by merging one end of the path $ P_{s} $ with one vertex of a cycle $ C_{a} $, and merging the other end of the path $ P_{s} $ with one vertex of $ \theta(b_{1}, b_{2}, b_{3}) $ of degree 3. If $ s = 1 $, denote $ \beta(a, b_{1}, b_{2}, b_{3}) = \alpha(a, b_{1}, b_{2}, b_{3}, 1) $. In this paper, we give the necessity and sufficiency condition for the singularity of $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ and $ \beta(a, b_{1}, b_{2}, b_{3}) $, and we also prove that the probability that any given $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ is a singular graph is equal to $ \frac{35}{64} $, the probability that any given $ \beta(a, b_{1}, b_{2}, b_{3}) $ is a singular graph is equal to $ \frac{9}{16} $. From our main results we can conclude that such a $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is singular if $ 4|a $ or three $ b_i (i = 1, 2, 3) $ are all odd numbers or exactly two of the three $ b_i (i = 1, 2, 3) $ are odd numbers and the length of the cycle formed by the two odd paths in $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is a multiple of 4. The theoretical probability of these graphs being singular is more than half.

The representation ideology has been incorporated in the New Civil Code without the acknowledgement and general acceptance of this substantial reform. The new representation ideology fundamentally rewrites the processes by which the private law moral person voluntarily participates in the civil circuit, a general unification of the technical legal mechanisms for the formation of the legal will to contract being noticeable. In the meantime, a reconstruction of the sanctions system for the irregularities tied to the due representation of moral persons by the holders of the representation power has taken place. The sanction of nullity survived only in exceptional cases. [ABSTRACT FROM AUTHOR]

This research paper is based on the investigation of over 1000 lawsuits for the declaration of nullity of marriage initiated at the Church Interdiocesan Tribunal in Split between 2000 and 2020. The main aim of this research paper was to focus on the lawsuits connected with the canon 1095, n. 3, especially on the problems of gambling addiction. The materials were analysed synthetically, and data were collected on a number of lawsuits conducted under this canon and on specific reasons for filing a lawsuit. Data on annulation of matrimony were also collected. The data are presented in an anonymous and aggregated manner. Descriptive and inferential statistical methods were used. The paper consists of a theoretical part and methodology, which includes the motive of the paper, the description of the sample used and the criteria of inclusion, materials, methods and procedures of paper, and the presentation of the results, together with the discussion of the results. [ABSTRACT FROM AUTHOR]

Let be an infinite dimensional complex Banach space, and let be the algebra of all bounded linear operators on. In this paper, given any positive integer n, we provide a characterization of all bijective maps on preserving the difference of semi-Fredholm operators with nullity less than n, and describe completely the structure of these maps. [ABSTRACT FROM AUTHOR]

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